From: Richard Kreckel Date: Mon, 28 Aug 2000 22:10:51 +0000 (+0000) Subject: * Argh, we are threshing CVS. Removed some non-sources. X-Git-Tag: cln_1-1-0~25 X-Git-Url: https://www.ginac.de/CLN/cln.git//cln.git?a=commitdiff_plain;h=f893f25f2d40c89d42e1d884b082966f8afeeb68;p=cln.git * Argh, we are threshing CVS. Removed some non-sources. --- diff --git a/doc/cln.dvi b/doc/cln.dvi deleted file mode 100644 index 78de3c1..0000000 Binary files a/doc/cln.dvi and /dev/null differ diff --git a/doc/cln.html b/doc/cln.html deleted file mode 100644 index 1c6aef9..0000000 --- a/doc/cln.html +++ /dev/null @@ -1,4898 +0,0 @@ - - - - -CLN, a Class Library for Numbers - - -

CLN, a Class Library for Numbers

-
by Bruno Haible
-

-

-

Table of Contents

- -

- - -

1. Introduction

- -

-CLN is a library for computations with all kinds of numbers. -It has a rich set of number classes: - - - -

- -

-The subtypes of the complex numbers among these are exactly the -types of numbers known to the Common Lisp language. Therefore -CLN can be used for Common Lisp implementations, giving -`CLN' another meaning: it becomes an abbreviation of -"Common Lisp Numbers". - - -

-The CLN package implements - - - -

- -

-CLN is a C++ library. Using C++ as an implementation language provides - - - -

- -

-CLN is memory efficient: - - - -

- -

-CLN is speed efficient: - - - -

- -

-CLN aims at being easily integrated into larger software packages: - - - -

- - - -

2. Installation

- -

-This section describes how to install the CLN package on your system. - - - - -

2.1 Prerequisites

- - - -

2.1.1 C++ compiler

- -

-To build CLN, you need a C++ compiler. -Actually, you need GNU g++ 2.90 or newer, the EGCS compilers will -do. -I recommend GNU g++ 2.95 or newer. - - -

-The following C++ features are used: -classes, member functions, overloading of functions and operators, -constructors and destructors, inline, const, multiple inheritance, -templates and namespaces. - - -

-The following C++ features are not used: -new, delete, virtual inheritance, exceptions. - - -

-CLN relies on semi-automatic ordering of initializations -of static and global variables, a feature which I could -implement for GNU g++ only. - - - - -

2.1.2 Make utility

-

- - - -

-To build CLN, you also need to have GNU make installed. - - - - -

2.1.3 Sed utility

-

- - - -

-To build CLN on HP-UX, you also need to have GNU sed installed. -This is because the libtool script, which creates the CLN library, relies -on sed, and the vendor's sed utility on these systems is too -limited. - - - - -

2.2 Building the library

- -

-As with any autoconfiguring GNU software, installation is as easy as this: - - - -

-$ ./configure
-$ make
-$ make check
-
- -

-If on your system, `make' is not GNU make, you have to use -`gmake' instead of `make' above. - - -

-The configure command checks out some features of your system and -C++ compiler and builds the Makefiles. The make command -builds the library. This step may take 4 hours on an average workstation. -The make check runs some test to check that no important subroutine -has been miscompiled. - - -

-The configure command accepts options. To get a summary of them, try - - - -

-$ ./configure --help
-
- -

-Some of the options are explained in detail in the `INSTALL.generic' file. - - -

-You can specify the C compiler, the C++ compiler and their options through -the following environment variables when running configure: - - -

- -
CC -
-Specifies the C compiler. - -
CFLAGS -
-Flags to be given to the C compiler when compiling programs (not when linking). - -
CXX -
-Specifies the C++ compiler. - -
CXXFLAGS -
-Flags to be given to the C++ compiler when compiling programs (not when linking). -
- -

-Examples: - - - -

-$ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
-$ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
-  CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
-$ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
-  CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
-
- -

-Note that for these environment variables to take effect, you have to set -them (assuming a Bourne-compatible shell) on the same line as the -configure command. If you made the settings in earlier shell -commands, you have to export the environment variables before -calling configure. In a csh shell, you have to use the -`setenv' command for setting each of the environment variables. - - -

-Currently CLN works only with the GNU g++ compiler, and only in -optimizing mode. So you should specify at least -O in the CXXFLAGS, -or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use -O.) - - -

-If you use g++ version 2.8.x or egcs-2.91.x (a.k.a. egcs-1.1) or -gcc-2.95.x, I recommend adding `-fno-exceptions' to the CXXFLAGS. -This will likely generate better code. - - -

-If you use g++ version egcs-2.91.x (egcs-1.1) or gcc-2.95.x on Sparc, -add either `-O', `-O1' or `-O2 -fno-schedule-insns' to the -CXXFLAGS. With full `-O2', g++ miscompiles the division routines. -Also, if you have g++ version egcs-1.1.1 or older on Sparc, you must -specify `--disable-shared' because g++ would miscompile parts of -the library. - - -

-By default, both a shared and a static library are built. You can build -CLN as a static (or shared) library only, by calling configure with -the option `--disable-shared' (or `--disable-static'). While -shared libraries are usually more convenient to use, they may not work -on all architectures. Try disabling them if you run into linker -problems. Also, they are generally somewhat slower than static -libraries so runtime-critical applications should be linked statically. - - - - -

2.2.1 Using the GNU MP Library

-

- - - -

-Starting with version 1.1, CLN may be configured to make use of a -preinstalled gmp library. Please make sure that you have at -least gmp version 3.0 installed since earlier versions are -unsupported and likely not to work. Enabling this feature by calling -configure with the option `--with-gmp' is known to be quite -a boost for CLN's performance. - - -

-If you have installed the gmp library and its header file in -some place where your compiler cannot find it by default, you must help -configure by setting CPPFLAGS and LDFLAGS. Here is -an example: - - - -

-$ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
-  CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
-
- - - -

2.3 Installing the library

-

- - - -

-As with any autoconfiguring GNU software, installation is as easy as this: - - - -

-$ make install
-
- -

-The `make install' command installs the library and the include files -into public places (`/usr/local/lib/' and `/usr/local/include/', -if you haven't specified a --prefix option to configure). -This step may require superuser privileges. - - -

-If you have already built the library and wish to install it, but didn't -specify --prefix=... at configure time, just re-run -configure, giving it the same options as the first time, plus -the --prefix=... option. - - - - -

2.4 Cleaning up

- -

-You can remove system-dependent files generated by make through - - - -

-$ make clean
-
- -

-You can remove all files generated by make, thus reverting to a -virgin distribution of CLN, through - - - -

-$ make distclean
-
- - - -

3. Ordinary number types

- -

-CLN implements the following class hierarchy: - - - -

-                        Number
-                      cl_number
-                    <cln/number.h>
-                          |
-                          |
-                 Real or complex number
-                        cl_N
-                    <cln/complex.h>
-                          |
-                          |
-                     Real number
-                        cl_R
-                     <cln/real.h>
-                          |
-      +-------------------+-------------------+
-      |                                       |
-Rational number                     Floating-point number
-    cl_RA                                   cl_F
-<cln/rational.h>                         <cln/float.h>
-      |                                       |
-      |                +--------------+--------------+--------------+
-   Integer             |              |              |              |
-    cl_I          Short-Float    Single-Float   Double-Float    Long-Float
-<cln/integer.h>      cl_SF          cl_FF          cl_DF          cl_LF
-                 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
-
- -

- - -The base class cl_number is an abstract base class. -It is not useful to declare a variable of this type except if you want -to completely disable compile-time type checking and use run-time type -checking instead. - - -

- - - -The class cl_N comprises real and complex numbers. There is -no special class for complex numbers since complex numbers with imaginary -part 0 are automatically converted to real numbers. - - -

- -The class cl_R comprises real numbers of different kinds. It is an -abstract class. - - -

- - - -The class cl_RA comprises exact real numbers: rational numbers, including -integers. There is no special class for non-integral rational numbers -since rational numbers with denominator 1 are automatically converted -to integers. - - -

- -The class cl_F implements floating-point approximations to real numbers. -It is an abstract class. - - - - -

3.1 Exact numbers

-

- - - -

-Some numbers are represented as exact numbers: there is no loss of information -when such a number is converted from its mathematical value to its internal -representation. On exact numbers, the elementary operations (+, --, *, /, comparisons, ...) compute the completely -correct result. - - -

-In CLN, the exact numbers are: - - - -

- -

-Rational numbers are always normalized to the form -numerator/denominator where the numerator and denominator -are coprime integers and the denominator is positive. If the resulting -denominator is 1, the rational number is converted to an integer. - - -

-Small integers (typically in the range -2^30...2^30-1, -for 32-bit machines) are especially efficient, because they consume no heap -allocation. Otherwise the distinction between these immediate integers -(called "fixnums") and heap allocated integers (called "bignums") -is completely transparent. - - - - -

3.2 Floating-point numbers

-

- - - -

-Not all real numbers can be represented exactly. (There is an easy mathematical -proof for this: Only a countable set of numbers can be stored exactly in -a computer, even if one assumes that it has unlimited storage. But there -are uncountably many real numbers.) So some approximation is needed. -CLN implements ordinary floating-point numbers, with mantissa and exponent. - - -

- -The elementary operations (+, -, *, /, ...) -only return approximate results. For example, the value of the expression -(cl_F) 0.3 + (cl_F) 0.4 prints as `0.70000005', not as -`0.7'. Rounding errors like this one are inevitable when computing -with floating-point numbers. - - -

-Nevertheless, CLN rounds the floating-point results of the operations +, --, *, /, sqrt according to the "round-to-even" -rule: It first computes the exact mathematical result and then returns the -floating-point number which is nearest to this. If two floating-point numbers -are equally distant from the ideal result, the one with a 0 in its least -significant mantissa bit is chosen. - - -

-Similarly, testing floating point numbers for equality `x == y' -is gambling with random errors. Better check for `abs(x - y) < epsilon' -for some well-chosen epsilon. - - -

-Floating point numbers come in four flavors: - - - -

- -

-Of course, computations with long floats are more expensive than those -with smaller floating-point formats. - - -

-CLN does not implement features like NaNs, denormalized numbers and -gradual underflow. If the exponent range of some floating-point type -is too limited for your application, choose another floating-point type -with larger exponent range. - - -

- -As a user of CLN, you can forget about the differences between the -four floating-point types and just declare all your floating-point -variables as being of type cl_F. This has the advantage that -when you change the precision of some computation (say, from cl_DF -to cl_LF), you don't have to change the code, only the precision -of the initial values. Also, many transcendental functions have been -declared as returning a cl_F when the argument is a cl_F, -but such declarations are missing for the types cl_SF, cl_FF, -cl_DF, cl_LF. (Such declarations would be wrong if -the floating point contagion rule happened to change in the future.) - - - - -

3.3 Complex numbers

-

- - - -

-Complex numbers, as implemented by the class cl_N, have a real -part and an imaginary part, both real numbers. A complex number whose -imaginary part is the exact number 0 is automatically converted -to a real number. - - -

-Complex numbers can arise from real numbers alone, for example -through application of sqrt or transcendental functions. - - - - -

3.4 Conversions

-

- - - -

-Conversions from any class to any its superclasses ("base classes" in -C++ terminology) is done automatically. - - -

-Conversions from the C built-in types `long' and `unsigned long' -are provided for the classes cl_I, cl_RA, cl_R, -cl_N and cl_number. - - -

-Conversions from the C built-in types `int' and `unsigned int' -are provided for the classes cl_I, cl_RA, cl_R, -cl_N and cl_number. However, these conversions emphasize -efficiency. Their range is therefore limited: - - - -

- -

-In a declaration like `cl_I x = 10;' the C++ compiler is able to -do the conversion of 10 from `int' to `cl_I' at compile time -already. On the other hand, code like `cl_I x = 1000000000;' is -in error. -So, if you want to be sure that an `int' whose magnitude is not guaranteed -to be < 2^29 is correctly converted to a `cl_I', first convert it to a -`long'. Similarly, if a large `unsigned int' is to be converted to a -`cl_I', first convert it to an `unsigned long'. - - -

-Conversions from the C built-in type `float' are provided for the classes -cl_FF, cl_F, cl_R, cl_N and cl_number. - - -

-Conversions from the C built-in type `double' are provided for the classes -cl_DF, cl_F, cl_R, cl_N and cl_number. - - -

-Conversions from `const char *' are provided for the classes -cl_I, cl_RA, -cl_SF, cl_FF, cl_DF, cl_LF, cl_F, -cl_R, cl_N. -The easiest way to specify a value which is outside of the range of the -C++ built-in types is therefore to specify it as a string, like this: - - -

-   cl_I order_of_rubiks_cube_group = "43252003274489856000";
-
- -

-Note that this conversion is done at runtime, not at compile-time. - - -

-Conversions from cl_I to the C built-in types `int', -`unsigned int', `long', `unsigned long' are provided through -the functions - - -

- -
int cl_I_to_int (const cl_I& x) -
- -
unsigned int cl_I_to_uint (const cl_I& x) -
- -
long cl_I_to_long (const cl_I& x) -
- -
unsigned long cl_I_to_ulong (const cl_I& x) -
- -Returns x as element of the C type ctype. If x is not -representable in the range of ctype, a runtime error occurs. -
- -

-Conversions from the classes cl_I, cl_RA, -cl_SF, cl_FF, cl_DF, cl_LF, cl_F and -cl_R -to the C built-in types `float' and `double' are provided through -the functions - - -

- -
float float_approx (const type& x) -
- -
double double_approx (const type& x) -
- -Returns an approximation of x of C type ctype. -If abs(x) is too close to 0 (underflow), 0 is returned. -If abs(x) is too large (overflow), an IEEE infinity is returned. -
- -

-Conversions from any class to any of its subclasses ("derived classes" in -C++ terminology) are not provided. Instead, you can assert and check -that a value belongs to a certain subclass, and return it as element of that -class, using the `As' and `The' macros. - -As(type)(value) checks that value belongs to -type and returns it as such. - -The(type)(value) assumes that value belongs to -type and returns it as such. It is your responsibility to ensure -that this assumption is valid. -Example: - - - -

-   cl_I x = ...;
-   if (!(x >= 0)) abort();
-   cl_I ten_x = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
-                // In general, it would be a rational number.
-
- - - -

4. Functions on numbers

- -

-Each of the number classes declares its mathematical operations in the -corresponding include file. For example, if your code operates with -objects of type cl_I, it should #include <cln/integer.h>. - - - - -

4.1 Constructing numbers

- -

-Here is how to create number objects "from nothing". - - - - -

4.1.1 Constructing integers

- -

-cl_I objects are most easily constructed from C integers and from -strings. See section 3.4 Conversions. - - - - -

4.1.2 Constructing rational numbers

- -

-cl_RA objects can be constructed from strings. The syntax -for rational numbers is described in section 5.1 Internal and printed representation. -Another standard way to produce a rational number is through application -of `operator /' or `recip' on integers. - - - - -

4.1.3 Constructing floating-point numbers

- -

-cl_F objects with low precision are most easily constructed from -C `float' and `double'. See section 3.4 Conversions. - - -

-To construct a cl_F with high precision, you can use the conversion -from `const char *', but you have to specify the desired precision -within the string. (See section 5.1 Internal and printed representation.) -Example: - -

-   cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
-
- -

-will set `e' to the given value, with a precision of 40 decimal digits. - - -

-The programmatic way to construct a cl_F with high precision is -through the cl_float conversion function, see -section 4.11.1 Conversion to floating-point numbers. For example, to compute -e to 40 decimal places, first construct 1.0 to 40 decimal places -and then apply the exponential function: - -

-   cl_float_format_t precision = cl_float_format(40);
-   cl_F e = exp(cl_float(1,precision));
-
- - - -

4.1.4 Constructing complex numbers

- -

-Non-real cl_N objects are normally constructed through the function - -

-   cl_N complex (const cl_R& realpart, const cl_R& imagpart)
-
- -

-See section 4.4 Elementary complex functions. - - - - -

4.2 Elementary functions

- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
type operator + (const type&, const type&) -
- -Addition. - -
type operator - (const type&, const type&) -
- -Subtraction. - -
type operator - (const type&) -
-Returns the negative of the argument. - -
type plus1 (const type& x) -
- -Returns x + 1. - -
type minus1 (const type& x) -
- -Returns x - 1. - -
type operator * (const type&, const type&) -
- -Multiplication. - -
type square (const type& x) -
- -Returns x * x. -
- -

-Each of the classes cl_N, cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
type operator / (const type&, const type&) -
- -Division. - -
type recip (const type&) -
- -Returns the reciprocal of the argument. -
- -

-The class cl_I doesn't define a `/' operation because -in the C/C++ language this operator, applied to integral types, -denotes the `floor' or `truncate' operation (which one of these, -is implementation dependent). (See section 4.6 Rounding functions.) -Instead, cl_I defines an "exact quotient" function: - - -

- -
cl_I exquo (const cl_I& x, const cl_I& y) -
- -Checks that y divides x, and returns the quotient x/y. -
- -

-The following exponentiation functions are defined: - - -

- -
cl_I expt_pos (const cl_I& x, const cl_I& y) -
- -
cl_RA expt_pos (const cl_RA& x, const cl_I& y) -
-y must be > 0. Returns x^y. - -
cl_RA expt (const cl_RA& x, const cl_I& y) -
- -
cl_R expt (const cl_R& x, const cl_I& y) -
-
cl_N expt (const cl_N& x, const cl_I& y) -
-Returns x^y. -
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operation: - - -

- -
type abs (const type& x) -
- -Returns the absolute value of x. -This is x if x >= 0, and -x if x <= 0. -
- -

-The class cl_N implements this as follows: - - -

- -
cl_R abs (const cl_N x) -
-Returns the absolute value of x. -
- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operation: - - -

- -
type signum (const type& x) -
- -Returns the sign of x, in the same number format as x. -This is defined as x / abs(x) if x is non-zero, and -x if x is zero. If x is real, the value is either -0 or 1 or -1. -
- - - -

4.3 Elementary rational functions

- -

-Each of the classes cl_RA, cl_I defines the following operations: - - -

- -
cl_I numerator (const type& x) -
- -Returns the numerator of x. - -
cl_I denominator (const type& x) -
- -Returns the denominator of x. -
- -

-The numerator and denominator of a rational number are normalized in such -a way that they have no factor in common and the denominator is positive. - - - - -

4.4 Elementary complex functions

- -

-The class cl_N defines the following operation: - - -

- -
cl_N complex (const cl_R& a, const cl_R& b) -
- -Returns the complex number a+bi, that is, the complex number with -real part a and imaginary part b. -
- -

-Each of the classes cl_N, cl_R defines the following operations: - - -

- -
cl_R realpart (const type& x) -
- -Returns the real part of x. - -
cl_R imagpart (const type& x) -
- -Returns the imaginary part of x. - -
type conjugate (const type& x) -
- -Returns the complex conjugate of x. -
- -

-We have the relations - - - -

- - - -

4.5 Comparisons

-

- - - -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
bool operator == (const type&, const type&) -
- -
bool operator != (const type&, const type&) -
- -Comparison, as in C and C++. - -
uint32 equal_hashcode (const type&) -
- -Returns a 32-bit hash code that is the same for any two numbers which are -the same according to ==. This hash code depends on the number's value, -not its type or precision. - -
cl_boolean zerop (const type& x) -
- -Compare against zero: x == 0 -
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
cl_signean compare (const type& x, const type& y) -
- -Compares x and y. Returns +1 if x>y, --1 if x<y, 0 if x=y. - -
bool operator <= (const type&, const type&) -
- -
bool operator < (const type&, const type&) -
- -
bool operator >= (const type&, const type&) -
- -
bool operator > (const type&, const type&) -
- -Comparison, as in C and C++. - -
cl_boolean minusp (const type& x) -
- -Compare against zero: x < 0 - -
cl_boolean plusp (const type& x) -
- -Compare against zero: x > 0 - -
type max (const type& x, const type& y) -
- -Return the maximum of x and y. - -
type min (const type& x, const type& y) -
- -Return the minimum of x and y. -
- -

-When a floating point number and a rational number are compared, the float -is first converted to a rational number using the function rational. -Since a floating point number actually represents an interval of real numbers, -the result might be surprising. -For example, (cl_F)(cl_R)"1/3" == (cl_R)"1/3" returns false because -there is no floating point number whose value is exactly 1/3. - - - - -

4.6 Rounding functions

-

- - - -

-When a real number is to be converted to an integer, there is no "best" -rounding. The desired rounding function depends on the application. -The Common Lisp and ISO Lisp standards offer four rounding functions: - - -

- -
floor(x) -
-This is the largest integer <=x. - -
ceiling(x) -
-This is the smallest integer >=x. - -
truncate(x) -
-Among the integers between 0 and x (inclusive) the one nearest to x. - -
round(x) -
-The integer nearest to x. If x is exactly halfway between two -integers, choose the even one. -
- -

-These functions have different advantages: - - -

-floor and ceiling are translation invariant: -floor(x+n) = floor(x) + n and ceiling(x+n) = ceiling(x) + n -for every x and every integer n. - - -

-On the other hand, truncate and round are symmetric: -truncate(-x) = -truncate(x) and round(-x) = -round(x), -and furthermore round is unbiased: on the "average", it rounds -down exactly as often as it rounds up. - - -

-The functions are related like this: - - - -

- -

-Each of the classes cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
cl_I floor1 (const type& x) -
- -Returns floor(x). -
cl_I ceiling1 (const type& x) -
- -Returns ceiling(x). -
cl_I truncate1 (const type& x) -
- -Returns truncate(x). -
cl_I round1 (const type& x) -
- -Returns round(x). -
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
cl_I floor1 (const type& x, const type& y) -
-Returns floor(x/y). -
cl_I ceiling1 (const type& x, const type& y) -
-Returns ceiling(x/y). -
cl_I truncate1 (const type& x, const type& y) -
-Returns truncate(x/y). -
cl_I round1 (const type& x, const type& y) -
-Returns round(x/y). -
- -

-These functions are called `floor1', ... here instead of -`floor', ..., because on some systems, system dependent include -files define `floor' and `ceiling' as macros. - - -

-In many cases, one needs both the quotient and the remainder of a division. -It is more efficient to compute both at the same time than to perform -two divisions, one for quotient and the next one for the remainder. -The following functions therefore return a structure containing both -the quotient and the remainder. The suffix `2' indicates the number -of "return values". The remainder is defined as follows: - - - -

- -

-and similarly for the other three operations. - - -

-Each of the classes cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
struct type_div_t { cl_I quotient; type remainder; }; -
-
type_div_t floor2 (const type& x) -
-
type_div_t ceiling2 (const type& x) -
-
type_div_t truncate2 (const type& x) -
-
type_div_t round2 (const type& x) -
-
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
struct type_div_t { cl_I quotient; type remainder; }; -
-
type_div_t floor2 (const type& x, const type& y) -
- -
type_div_t ceiling2 (const type& x, const type& y) -
- -
type_div_t truncate2 (const type& x, const type& y) -
- -
type_div_t round2 (const type& x, const type& y) -
- -
- -

-Sometimes, one wants the quotient as a floating-point number (of the -same format as the argument, if the argument is a float) instead of as -an integer. The prefix `f' indicates this. - - -

-Each of the classes -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
type ffloor (const type& x) -
- -
type fceiling (const type& x) -
- -
type ftruncate (const type& x) -
- -
type fround (const type& x) -
- -
- -

-and similarly for class cl_R, but with return type cl_F. - - -

-The class cl_R defines the following operations: - - -

- -
cl_F ffloor (const type& x, const type& y) -
-
cl_F fceiling (const type& x, const type& y) -
-
cl_F ftruncate (const type& x, const type& y) -
-
cl_F fround (const type& x, const type& y) -
-
- -

-These functions also exist in versions which return both the quotient -and the remainder. The suffix `2' indicates this. - - -

-Each of the classes -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - - - - - - -

- -
struct type_fdiv_t { type quotient; type remainder; }; -
-
type_fdiv_t ffloor2 (const type& x) -
- -
type_fdiv_t fceiling2 (const type& x) -
- -
type_fdiv_t ftruncate2 (const type& x) -
- -
type_fdiv_t fround2 (const type& x) -
- -
-

-and similarly for class cl_R, but with quotient type cl_F. - - - -

-The class cl_R defines the following operations: - - -

- -
struct type_fdiv_t { cl_F quotient; cl_R remainder; }; -
-
type_fdiv_t ffloor2 (const type& x, const type& y) -
-
type_fdiv_t fceiling2 (const type& x, const type& y) -
-
type_fdiv_t ftruncate2 (const type& x, const type& y) -
-
type_fdiv_t fround2 (const type& x, const type& y) -
-
- -

-Other applications need only the remainder of a division. -The remainder of `floor' and `ffloor' is called `mod' -(abbreviation of "modulo"). The remainder `truncate' and -`ftruncate' is called `rem' (abbreviation of "remainder"). - - - -

- -

-If x and y are both >= 0, mod(x,y) = rem(x,y) >= 0. -In general, mod(x,y) has the sign of y or is zero, -and rem(x,y) has the sign of x or is zero. - - -

-The classes cl_R, cl_I define the following operations: - - -

- -
type mod (const type& x, const type& y) -
- -
type rem (const type& x, const type& y) -
- -
- - - -

4.7 Roots

- -

-Each of the classes cl_R, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operation: - - -

- -
type sqrt (const type& x) -
- -x must be >= 0. This function returns the square root of x, -normalized to be >= 0. If x is the square of a rational number, -sqrt(x) will be a rational number, else it will return a -floating-point approximation. -
- -

-The classes cl_RA, cl_I define the following operation: - - -

- -
cl_boolean sqrtp (const type& x, type* root) -
- -This tests whether x is a perfect square. If so, it returns true -and the exact square root in *root, else it returns false. -
- -

-Furthermore, for integers, similarly: - - -

- -
cl_boolean isqrt (const type& x, type* root) -
- -x should be >= 0. This function sets *root to -floor(sqrt(x)) and returns the same value as sqrtp: -the boolean value (expt(*root,2) == x). -
- -

-For nth roots, the classes cl_RA, cl_I -define the following operation: - - -

- -
cl_boolean rootp (const type& x, const cl_I& n, type* root) -
- -x must be >= 0. n must be > 0. -This tests whether x is an nth power of a rational number. -If so, it returns true and the exact root in *root, else it returns -false. -
- -

-The only square root function which accepts negative numbers is the one -for class cl_N: - - -

- -
cl_N sqrt (const cl_N& z) -
- -Returns the square root of z, as defined by the formula -sqrt(z) = exp(log(z)/2). Conversion to a floating-point type -or to a complex number are done if necessary. The range of the result is the -right half plane realpart(sqrt(z)) >= 0 -including the positive imaginary axis and 0, but excluding -the negative imaginary axis. -The result is an exact number only if z is an exact number. -
- - - -

4.8 Transcendental functions

-

- - - -

-The transcendental functions return an exact result if the argument -is exact and the result is exact as well. Otherwise they must return -inexact numbers even if the argument is exact. -For example, cos(0) = 1 returns the rational number 1. - - - - -

4.8.1 Exponential and logarithmic functions

- -
- -
cl_R exp (const cl_R& x) -
- -
cl_N exp (const cl_N& x) -
-Returns the exponential function of x. This is e^x where -e is the base of the natural logarithms. The range of the result -is the entire complex plane excluding 0. - -
cl_R ln (const cl_R& x) -
- -x must be > 0. Returns the (natural) logarithm of x. - -
cl_N log (const cl_N& x) -
- -Returns the (natural) logarithm of x. If x is real and positive, -this is ln(x). In general, log(x) = log(abs(x)) + i*phase(x). -The range of the result is the strip in the complex plane --pi < imagpart(log(x)) <= pi. - -
cl_R phase (const cl_N& x) -
- -Returns the angle part of x in its polar representation as a -complex number. That is, phase(x) = atan(realpart(x),imagpart(x)). -This is also the imaginary part of log(x). -The range of the result is the interval -pi < phase(x) <= pi. -The result will be an exact number only if zerop(x) or -if x is real and positive. - -
cl_R log (const cl_R& a, const cl_R& b) -
-a and b must be > 0. Returns the logarithm of a with -respect to base b. log(a,b) = ln(a)/ln(b). -The result can be exact only if a = 1 or if a and b -are both rational. - -
cl_N log (const cl_N& a, const cl_N& b) -
-Returns the logarithm of a with respect to base b. -log(a,b) = log(a)/log(b). - -
cl_N expt (const cl_N& x, const cl_N& y) -
- -Exponentiation: Returns x^y = exp(y*log(x)). -
- -

-The constant e = exp(1) = 2.71828... is returned by the following functions: - - -

- -
cl_F exp1 (cl_float_format_t f) -
- -Returns e as a float of format f. - -
cl_F exp1 (const cl_F& y) -
-Returns e in the float format of y. - -
cl_F exp1 (void) -
-Returns e as a float of format default_float_format. -
- - - -

4.8.2 Trigonometric functions

- -
- -
cl_R sin (const cl_R& x) -
- -Returns sin(x). The range of the result is the interval --1 <= sin(x) <= 1. - -
cl_N sin (const cl_N& z) -
-Returns sin(z). The range of the result is the entire complex plane. - -
cl_R cos (const cl_R& x) -
- -Returns cos(x). The range of the result is the interval --1 <= cos(x) <= 1. - -
cl_N cos (const cl_N& x) -
-Returns cos(z). The range of the result is the entire complex plane. - -
struct cos_sin_t { cl_R cos; cl_R sin; }; -
- -
cos_sin_t cos_sin (const cl_R& x) -
-Returns both sin(x) and cos(x). This is more efficient than - -computing them separately. The relation cos^2 + sin^2 = 1 will -hold only approximately. - -
cl_R tan (const cl_R& x) -
- -
cl_N tan (const cl_N& x) -
-Returns tan(x) = sin(x)/cos(x). - -
cl_N cis (const cl_R& x) -
- -
cl_N cis (const cl_N& x) -
-Returns exp(i*x). The name `cis' means "cos + i sin", because -e^(i*x) = cos(x) + i*sin(x). - - - -
cl_N asin (const cl_N& z) -
-Returns arcsin(z). This is defined as -arcsin(z) = log(iz+sqrt(1-z^2))/i and satisfies -arcsin(-z) = -arcsin(z). -The range of the result is the strip in the complex domain --pi/2 <= realpart(arcsin(z)) <= pi/2, excluding the numbers -with realpart = -pi/2 and imagpart < 0 and the numbers -with realpart = pi/2 and imagpart > 0. - -
cl_N acos (const cl_N& z) -
- -Returns arccos(z). This is defined as -arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i -and satisfies arccos(-z) = pi - arccos(z). -The range of the result is the strip in the complex domain -0 <= realpart(arcsin(z)) <= pi, excluding the numbers -with realpart = 0 and imagpart < 0 and the numbers -with realpart = pi and imagpart > 0. - - - -
cl_R atan (const cl_R& x, const cl_R& y) -
-Returns the angle of the polar representation of the complex number -x+iy. This is atan(y/x) if x>0. The range of -the result is the interval -pi < atan(x,y) <= pi. The result will -be an exact number only if x > 0 and y is the exact 0. -WARNING: In Common Lisp, this function is called as (atan y x), -with reversed order of arguments. - -
cl_R atan (const cl_R& x) -
-Returns arctan(x). This is the same as atan(1,x). The range -of the result is the interval -pi/2 < atan(x) < pi/2. The result -will be an exact number only if x is the exact 0. - -
cl_N atan (const cl_N& z) -
-Returns arctan(z). This is defined as -arctan(z) = (log(1+iz)-log(1-iz)) / 2i and satisfies -arctan(-z) = -arctan(z). The range of the result is -the strip in the complex domain --pi/2 <= realpart(arctan(z)) <= pi/2, excluding the numbers -with realpart = -pi/2 and imagpart >= 0 and the numbers -with realpart = pi/2 and imagpart <= 0. - -
- -

- - -Archimedes' constant pi = 3.14... is returned by the following functions: - - -

- -
cl_F pi (cl_float_format_t f) -
- -Returns pi as a float of format f. - -
cl_F pi (const cl_F& y) -
-Returns pi in the float format of y. - -
cl_F pi (void) -
-Returns pi as a float of format default_float_format. -
- - - -

4.8.3 Hyperbolic functions

- -
- -
cl_R sinh (const cl_R& x) -
- -Returns sinh(x). - -
cl_N sinh (const cl_N& z) -
-Returns sinh(z). The range of the result is the entire complex plane. - -
cl_R cosh (const cl_R& x) -
- -Returns cosh(x). The range of the result is the interval -cosh(x) >= 1. - -
cl_N cosh (const cl_N& z) -
-Returns cosh(z). The range of the result is the entire complex plane. - -
struct cosh_sinh_t { cl_R cosh; cl_R sinh; }; -
- -
cosh_sinh_t cosh_sinh (const cl_R& x) -
- -Returns both sinh(x) and cosh(x). This is more efficient than -computing them separately. The relation cosh^2 - sinh^2 = 1 will -hold only approximately. - -
cl_R tanh (const cl_R& x) -
- -
cl_N tanh (const cl_N& x) -
-Returns tanh(x) = sinh(x)/cosh(x). - -
cl_N asinh (const cl_N& z) -
- -Returns arsinh(z). This is defined as -arsinh(z) = log(z+sqrt(1+z^2)) and satisfies -arsinh(-z) = -arsinh(z). -The range of the result is the strip in the complex domain --pi/2 <= imagpart(arsinh(z)) <= pi/2, excluding the numbers -with imagpart = -pi/2 and realpart > 0 and the numbers -with imagpart = pi/2 and realpart < 0. - -
cl_N acosh (const cl_N& z) -
- -Returns arcosh(z). This is defined as -arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2)). -The range of the result is the half-strip in the complex domain --pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0, -excluding the numbers with realpart = 0 and -pi < imagpart < 0. - -
cl_N atanh (const cl_N& z) -
- -Returns artanh(z). This is defined as -artanh(z) = (log(1+z)-log(1-z)) / 2 and satisfies -artanh(-z) = -artanh(z). The range of the result is -the strip in the complex domain --pi/2 <= imagpart(artanh(z)) <= pi/2, excluding the numbers -with imagpart = -pi/2 and realpart <= 0 and the numbers -with imagpart = pi/2 and realpart >= 0. -
- - - -

4.8.4 Euler gamma

-

- - - -

-Euler's constant C = 0.577... is returned by the following functions: - - -

- -
cl_F eulerconst (cl_float_format_t f) -
- -Returns Euler's constant as a float of format f. - -
cl_F eulerconst (const cl_F& y) -
-Returns Euler's constant in the float format of y. - -
cl_F eulerconst (void) -
-Returns Euler's constant as a float of format default_float_format. -
- -

-Catalan's constant G = 0.915... is returned by the following functions: - - - -

- -
cl_F catalanconst (cl_float_format_t f) -
- -Returns Catalan's constant as a float of format f. - -
cl_F catalanconst (const cl_F& y) -
-Returns Catalan's constant in the float format of y. - -
cl_F catalanconst (void) -
-Returns Catalan's constant as a float of format default_float_format. -
- - - -

4.8.5 Riemann zeta

-

- - - -

-Riemann's zeta function at an integral point s>1 is returned by the -following functions: - - -

- -
cl_F zeta (int s, cl_float_format_t f) -
- -Returns Riemann's zeta function at s as a float of format f. - -
cl_F zeta (int s, const cl_F& y) -
-Returns Riemann's zeta function at s in the float format of y. - -
cl_F zeta (int s) -
-Returns Riemann's zeta function at s as a float of format -default_float_format. -
- - - -

4.9 Functions on integers

- - - -

4.9.1 Logical functions

- -

-Integers, when viewed as in two's complement notation, can be thought as -infinite bit strings where the bits' values eventually are constant. -For example, - -

-    17 = ......00010001
-    -6 = ......11111010
-
- -

-The logical operations view integers as such bit strings and operate -on each of the bit positions in parallel. - - -

- -
cl_I lognot (const cl_I& x) -
- -
cl_I operator ~ (const cl_I& x) -
- -Logical not, like ~x in C. This is the same as -1-x. - -
cl_I logand (const cl_I& x, const cl_I& y) -
- -
cl_I operator & (const cl_I& x, const cl_I& y) -
- -Logical and, like x & y in C. - -
cl_I logior (const cl_I& x, const cl_I& y) -
- -
cl_I operator | (const cl_I& x, const cl_I& y) -
- -Logical (inclusive) or, like x | y in C. - -
cl_I logxor (const cl_I& x, const cl_I& y) -
- -
cl_I operator ^ (const cl_I& x, const cl_I& y) -
- -Exclusive or, like x ^ y in C. - -
cl_I logeqv (const cl_I& x, const cl_I& y) -
- -Bitwise equivalence, like ~(x ^ y) in C. - -
cl_I lognand (const cl_I& x, const cl_I& y) -
- -Bitwise not and, like ~(x & y) in C. - -
cl_I lognor (const cl_I& x, const cl_I& y) -
- -Bitwise not or, like ~(x | y) in C. - -
cl_I logandc1 (const cl_I& x, const cl_I& y) -
- -Logical and, complementing the first argument, like ~x & y in C. - -
cl_I logandc2 (const cl_I& x, const cl_I& y) -
- -Logical and, complementing the second argument, like x & ~y in C. - -
cl_I logorc1 (const cl_I& x, const cl_I& y) -
- -Logical or, complementing the first argument, like ~x | y in C. - -
cl_I logorc2 (const cl_I& x, const cl_I& y) -
- -Logical or, complementing the second argument, like x | ~y in C. -
- -

-These operations are all available though the function -

- -
cl_I boole (cl_boole op, const cl_I& x, const cl_I& y) -
- -
-

-where op must have one of the 16 values (each one stands for a function -which combines two bits into one bit): boole_clr, boole_set, -boole_1, boole_2, boole_c1, boole_c2, -boole_and, boole_ior, boole_xor, boole_eqv, -boole_nand, boole_nor, boole_andc1, boole_andc2, -boole_orc1, boole_orc2. - - - - - - - - - - - - - - - - - -

-Other functions that view integers as bit strings: - - -

- -
cl_boolean logtest (const cl_I& x, const cl_I& y) -
- -Returns true if some bit is set in both x and y, i.e. if -logand(x,y) != 0. - -
cl_boolean logbitp (const cl_I& n, const cl_I& x) -
- -Returns true if the nth bit (from the right) of x is set. -Bit 0 is the least significant bit. - -
uintL logcount (const cl_I& x) -
- -Returns the number of one bits in x, if x >= 0, or -the number of zero bits in x, if x < 0. -
- -

-The following functions operate on intervals of bits in integers. -The type - -

-struct cl_byte { uintL size; uintL position; };
-
- -

- -represents the bit interval containing the bits -position...position+size-1 of an integer. -The constructor cl_byte(size,position) constructs a cl_byte. - - -

- -
cl_I ldb (const cl_I& n, const cl_byte& b) -
- -extracts the bits of n described by the bit interval b -and returns them as a nonnegative integer with b.size bits. - -
cl_boolean ldb_test (const cl_I& n, const cl_byte& b) -
- -Returns true if some bit described by the bit interval b is set in -n. - -
cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b) -
- -Returns n, with the bits described by the bit interval b -replaced by newbyte. Only the lowest b.size bits of -newbyte are relevant. -
- -

-The functions ldb and dpb implicitly shift. The following -functions are their counterparts without shifting: - - -

- -
cl_I mask_field (const cl_I& n, const cl_byte& b) -
- -returns an integer with the bits described by the bit interval b -copied from the corresponding bits in n, the other bits zero. - -
cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b) -
- -returns an integer where the bits described by the bit interval b -come from newbyte and the other bits come from n. -
- -

-The following relations hold: - - - -

- -

-The following operations on integers as bit strings are efficient shortcuts -for common arithmetic operations: - - -

- -
cl_boolean oddp (const cl_I& x) -
- -Returns true if the least significant bit of x is 1. Equivalent to -mod(x,2) != 0. - -
cl_boolean evenp (const cl_I& x) -
- -Returns true if the least significant bit of x is 0. Equivalent to -mod(x,2) == 0. - -
cl_I operator << (const cl_I& x, const cl_I& n) -
- -Shifts x by n bits to the left. n should be >=0. -Equivalent to x * expt(2,n). - -
cl_I operator >> (const cl_I& x, const cl_I& n) -
- -Shifts x by n bits to the right. n should be >=0. -Bits shifted out to the right are thrown away. -Equivalent to floor(x / expt(2,n)). - -
cl_I ash (const cl_I& x, const cl_I& y) -
- -Shifts x by y bits to the left (if y>=0) or -by -y bits to the right (if y<=0). In other words, this -returns floor(x * expt(2,y)). - -
uintL integer_length (const cl_I& x) -
- -Returns the number of bits (excluding the sign bit) needed to represent x -in two's complement notation. This is the smallest n >= 0 such that --2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that -2^(n-1) <= x < 2^n. - -
uintL ord2 (const cl_I& x) -
- -x must be non-zero. This function returns the number of 0 bits at the -right of x in two's complement notation. This is the largest n >= 0 -such that 2^n divides x. - -
uintL power2p (const cl_I& x) -
- -x must be > 0. This function checks whether x is a power of 2. -If x = 2^(n-1), it returns n. Else it returns 0. -(See also the function logp.) -
- - - -

4.9.2 Number theoretic functions

- -
- -
uint32 gcd (uint32 a, uint32 b) -
- -
cl_I gcd (const cl_I& a, const cl_I& b) -
-This function returns the greatest common divisor of a and b, -normalized to be >= 0. - -
cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v) -
- -This function ("extended gcd") returns the greatest common divisor g of -a and b and at the same time the representation of g -as an integral linear combination of a and b: -u and v with u*a+v*b = g, g >= 0. -u and v will be normalized to be of smallest possible absolute -value, in the following sense: If a and b are non-zero, and -abs(a) != abs(b), u and v will satisfy the inequalities -abs(u) <= abs(b)/(2*g), abs(v) <= abs(a)/(2*g). - -
cl_I lcm (const cl_I& a, const cl_I& b) -
- -This function returns the least common multiple of a and b, -normalized to be >= 0. - -
cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l) -
- -
cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l) -
-a must be > 0. b must be >0 and != 1. If log(a,b) is -rational number, this function returns true and sets *l = log(a,b), else -it returns false. -
- - - -

4.9.3 Combinatorial functions

- -
- -
cl_I factorial (uintL n) -
- -n must be a small integer >= 0. This function returns the factorial -n! = 1*2*...*n. - -
cl_I doublefactorial (uintL n) -
- -n must be a small integer >= 0. This function returns the -doublefactorial n!! = 1*3*...*n or -n!! = 2*4*...*n, respectively. - -
cl_I binomial (uintL n, uintL k) -
- -n and k must be small integers >= 0. This function returns the -binomial coefficient -(n choose k) = n! / k! (n-k)! -for 0 <= k <= n, 0 else. -
- - - -

4.10 Functions on floating-point numbers

- -

-Recall that a floating-point number consists of a sign s, an -exponent e and a mantissa m. The value of the number is -(-1)^s * 2^e * m. - - -

-Each of the classes -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations. - - -

- -
type scale_float (const type& x, sintL delta) -
- -
type scale_float (const type& x, const cl_I& delta) -
-Returns x*2^delta. This is more efficient than an explicit multiplication -because it copies x and modifies the exponent. -
- -

-The following functions provide an abstract interface to the underlying -representation of floating-point numbers. - - -

- -
sintL float_exponent (const type& x) -
- -Returns the exponent e of x. -For x = 0.0, this is 0. For x non-zero, this is the unique -integer with 2^(e-1) <= abs(x) < 2^e. - -
sintL float_radix (const type& x) -
- -Returns the base of the floating-point representation. This is always 2. - -
type float_sign (const type& x) -
- -Returns the sign s of x as a float. The value is 1 for -x >= 0, -1 for x < 0. - -
uintL float_digits (const type& x) -
- -Returns the number of mantissa bits in the floating-point representation -of x, including the hidden bit. The value only depends on the type -of x, not on its value. - -
uintL float_precision (const type& x) -
- -Returns the number of significant mantissa bits in the floating-point -representation of x. Since denormalized numbers are not supported, -this is the same as float_digits(x) if x is non-zero, and -0 if x = 0. -
- -

-The complete internal representation of a float is encoded in the type - - - - - -decoded_float (or decoded_sfloat, decoded_ffloat, -decoded_dfloat, decoded_lfloat, respectively), defined by - -

-struct decoded_typefloat {
-        type mantissa; cl_I exponent; type sign;
-};
-
- -

-and returned by the function - - -

- -
decoded_typefloat decode_float (const type& x) -
- -For x non-zero, this returns (-1)^s, e, m with -x = (-1)^s * 2^e * m and 0.5 <= m < 1.0. For x = 0, -it returns (-1)^s=1, e=0, m=0. -e is the same as returned by the function float_exponent. -
- -

-A complete decoding in terms of integers is provided as type - - -

-struct cl_idecoded_float {
-        cl_I mantissa; cl_I exponent; cl_I sign;
-};
-
- -

-by the following function: - - -

- -
cl_idecoded_float integer_decode_float (const type& x) -
- -For x non-zero, this returns (-1)^s, e, m with -x = (-1)^s * 2^e * m and m an integer with float_digits(x) -bits. For x = 0, it returns (-1)^s=1, e=0, m=0. -WARNING: The exponent e is not the same as the one returned by -the functions decode_float and float_exponent. -
- -

-Some other function, implemented only for class cl_F: - - -

- -
cl_F float_sign (const cl_F& x, const cl_F& y) -
- -This returns a floating point number whose precision and absolute value -is that of y and whose sign is that of x. If x is -zero, it is treated as positive. Same for y. -
- - - -

4.11 Conversion functions

-

- - - - - -

4.11.1 Conversion to floating-point numbers

- -

-The type cl_float_format_t describes a floating-point format. - - - -

- -
cl_float_format_t cl_float_format (uintL n) -
- -Returns the smallest float format which guarantees at least n -decimal digits in the mantissa (after the decimal point). - -
cl_float_format_t cl_float_format (const cl_F& x) -
-Returns the floating point format of x. - -
cl_float_format_t default_float_format -
- -Global variable: the default float format used when converting rational numbers -to floats. -
- -

-To convert a real number to a float, each of the types -cl_R, cl_F, cl_I, cl_RA, -int, unsigned int, float, double -defines the following operations: - - -

- -
cl_F cl_float (const type&x, cl_float_format_t f) -
- -Returns x as a float of format f. -
cl_F cl_float (const type&x, const cl_F& y) -
-Returns x in the float format of y. -
cl_F cl_float (const type&x) -
-Returns x as a float of format default_float_format if -it is an exact number, or x itself if it is already a float. -
- -

-Of course, converting a number to a float can lose precision. - - -

-Every floating-point format has some characteristic numbers: - - -

- -
cl_F most_positive_float (cl_float_format_t f) -
- -Returns the largest (most positive) floating point number in float format f. - -
cl_F most_negative_float (cl_float_format_t f) -
- -Returns the smallest (most negative) floating point number in float format f. - -
cl_F least_positive_float (cl_float_format_t f) -
- -Returns the least positive floating point number (i.e. > 0 but closest to 0) -in float format f. - -
cl_F least_negative_float (cl_float_format_t f) -
- -Returns the least negative floating point number (i.e. < 0 but closest to 0) -in float format f. - -
cl_F float_epsilon (cl_float_format_t f) -
- -Returns the smallest floating point number e > 0 such that 1+e != 1. - -
cl_F float_negative_epsilon (cl_float_format_t f) -
- -Returns the smallest floating point number e > 0 such that 1-e != 1. -
- - - -

4.11.2 Conversion to rational numbers

- -

-Each of the classes cl_R, cl_RA, cl_F -defines the following operation: - - -

- -
cl_RA rational (const type& x) -
- -Returns the value of x as an exact number. If x is already -an exact number, this is x. If x is a floating-point number, -the value is a rational number whose denominator is a power of 2. -
- -

-In order to convert back, say, (cl_F)(cl_R)"1/3" to 1/3, there is -the function - - -

- -
cl_RA rationalize (const cl_R& x) -
- -If x is a floating-point number, it actually represents an interval -of real numbers, and this function returns the rational number with -smallest denominator (and smallest numerator, in magnitude) -which lies in this interval. -If x is already an exact number, this function returns x. -
- -

-If x is any float, one has - - - -

- - - -

4.12 Random number generators

- -

-A random generator is a machine which produces (pseudo-)random numbers. -The include file <cln/random.h> defines a class random_state -which contains the state of a random generator. If you make a copy -of the random number generator, the original one and the copy will produce -the same sequence of random numbers. - - -

-The following functions return (pseudo-)random numbers in different formats. -Calling one of these modifies the state of the random number generator in -a complicated but deterministic way. - - -

-The global variable - - - -

-random_state default_random_state
-
- -

-contains a default random number generator. It is used when the functions -below are called without random_state argument. - - -

- -
uint32 random32 (random_state& randomstate) -
-
uint32 random32 () -
- -Returns a random unsigned 32-bit number. All bits are equally random. - -
cl_I random_I (random_state& randomstate, const cl_I& n) -
-
cl_I random_I (const cl_I& n) -
- -n must be an integer > 0. This function returns a random integer x -in the range 0 <= x < n. - -
cl_F random_F (random_state& randomstate, const cl_F& n) -
-
cl_F random_F (const cl_F& n) -
- -n must be a float > 0. This function returns a random floating-point -number of the same format as n in the range 0 <= x < n. - -
cl_R random_R (random_state& randomstate, const cl_R& n) -
-
cl_R random_R (const cl_R& n) -
- -Behaves like random_I if n is an integer and like random_F -if n is a float. -
- - - -

4.13 Obfuscating operators

-

- - - -

-The modifying C/C++ operators +=, -=, *=, /=, -&=, |=, ^=, <<=, >>= -are not available by default because their -use tends to make programs unreadable. It is trivial to get away without -them. However, if you feel that you absolutely need these operators -to get happy, then add - -

-#define WANT_OBFUSCATING_OPERATORS
-
- -

- -to the beginning of your source files, before the inclusion of any CLN -include files. This flag will enable the following operators: - - -

-For the classes cl_N, cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF: - - -

- -
type& operator += (type&, const type&) -
- -
type& operator -= (type&, const type&) -
- -
type& operator *= (type&, const type&) -
- -
type& operator /= (type&, const type&) -
- -
- -

-For the class cl_I: - - -

- -
type& operator += (type&, const type&) -
-
type& operator -= (type&, const type&) -
-
type& operator *= (type&, const type&) -
-
type& operator &= (type&, const type&) -
- -
type& operator |= (type&, const type&) -
- -
type& operator ^= (type&, const type&) -
- -
type& operator <<= (type&, const type&) -
- -
type& operator >>= (type&, const type&) -
- -
- -

-For the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF: - - -

- -
type& operator ++ (type& x) -
- -The prefix operator ++x. - -
void operator ++ (type& x, int) -
-The postfix operator x++. - -
type& operator -- (type& x) -
- -The prefix operator --x. - -
void operator -- (type& x, int) -
-The postfix operator x--. -
- -

-Note that by using these obfuscating operators, you wouldn't gain efficiency: -In CLN `x += y;' is exactly the same as `x = x+y;', not more -efficient. - - - - -

5. Input/Output

-

- - - - - -

5.1 Internal and printed representation

-

- - - -

-All computations deal with the internal representations of the numbers. - - -

-Every number has an external representation as a sequence of ASCII characters. -Several external representations may denote the same number, for example, -"20.0" and "20.000". - - -

-Converting an internal to an external representation is called "printing", - -converting an external to an internal representation is called "reading". - -In CLN, it is always true that conversion of an internal to an external -representation and then back to an internal representation will yield the -same internal representation. Symbolically: read(print(x)) == x. -This is called "print-read consistency". - - -

-Different types of numbers have different external representations (case -is insignificant): - - -

- -
Integers -
-External representation: sign{digit}+. The reader also accepts the -Common Lisp syntaxes sign{digit}+. with a trailing dot -for decimal integers -and the #nR, #b, #o, #x prefixes. - -
Rational numbers -
-External representation: sign{digit}+/{digit}+. -The #nR, #b, #o, #x prefixes are allowed -here as well. - -
Floating-point numbers -
-External representation: sign{digit}*exponent or -sign{digit}*.{digit}*exponent or -sign{digit}*.{digit}+. A precision specifier -of the form _prec may be appended. There must be at least -one digit in the non-exponent part. The exponent has the syntax -expmarker expsign {digit}+. -The exponent marker is - - -
    -
  • - -`s' for short-floats, -
  • - -`f' for single-floats, -
  • - -`d' for double-floats, -
  • - -`L' for long-floats, -
- -or `e', which denotes a default float format. The precision specifying -suffix has the syntax _prec where prec denotes the number of -valid mantissa digits (in decimal, excluding leading zeroes), cf. also -function `cl_float_format'. - -
Complex numbers -
-External representation: - -
    -
  • - -In algebraic notation: realpart+imagparti. Of course, -if imagpart is negative, its printed representation begins with -a `-', and the `+' between realpart and imagpart -may be omitted. Note that this notation cannot be used when the imagpart -is rational and the rational number's base is >18, because the `i' -is then read as a digit. -
  • - -In Common Lisp notation: #C(realpart imagpart). -
- -
- - - -

5.2 Input functions

- -

-Including <cln/io.h> defines a type cl_istream, which is -the type of the first argument to all input functions. cl_istream -is the same as std::istream&. - - -

-The variable - -

- -

-contains the standard input stream. - - -

-These are the simple input functions: - - -

- -
int freadchar (cl_istream stream) -
-Reads a character from stream. Returns cl_EOF (not a `char'!) -if the end of stream was encountered or an error occurred. - -
int funreadchar (cl_istream stream, int c) -
-Puts back c onto stream. c must be the result of the -last freadchar operation on stream. -
- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines, in <cln/type_io.h>, the following input function: - - -

- -
cl_istream operator>> (cl_istream stream, type& result) -
-Reads a number from stream and stores it in the result. -
- -

-The most flexible input functions, defined in <cln/type_io.h>, -are the following: - - -

- -
cl_N read_complex (cl_istream stream, const cl_read_flags& flags) -
-
cl_R read_real (cl_istream stream, const cl_read_flags& flags) -
-
cl_F read_float (cl_istream stream, const cl_read_flags& flags) -
-
cl_RA read_rational (cl_istream stream, const cl_read_flags& flags) -
-
cl_I read_integer (cl_istream stream, const cl_read_flags& flags) -
-Reads a number from stream. The flags are parameters which -affect the input syntax. Whitespace before the number is silently skipped. - -
cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-Reads a number from a string in memory. The flags are parameters which -affect the input syntax. The string starts at string and ends at -string_limit (exclusive limit). string_limit may also be -NULL, denoting the entire string, i.e. equivalent to -string_limit = string + strlen(string). If end_of_parse is -NULL, the string in memory must contain exactly one number and nothing -more, else a fatal error will be signalled. If end_of_parse -is not NULL, *end_of_parse will be assigned a pointer past -the last parsed character (i.e. string_limit if nothing came after -the number). Whitespace is not allowed. -
- -

-The structure cl_read_flags contains the following fields: - - -

- -
cl_read_syntax_t syntax -
-The possible results of the read operation. Possible values are -syntax_number, syntax_real, syntax_rational, -syntax_integer, syntax_float, syntax_sfloat, -syntax_ffloat, syntax_dfloat, syntax_lfloat. - -
cl_read_lsyntax_t lsyntax -
-Specifies the language-dependent syntax variant for the read operation. -Possible values are - -
- -
lsyntax_standard -
-accept standard algebraic notation only, no complex numbers, -
lsyntax_algebraic -
-accept the algebraic notation x+yi for complex numbers, -
lsyntax_commonlisp -
-accept the #b, #o, #x syntaxes for binary, octal, -hexadecimal numbers, -#baseR for rational numbers in a given base, -#c(realpart imagpart) for complex numbers, -
lsyntax_all -
-accept all of these extensions. -
- -
unsigned int rational_base -
-The base in which rational numbers are read. - -
cl_float_format_t float_flags.default_float_format -
-The float format used when reading floats with exponent marker `e'. - -
cl_float_format_t float_flags.default_lfloat_format -
-The float format used when reading floats with exponent marker `l'. - -
cl_boolean float_flags.mantissa_dependent_float_format -
-When this flag is true, floats specified with more digits than corresponding -to the exponent marker they contain, but without _nnn suffix, will get a -precision corresponding to their number of significant digits. -
- - - -

5.3 Output functions

- -

-Including <cln/io.h> defines a type cl_ostream, which is -the type of the first argument to all output functions. cl_ostream -is the same as std::ostream&. - - -

-The variable - -

- -

-contains the standard output stream. - - -

-The variable - -

- -

-contains the standard error output stream. - - -

-These are the simple output functions: - - -

- -
void fprintchar (cl_ostream stream, char c) -
-Prints the character x literally on the stream. - -
void fprint (cl_ostream stream, const char * string) -
-Prints the string literally on the stream. - -
void fprintdecimal (cl_ostream stream, int x) -
-
void fprintdecimal (cl_ostream stream, const cl_I& x) -
-Prints the integer x in decimal on the stream. - -
void fprintbinary (cl_ostream stream, const cl_I& x) -
-Prints the integer x in binary (base 2, without prefix) -on the stream. - -
void fprintoctal (cl_ostream stream, const cl_I& x) -
-Prints the integer x in octal (base 8, without prefix) -on the stream. - -
void fprinthexadecimal (cl_ostream stream, const cl_I& x) -
-Prints the integer x in hexadecimal (base 16, without prefix) -on the stream. -
- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines, in <cln/type_io.h>, the following output functions: - - -

- -
void fprint (cl_ostream stream, const type& x) -
-
cl_ostream operator<< (cl_ostream stream, const type& x) -
-Prints the number x on the stream. The output may depend -on the global printer settings in the variable default_print_flags. -The ostream flags and settings (flags, width and locale) are -ignored. -
- -

-The most flexible output function, defined in <cln/type_io.h>, -are the following: - -

-void print_complex  (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_N& z);
-void print_real     (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_R& z);
-void print_float    (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_F& z);
-void print_rational (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_RA& z);
-void print_integer  (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_I& z);
-
- -

-Prints the number x on the stream. The flags are -parameters which affect the output. - - -

-The structure type cl_print_flags contains the following fields: - - -

- -
unsigned int rational_base -
-The base in which rational numbers are printed. Default is 10. - -
cl_boolean rational_readably -
-If this flag is true, rational numbers are printed with radix specifiers in -Common Lisp syntax (#nR or #b or #o or #x -prefixes, trailing dot). Default is false. - -
cl_boolean float_readably -
-If this flag is true, type specific exponent markers have precedence over 'E'. -Default is false. - -
cl_float_format_t default_float_format -
-Floating point numbers of this format will be printed using the 'E' exponent -marker. Default is cl_float_format_ffloat. - -
cl_boolean complex_readably -
-If this flag is true, complex numbers will be printed using the Common Lisp -syntax #C(realpart imagpart). Default is false. - -
cl_string univpoly_varname -
-Univariate polynomials with no explicit indeterminate name will be printed -using this variable name. Default is "x". -
- -

-The global variable default_print_flags contains the default values, -used by the function fprint. - - - - -

6. Rings

- -

-CLN has a class of abstract rings. - - - -

-                         Ring
-                       cl_ring
-                     <cln/ring.h>
-
- -

-Rings can be compared for equality: - - -

- -
bool operator== (const cl_ring&, const cl_ring&) -
-
bool operator!= (const cl_ring&, const cl_ring&) -
-These compare two rings for equality. -
- -

-Given a ring R, the following members can be used. - - -

- -
void R->fprint (cl_ostream stream, const cl_ring_element& x) -
- -
cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->zero () -
- -
cl_boolean R->zerop (const cl_ring_element& x) -
- -
cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->uminus (const cl_ring_element& x) -
- -
cl_ring_element R->one () -
- -
cl_ring_element R->canonhom (const cl_I& x) -
- -
cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->square (const cl_ring_element& x) -
- -
cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y) -
- -
- -

-The following rings are built-in. - - -

- -
cl_null_ring cl_0_ring -
-The null ring, containing only zero. - -
cl_complex_ring cl_C_ring -
-The ring of complex numbers. This corresponds to the type cl_N. - -
cl_real_ring cl_R_ring -
-The ring of real numbers. This corresponds to the type cl_R. - -
cl_rational_ring cl_RA_ring -
-The ring of rational numbers. This corresponds to the type cl_RA. - -
cl_integer_ring cl_I_ring -
-The ring of integers. This corresponds to the type cl_I. -
- -

-Type tests can be performed for any of cl_C_ring, cl_R_ring, -cl_RA_ring, cl_I_ring: - - -

- -
cl_boolean instanceof (const cl_number& x, const cl_number_ring& R) -
- -Tests whether the given number is an element of the number ring R. -
- - - -

7. Modular integers

-

- - - - - -

7.1 Modular integer rings

-

- - - -

-CLN implements modular integers, i.e. integers modulo a fixed integer N. -The modulus is explicitly part of every modular integer. CLN doesn't -allow you to (accidentally) mix elements of different modular rings, -e.g. (3 mod 4) + (2 mod 5) will result in a runtime error. -(Ideally one would imagine a generic data type cl_MI(N), but C++ -doesn't have generic types. So one has to live with runtime checks.) - - -

-The class of modular integer rings is - - - -

-                         Ring
-                       cl_ring
-                     <cln/ring.h>
-                          |
-                          |
-                 Modular integer ring
-                    cl_modint_ring
-                  <cln/modinteger.h>
-
- -

- - - -

-and the class of all modular integers (elements of modular integer rings) is - - - -

-                    Modular integer
-                         cl_MI
-                   <cln/modinteger.h>
-
- -

-Modular integer rings are constructed using the function - - -

- -
cl_modint_ring find_modint_ring (const cl_I& N) -
- -This function returns the modular ring `Z/NZ'. It takes care -of finding out about special cases of N, like powers of two -and odd numbers for which Montgomery multiplication will be a win, - -and precomputes any necessary auxiliary data for computing modulo N. -There is a cache table of rings, indexed by N (or, more precisely, -by abs(N)). This ensures that the precomputation costs are reduced -to a minimum. -
- -

-Modular integer rings can be compared for equality: - - -

- -
bool operator== (const cl_modint_ring&, const cl_modint_ring&) -
- -
bool operator!= (const cl_modint_ring&, const cl_modint_ring&) -
- -These compare two modular integer rings for equality. Two different calls -to find_modint_ring with the same argument necessarily return the -same ring because it is memoized in the cache table. -
- - - -

7.2 Functions on modular integers

- -

-Given a modular integer ring R, the following members can be used. - - -

- -
cl_I R->modulus -
- -This is the ring's modulus, normalized to be nonnegative: abs(N). - -
cl_MI R->zero() -
- -This returns 0 mod N. - -
cl_MI R->one() -
- -This returns 1 mod N. - -
cl_MI R->canonhom (const cl_I& x) -
- -This returns x mod N. - -
cl_I R->retract (const cl_MI& x) -
- -This is a partial inverse function to R->canonhom. It returns the -standard representative (>=0, <N) of x. - -
cl_MI R->random(random_state& randomstate) -
-
cl_MI R->random() -
- -This returns a random integer modulo N. -
- -

-The following operations are defined on modular integers. - - -

- -
cl_modint_ring x.ring () -
- -Returns the ring to which the modular integer x belongs. - -
cl_MI operator+ (const cl_MI&, const cl_MI&) -
- -Returns the sum of two modular integers. One of the arguments may also -be a plain integer. - -
cl_MI operator- (const cl_MI&, const cl_MI&) -
- -Returns the difference of two modular integers. One of the arguments may also -be a plain integer. - -
cl_MI operator- (const cl_MI&) -
-Returns the negative of a modular integer. - -
cl_MI operator* (const cl_MI&, const cl_MI&) -
- -Returns the product of two modular integers. One of the arguments may also -be a plain integer. - -
cl_MI square (const cl_MI&) -
- -Returns the square of a modular integer. - -
cl_MI recip (const cl_MI& x) -
- -Returns the reciprocal x^-1 of a modular integer x. x -must be coprime to the modulus, otherwise an error message is issued. - -
cl_MI div (const cl_MI& x, const cl_MI& y) -
- -Returns the quotient x*y^-1 of two modular integers x, y. -y must be coprime to the modulus, otherwise an error message is issued. - -
cl_MI expt_pos (const cl_MI& x, const cl_I& y) -
- -y must be > 0. Returns x^y. - -
cl_MI expt (const cl_MI& x, const cl_I& y) -
- -Returns x^y. If y is negative, x must be coprime to the -modulus, else an error message is issued. - -
cl_MI operator<< (const cl_MI& x, const cl_I& y) -
- -Returns x*2^y. - -
cl_MI operator>> (const cl_MI& x, const cl_I& y) -
- -Returns x*2^-y. When y is positive, the modulus must be odd, -or an error message is issued. - -
bool operator== (const cl_MI&, const cl_MI&) -
- -
bool operator!= (const cl_MI&, const cl_MI&) -
- -Compares two modular integers, belonging to the same modular integer ring, -for equality. - -
cl_boolean zerop (const cl_MI& x) -
- -Returns true if x is 0 mod N. -
- -

-The following output functions are defined (see also the chapter on -input/output). - - -

- -
void fprint (cl_ostream stream, const cl_MI& x) -
- -
cl_ostream operator<< (cl_ostream stream, const cl_MI& x) -
- -Prints the modular integer x on the stream. The output may depend -on the global printer settings in the variable default_print_flags. -
- - - -

8. Symbolic data types

-

- - - -

-CLN implements two symbolic (non-numeric) data types: strings and symbols. - - - - -

8.1 Strings

-

- - - - -

-The class - - - -

-                      String
-                     cl_string
-                   <cln/string.h>
-
- -

-implements immutable strings. - - -

-Strings are constructed through the following constructors: - - -

- -
cl_string (const char * s) -
-Returns an immutable copy of the (zero-terminated) C string s. - -
cl_string (const char * ptr, unsigned long len) -
-Returns an immutable copy of the len characters at -ptr[0], ..., ptr[len-1]. NUL characters are allowed. -
- -

-The following functions are available on strings: - - -

- -
operator = -
-Assignment from cl_string and const char *. - -
s.length() -
- -
strlen(s) -
- -Returns the length of the string s. - -
s[i] -
- -Returns the ith character of the string s. -i must be in the range 0 <= i < s.length(). - -
bool equal (const cl_string& s1, const cl_string& s2) -
- -Compares two strings for equality. One of the arguments may also be a -plain const char *. -
- - - -

8.2 Symbols

-

- - - - -

-Symbols are uniquified strings: all symbols with the same name are shared. -This means that comparison of two symbols is fast (effectively just a pointer -comparison), whereas comparison of two strings must in the worst case walk -both strings until their end. -Symbols are used, for example, as tags for properties, as names of variables -in polynomial rings, etc. - - -

-Symbols are constructed through the following constructor: - - -

- -
cl_symbol (const cl_string& s) -
-Looks up or creates a new symbol with a given name. -
- -

-The following operations are available on symbols: - - -

- -
cl_string (const cl_symbol& sym) -
-Conversion to cl_string: Returns the string which names the symbol -sym. - -
bool equal (const cl_symbol& sym1, const cl_symbol& sym2) -
- -Compares two symbols for equality. This is very fast. -
- - - -

9. Univariate polynomials

-

- - - - - - -

9.1 Univariate polynomial rings

- -

-CLN implements univariate polynomials (polynomials in one variable) over an -arbitrary ring. The indeterminate variable may be either unnamed (and will be -printed according to default_print_flags.univpoly_varname, which -defaults to `x') or carry a given name. The base ring and the -indeterminate are explicitly part of every polynomial. CLN doesn't allow you to -(accidentally) mix elements of different polynomial rings, e.g. -(a^2+1) * (b^3-1) will result in a runtime error. (Ideally this should -return a multivariate polynomial, but they are not yet implemented in CLN.) - - -

-The classes of univariate polynomial rings are - - - -

-                           Ring
-                         cl_ring
-                       <cln/ring.h>
-                            |
-                            |
-                 Univariate polynomial ring
-                      cl_univpoly_ring
-                      <cln/univpoly.h>
-                            |
-           +----------------+-------------------+
-           |                |                   |
- Complex polynomial ring    |    Modular integer polynomial ring
- cl_univpoly_complex_ring   |        cl_univpoly_modint_ring
- <cln/univpoly_complex.h>   |        <cln/univpoly_modint.h>
-                            |
-           +----------------+
-           |                |
-   Real polynomial ring     |
-   cl_univpoly_real_ring    |
-   <cln/univpoly_real.h>    |
-                            |
-           +----------------+
-           |                |
- Rational polynomial ring   |
- cl_univpoly_rational_ring  |
- <cln/univpoly_rational.h>  |
-                            |
-           +----------------+
-           |
- Integer polynomial ring
- cl_univpoly_integer_ring
- <cln/univpoly_integer.h>
-
- -

-and the corresponding classes of univariate polynomials are - - - -

-                   Univariate polynomial
-                          cl_UP
-                      <cln/univpoly.h>
-                            |
-           +----------------+-------------------+
-           |                |                   |
-   Complex polynomial       |      Modular integer polynomial
-        cl_UP_N             |                cl_UP_MI
- <cln/univpoly_complex.h>   |        <cln/univpoly_modint.h>
-                            |
-           +----------------+
-           |                |
-     Real polynomial        |
-        cl_UP_R             |
-  <cln/univpoly_real.h>     |
-                            |
-           +----------------+
-           |                |
-   Rational polynomial      |
-        cl_UP_RA            |
- <cln/univpoly_rational.h>  |
-                            |
-           +----------------+
-           |
-   Integer polynomial
-        cl_UP_I
- <cln/univpoly_integer.h>
-
- -

-Univariate polynomial rings are constructed using the functions - - -

- -
cl_univpoly_ring find_univpoly_ring (const cl_ring& R) -
-
cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname) -
-This function returns the polynomial ring `R[X]', unnamed or named. -R may be an arbitrary ring. This function takes care of finding out -about special cases of R, such as the rings of complex numbers, -real numbers, rational numbers, integers, or modular integer rings. -There is a cache table of rings, indexed by R and varname. -This ensures that two calls of this function with the same arguments will -return the same polynomial ring. - -
cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R) -
- -
cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname) -
-
cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R) -
-
cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname) -
-
cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R) -
-
cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname) -
-
cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R) -
-
cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname) -
-
cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R) -
-
cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname) -
-These functions are equivalent to the general find_univpoly_ring, -only the return type is more specific, according to the base ring's type. -
- - - -

9.2 Functions on univariate polynomials

- -

-Given a univariate polynomial ring R, the following members can be used. - - -

- -
cl_ring R->basering() -
- -This returns the base ring, as passed to `find_univpoly_ring'. - -
cl_UP R->zero() -
- -This returns 0 in R, a polynomial of degree -1. - -
cl_UP R->one() -
- -This returns 1 in R, a polynomial of degree <= 0. - -
cl_UP R->canonhom (const cl_I& x) -
- -This returns x in R, a polynomial of degree <= 0. - -
cl_UP R->monomial (const cl_ring_element& x, uintL e) -
- -This returns a sparse polynomial: x * X^e, where X is the -indeterminate. - -
cl_UP R->create (sintL degree) -
- -Creates a new polynomial with a given degree. The zero polynomial has degree --1. After creating the polynomial, you should put in the coefficients, -using the set_coeff member function, and then call the finalize -member function. -
- -

-The following are the only destructive operations on univariate polynomials. - - -

- -
void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y) -
- -This changes the coefficient of X^index in x to be y. -After changing a polynomial and before applying any "normal" operation on it, -you should call its finalize member function. - -
void finalize (cl_UP& x) -
- -This function marks the endpoint of destructive modifications of a polynomial. -It normalizes the internal representation so that subsequent computations have -less overhead. Doing normal computations on unnormalized polynomials may -produce wrong results or crash the program. -
- -

-The following operations are defined on univariate polynomials. - - -

- -
cl_univpoly_ring x.ring () -
- -Returns the ring to which the univariate polynomial x belongs. - -
cl_UP operator+ (const cl_UP&, const cl_UP&) -
- -Returns the sum of two univariate polynomials. - -
cl_UP operator- (const cl_UP&, const cl_UP&) -
- -Returns the difference of two univariate polynomials. - -
cl_UP operator- (const cl_UP&) -
-Returns the negative of a univariate polynomial. - -
cl_UP operator* (const cl_UP&, const cl_UP&) -
- -Returns the product of two univariate polynomials. One of the arguments may -also be a plain integer or an element of the base ring. - -
cl_UP square (const cl_UP&) -
- -Returns the square of a univariate polynomial. - -
cl_UP expt_pos (const cl_UP& x, const cl_I& y) -
- -y must be > 0. Returns x^y. - -
bool operator== (const cl_UP&, const cl_UP&) -
- -
bool operator!= (const cl_UP&, const cl_UP&) -
- -Compares two univariate polynomials, belonging to the same univariate -polynomial ring, for equality. - -
cl_boolean zerop (const cl_UP& x) -
- -Returns true if x is 0 in R. - -
sintL degree (const cl_UP& x) -
- -Returns the degree of the polynomial. The zero polynomial has degree -1. - -
cl_ring_element coeff (const cl_UP& x, uintL index) -
- -Returns the coefficient of X^index in the polynomial x. - -
cl_ring_element x (const cl_ring_element& y) -
- -Evaluation: If x is a polynomial and y belongs to the base ring, -then `x(y)' returns the value of the substitution of y into -x. - -
cl_UP deriv (const cl_UP& x) -
- -Returns the derivative of the polynomial x with respect to the -indeterminate X. -
- -

-The following output functions are defined (see also the chapter on -input/output). - - -

- -
void fprint (cl_ostream stream, const cl_UP& x) -
- -
cl_ostream operator<< (cl_ostream stream, const cl_UP& x) -
- -Prints the univariate polynomial x on the stream. The output may -depend on the global printer settings in the variable -default_print_flags. -
- - - -

9.3 Special polynomials

- -

-The following functions return special polynomials. - - -

- -
cl_UP_I tschebychev (sintL n) -
- - -Returns the n-th Chebyshev polynomial (n >= 0). - -
cl_UP_I hermite (sintL n) -
- - -Returns the n-th Hermite polynomial (n >= 0). - -
cl_UP_RA legendre (sintL n) -
- - -Returns the n-th Legendre polynomial (n >= 0). - -
cl_UP_I laguerre (sintL n) -
- - -Returns the n-th Laguerre polynomial (n >= 0). -
- -

-Information how to derive the differential equation satisfied by each -of these polynomials from their definition can be found in the -doc/polynomial/ directory. - - - - -

10. Internals

- - - -

10.1 Why C++ ?

-

- - - -

-Using C++ as an implementation language provides - - - -

- -

-With these language features, there is no need for two separate languages, -one for the implementation of the library and one in which the library's users -can program. This means that a prototype implementation of an algorithm -can be integrated into the library immediately after it has been tested and -debugged. No need to rewrite it in a low-level language after having prototyped -in a high-level language. - - - - -

10.2 Memory efficiency

- -

-In order to save memory allocations, CLN implements: - - - -

- - - -

10.3 Speed efficiency

- -

-Speed efficiency is obtained by the combination of the following tricks -and algorithms: - - - -

- - - -

10.4 Garbage collection

-

- - - -

-All the number classes are reference count classes: They only contain a pointer -to an object in the heap. Upon construction, assignment and destruction of -number objects, only the objects' reference count are manipulated. - - -

-Memory occupied by number objects are automatically reclaimed as soon as -their reference count drops to zero. - - -

-For number rings, another strategy is implemented: There is a cache of, -for example, the modular integer rings. A modular integer ring is destroyed -only if its reference count dropped to zero and the cache is about to be -resized. The effect of this strategy is that recently used rings remain -cached, whereas undue memory consumption through cached rings is avoided. - - - - -

11. Using the library

- -

-For the following discussion, we will assume that you have installed -the CLN source in $CLN_DIR and built it in $CLN_TARGETDIR. -For example, for me it's CLN_DIR="$HOME/cln" and -CLN_TARGETDIR="$HOME/cln/linuxelf". You might define these as -environment variables, or directly substitute the appropriate values. - - - - -

11.1 Compiler options

-

- - - -

-Until you have installed CLN in a public place, the following options are -needed: - - -

-When you compile CLN application code, add the flags - -

-   -I$CLN_DIR/include -I$CLN_TARGETDIR/include
-
- -

-to the C++ compiler's command line (make variable CFLAGS or CXXFLAGS). -When you link CLN application code to form an executable, add the flags - -

-   $CLN_TARGETDIR/src/libcln.a
-
- -

-to the C/C++ compiler's command line (make variable LIBS). - - -

-If you did a make install, the include files are installed in a -public directory (normally /usr/local/include), hence you don't -need special flags for compiling. The library has been installed to a -public directory as well (normally /usr/local/lib), hence when -linking a CLN application it is sufficient to give the flag -lcln. - - - - -

11.2 Compatibility to old CLN versions

-

- - - - -

-As of CLN version 1.1 all non-macro identifiers were hidden in namespace -cln in order to avoid potential name clashes with other C++ -libraries. If you have an old application, you will have to manually -port it to the new scheme. The following principles will help during -the transition: - -

- -

-When developing other libraries, please keep in mind not to import the -namespace cln in one of your public header files by saying -using namespace cln;. This would propagate to other applications -and can cause name clashes there. - - - - -

11.3 Include files

-

- - - - -

-Here is a summary of the include files and their contents. - - -

- -
<cln/object.h> -
-General definitions, reference counting, garbage collection. -
<cln/number.h> -
-The class cl_number. -
<cln/complex.h> -
-Functions for class cl_N, the complex numbers. -
<cln/real.h> -
-Functions for class cl_R, the real numbers. -
<cln/float.h> -
-Functions for class cl_F, the floats. -
<cln/sfloat.h> -
-Functions for class cl_SF, the short-floats. -
<cln/ffloat.h> -
-Functions for class cl_FF, the single-floats. -
<cln/dfloat.h> -
-Functions for class cl_DF, the double-floats. -
<cln/lfloat.h> -
-Functions for class cl_LF, the long-floats. -
<cln/rational.h> -
-Functions for class cl_RA, the rational numbers. -
<cln/integer.h> -
-Functions for class cl_I, the integers. -
<cln/io.h> -
-Input/Output. -
<cln/complex_io.h> -
-Input/Output for class cl_N, the complex numbers. -
<cln/real_io.h> -
-Input/Output for class cl_R, the real numbers. -
<cln/float_io.h> -
-Input/Output for class cl_F, the floats. -
<cln/sfloat_io.h> -
-Input/Output for class cl_SF, the short-floats. -
<cln/ffloat_io.h> -
-Input/Output for class cl_FF, the single-floats. -
<cln/dfloat_io.h> -
-Input/Output for class cl_DF, the double-floats. -
<cln/lfloat_io.h> -
-Input/Output for class cl_LF, the long-floats. -
<cln/rational_io.h> -
-Input/Output for class cl_RA, the rational numbers. -
<cln/integer_io.h> -
-Input/Output for class cl_I, the integers. -
<cln/input.h> -
-Flags for customizing input operations. -
<cln/output.h> -
-Flags for customizing output operations. -
<cln/malloc.h> -
-malloc_hook, free_hook. -
<cln/abort.h> -
-cl_abort. -
<cln/condition.h> -
-Conditions/exceptions. -
<cln/string.h> -
-Strings. -
<cln/symbol.h> -
-Symbols. -
<cln/proplist.h> -
-Property lists. -
<cln/ring.h> -
-General rings. -
<cln/null_ring.h> -
-The null ring. -
<cln/complex_ring.h> -
-The ring of complex numbers. -
<cln/real_ring.h> -
-The ring of real numbers. -
<cln/rational_ring.h> -
-The ring of rational numbers. -
<cln/integer_ring.h> -
-The ring of integers. -
<cln/numtheory.h> -
-Number threory functions. -
<cln/modinteger.h> -
-Modular integers. -
<cln/V.h> -
-Vectors. -
<cln/GV.h> -
-General vectors. -
<cln/GV_number.h> -
-General vectors over cl_number. -
<cln/GV_complex.h> -
-General vectors over cl_N. -
<cln/GV_real.h> -
-General vectors over cl_R. -
<cln/GV_rational.h> -
-General vectors over cl_RA. -
<cln/GV_integer.h> -
-General vectors over cl_I. -
<cln/GV_modinteger.h> -
-General vectors of modular integers. -
<cln/SV.h> -
-Simple vectors. -
<cln/SV_number.h> -
-Simple vectors over cl_number. -
<cln/SV_complex.h> -
-Simple vectors over cl_N. -
<cln/SV_real.h> -
-Simple vectors over cl_R. -
<cln/SV_rational.h> -
-Simple vectors over cl_RA. -
<cln/SV_integer.h> -
-Simple vectors over cl_I. -
<cln/SV_ringelt.h> -
-Simple vectors of general ring elements. -
<cln/univpoly.h> -
-Univariate polynomials. -
<cln/univpoly_integer.h> -
-Univariate polynomials over the integers. -
<cln/univpoly_rational.h> -
-Univariate polynomials over the rational numbers. -
<cln/univpoly_real.h> -
-Univariate polynomials over the real numbers. -
<cln/univpoly_complex.h> -
-Univariate polynomials over the complex numbers. -
<cln/univpoly_modint.h> -
-Univariate polynomials over modular integer rings. -
<cln/timing.h> -
-Timing facilities. -
<cln/cln.h> -
-Includes all of the above. -
- - - -

11.4 An Example

- -

-A function which computes the nth Fibonacci number can be written as follows. - - - - -

-#include <cln/integer.h>
-#include <cln/real.h>
-using namespace cln;
-
-// Returns F_n, computed as the nearest integer to
-// ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
-const cl_I fibonacci (int n)
-{
-        // Need a precision of ((1+sqrt(5))/2)^-n.
-        cl_float_format_t prec = cl_float_format((int)(0.208987641*n+5));
-        cl_R sqrt5 = sqrt(cl_float(5,prec));
-        cl_R phi = (1+sqrt5)/2;
-        return round1( expt(phi,n)/sqrt5 );
-}
-
- -

-Let's explain what is going on in detail. - - -

-The include file <cln/integer.h> is necessary because the type -cl_I is used in the function, and the include file <cln/real.h> -is needed for the type cl_R and the floating point number functions. -The order of the include files does not matter. In order not to write out -cln::foo we can safely import the whole namespace cln. - - -

-Then comes the function declaration. The argument is an int, the -result an integer. The return type is defined as `const cl_I', not -simply `cl_I', because that allows the compiler to detect typos like -`fibonacci(n) = 100'. It would be possible to declare the return -type as const cl_R (real number) or even const cl_N (complex -number). We use the most specialized possible return type because functions -which call `fibonacci' will be able to profit from the compiler's type -analysis: Adding two integers is slightly more efficient than adding the -same objects declared as complex numbers, because it needs less type -dispatch. Also, when linking to CLN as a non-shared library, this minimizes -the size of the resulting executable program. - - -

-The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest -integer. In order to get a correct result, the absolute error should be less -than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)). -To this end, the first line computes a floating point precision for sqrt(5) -and phi. - - -

-Then sqrt(5) is computed by first converting the integer 5 to a floating point -number and than taking the square root. The converse, first taking the square -root of 5, and then converting to the desired precision, would not work in -CLN: The square root would be computed to a default precision (normally -single-float precision), and the following conversion could not help about -the lacking accuracy. This is because CLN is not a symbolic computer algebra -system and does not represent sqrt(5) in a non-numeric way. - - -

-The type cl_R for sqrt5 and, in the following line, phi is the only -possible choice. You cannot write cl_F because the C++ compiler can -only infer that cl_float(5,prec) is a real number. You cannot write -cl_N because a `round1' does not exist for general complex -numbers. - - -

-When the function returns, all the local variables in the function are -automatically reclaimed (garbage collected). Only the result survives and -gets passed to the caller. - - -

-The file fibonacci.cc in the subdirectory examples -contains this implementation together with an even faster algorithm. - - - - -

11.5 Debugging support

-

- - - -

-When debugging a CLN application with GNU gdb, two facilities are -available from the library: - - - -

- - - -

12. Customizing

-

- - - - - -

12.1 Error handling

- -

-When a fatal error occurs, an error message is output to the standard error -output stream, and the function cl_abort is called. The default -version of this function (provided in the library) terminates the application. -To catch such a fatal error, you need to define the function cl_abort -yourself, with the prototype - -

-#include <cln/abort.h>
-void cl_abort (void);
-
- -

- -This function must not return control to its caller. - - - - -

12.2 Floating-point underflow

-

- - - -

-Floating point underflow denotes the situation when a floating-point number -is to be created which is so close to 0 that its exponent is too -low to be represented internally. By default, this causes a fatal error. -If you set the global variable - -

-cl_boolean cl_inhibit_floating_point_underflow
-
- -

-to cl_true, the error will be inhibited, and a floating-point zero -will be generated instead. The default value of -cl_inhibit_floating_point_underflow is cl_false. - - - - -

12.3 Customizing I/O

- -

-The output of the function fprint may be customized by changing the -value of the global variable default_print_flags. - - - - - -

12.4 Customizing the memory allocator

- -

-Every memory allocation of CLN is done through the function pointer -malloc_hook. Freeing of this memory is done through the function -pointer free_hook. The default versions of these functions, -provided in the library, call malloc and free and check -the malloc result against NULL. -If you want to provide another memory allocator, you need to define -the variables malloc_hook and free_hook yourself, -like this: - -

-#include <cln/malloc.h>
-namespace cln {
-        void* (*malloc_hook) (size_t size) = ...;
-        void (*free_hook) (void* ptr)      = ...;
-}
-
- -

- - -The cl_malloc_hook function must not return a NULL pointer. - - -

-It is not possible to change the memory allocator at runtime, because -it is already called at program startup by the constructors of some -global variables. - - - - -

Index

- -

-Jump to: -

- - -

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b(alid.)41 b(Example:)353 -2330 y Fp(cl_I)47 b(x)g(=)h(...)o(;)353 2434 y(if)f(\(!\(x)g(>=)g -(0\)\))g(abort\(\);)353 2538 y(cl_I)g(ten_x)f(=)i -(The\(cl_I\)\(expt\(10,x\)\);)41 b(//)48 b(If)f(x)g(>=)g(0,)h(10^x)e -(is)h(an)h(integer.)974 2642 y(//)f(In)g(general,)f(it)h(would)f(be)h -(a)h(rational)d(number.)p eop -%%Page: 10 12 -10 11 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2523 b(10)-30 299 y Fo(4)80 b(F)-13 -b(unctions)53 b(on)g(n)l(um)l(b)t(ers)-30 600 y Fr(Eac)m(h)36 -b(of)g(the)g(n)m(um)m(b)s(er)d(classes)k(declares)f(its)g(mathematical) -g(op)s(erations)g(in)f(the)h(corresp)s(onding)e(include)i(\014le.)-30 -709 y(F)-8 b(or)27 b(example,)g(if)f(y)m(our)g(co)s(de)h(op)s(erates)g -(with)f(ob)5 b(jects)27 b(of)f(t)m(yp)s(e)g Fp(cl_I)p -Fr(,)h(it)f(should)g Fp(#include)i()p -Fr(.)-30 1022 y Fs(4.1)68 b(Constructing)45 b(n)l(um)l(b)t(ers)-30 -1237 y Fr(Here)31 b(is)f(ho)m(w)h(to)g(create)h(n)m(um)m(b)s(er)c(ob)5 -b(jects)31 b(\\from)e(nothing".)-30 1505 y Fn(4.1.1)63 -b(Constructing)41 b(in)m(tegers)-30 1720 y Fp(cl_I)28 -b Fr(ob)5 b(jects)30 b(are)f(most)g(easily)h(constructed)f(from)f(C)g -(in)m(tegers)i(and)f(from)f(strings.)40 b(See)29 b(Section)h(3.4)g -([Con)m(v)m(er-)-30 1830 y(sions],)h(page)g(8.)-30 2098 -y Fn(4.1.2)63 b(Constructing)41 b(rational)g(n)m(um)m(b)s(ers)-30 -2313 y Fp(cl_RA)e Fr(ob)5 b(jects)41 b(can)g(b)s(e)f(constructed)h -(from)e(strings.)71 b(The)40 b(syn)m(tax)h(for)f(rational)i(n)m(um)m(b) -s(ers)c(is)j(describ)s(ed)e(in)-30 2423 y(Section)44 -b(5.1)g([In)m(ternal)g(and)f(prin)m(ted)g(represen)m(tation],)48 -b(page)c(28.)80 b(Another)44 b(standard)e(w)m(a)m(y)i(to)g(pro)s(duce)f -(a)-30 2532 y(rational)32 b(n)m(um)m(b)s(er)c(is)i(through)g -(application)h(of)g(`)p Fp(operator)d(/)p Fr(')j(or)f(`)p -Fp(recip)p Fr(')f(on)i(in)m(tegers.)-30 2801 y Fn(4.1.3)63 -b(Constructing)41 b(\015oating-p)s(oin)m(t)h(n)m(um)m(b)s(ers)-30 -3016 y Fp(cl_F)h Fr(ob)5 b(jects)45 b(with)f(lo)m(w)h(precision)f(are)g -(most)g(easily)h(constructed)g(from)d(C)i(`)p Fp(float)p -Fr(')f(and)h(`)p Fp(double)p Fr('.)81 b(See)-30 3125 -y(Section)31 b(3.4)h([Con)m(v)m(ersions],)f(page)g(8.)-30 -3282 y(T)-8 b(o)34 b(construct)h(a)f Fp(cl_F)e Fr(with)i(high)g -(precision,)h(y)m(ou)f(can)g(use)g(the)g(con)m(v)m(ersion)h(from)d(`)p -Fp(const)d(char)h(*)p Fr(',)k(but)g(y)m(ou)-30 3392 y(ha)m(v)m(e)e(to)f -(sp)s(ecify)f(the)h(desired)f(precision)h(within)f(the)h(string.)41 -b(\(See)32 b(Section)f(5.1)h([In)m(ternal)f(and)f(prin)m(ted)g(repre-) --30 3501 y(sen)m(tation],)i(page)g(28.\))41 b(Example:)353 -3652 y Fp(cl_F)47 b(e)g(=)h("0.27182818284590452353)o(6028)o(7471)o -(352)o(6624)o(9775)o(724)o(7093)o(6999)o(6e+)o(1_40)o(";)-30 -3809 y Fr(will)31 b(set)g(`)p Fp(e)p Fr(')f(to)h(the)g(giv)m(en)g(v)-5 -b(alue,)31 b(with)f(a)h(precision)g(of)f(40)h(decimal)g(digits.)-30 -3966 y(The)e(programmatic)g(w)m(a)m(y)h(to)h(construct)f(a)g -Fp(cl_F)e 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Fl(t)m(yp)s(e)36 -b Fp(minus1)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 -b(x\))450 1945 y Fr(Returns)f Fp(x)h(-)h(1)p Fr(.)-30 -2104 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(*)i(\(const)e -Fl(t)m(yp)s(e)5 b Fp(&,)31 b(const)d Fl(t)m(yp)s(e)5 -b Fp(&\))450 2214 y Fr(Multiplication.)-30 2373 y Fl(t)m(yp)s(e)36 -b Fp(square)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 -b(x\))450 2483 y Fr(Returns)f Fp(x)h(*)h(x)p Fr(.)-30 -2642 y(Eac)m(h)43 b(of)g(the)f(classes)i Fp(cl_N)p Fr(,)h -Fp(cl_R)p Fr(,)f Fp(cl_RA)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p -Fr(,)g Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)d -Fr(de\014nes)h(the)g(follo)m(wing)-30 2751 y(op)s(erations:)-30 -2911 y Fl(t)m(yp)s(e)36 b Fp(operator)28 b(/)i(\(const)e -Fl(t)m(yp)s(e)5 b Fp(&,)31 b(const)d Fl(t)m(yp)s(e)5 -b Fp(&\))450 3020 y Fr(Division.)-30 3179 y Fl(t)m(yp)s(e)36 -b Fp(recip)29 b(\(const)f Fl(t)m(yp)s(e)5 b Fp(&\))450 -3289 y Fr(Returns)29 b(the)i(recipro)s(cal)g(of)g(the)f(argumen)m(t.) --30 3448 y(The)h(class)h Fp(cl_I)d Fr(do)s(esn't)i(de\014ne)g(a)g(`)p -Fp(/)p Fr(')h(op)s(eration)f(b)s(ecause)g(in)g(the)g(C/C)p -Fp(++)g Fr(language)h(this)f(op)s(erator,)h(applied)-30 -3558 y(to)j(in)m(tegral)h(t)m(yp)s(es,)g(denotes)f(the)g(`)p -Fp(floor)p Fr(')e(or)h(`)p Fp(truncate)p Fr(')f(op)s(eration)i(\(whic)m -(h)g(one)f(of)h(these,)h(is)f(implemen)m(ta-)-30 3667 -y(tion)g(dep)s(enden)m(t\).)51 b(\(See)35 b(Section)g(4.6)g([Rounding)f -(functions],)h(page)g(13.\))53 b(Instead,)36 b Fp(cl_I)d -Fr(de\014nes)g(an)h(\\exact)-30 3777 y(quotien)m(t")e(function:)-30 -3936 y Fp(cl_I)d(exquo)g(\(const)g(cl_I&)g(x,)h(const)e(cl_I&)h(y\))450 -4046 y Fr(Chec)m(ks)i(that)g Fp(y)f Fr(divides)g Fp(x)p -Fr(,)g(and)g(returns)f(the)h(quotien)m(t)i Fp(x)p Fr(/)p -Fp(y)p Fr(.)-30 4205 y(The)e(follo)m(wing)i(exp)s(onen)m(tiation)f -(functions)f(are)h(de\014ned:)-30 4364 y Fp(cl_I)e(expt_pos)f(\(const)h -(cl_I&)g(x,)h(const)f(cl_I&)g(y\))-30 4474 y(cl_RA)g(expt_pos)f -(\(const)h(cl_RA&)f(x,)i(const)f(cl_I&)g(y\))450 4583 -y(y)h 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Fr(implemen)m(ts)g(this)h(as)h(follo)m -(ws:)-30 767 y Fp(cl_R)e(abs)h(\(const)e(cl_N)i(x\))450 -877 y Fr(Returns)f(the)i(absolute)g(v)-5 b(alue)31 b(of)f -Fp(x)p Fr(.)-30 1057 y(Eac)m(h)d(of)f(the)h(classes)g -Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)g Fp(cl_I)p -Fr(,)g Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)g Fp(cl_FF)p -Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f Fr(de\014nes)g(the)h(follo)m(wing) --30 1166 y(op)s(eration:)-30 1346 y Fl(t)m(yp)s(e)36 -b Fp(signum)28 b(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 -b(x\))450 1455 y Fr(Returns)f(the)h(sign)g(of)g Fp(x)p -Fr(,)g(in)f(the)i(same)e(n)m(um)m(b)s(er)e(format)j(as)g -Fp(x)p Fr(.)40 b(This)29 b(is)h(de\014ned)f(as)h Fp(x)g(/)g(abs\(x\))e -Fr(if)450 1565 y Fp(x)i Fr(is)g(non-zero,)i(and)d Fp(x)h -Fr(if)h Fp(x)f Fr(is)g(zero.)42 b(If)30 b Fp(x)g Fr(is)g(real,)h(the)g -(v)-5 b(alue)31 b(is)f(either)h(0)g(or)f(1)h(or)f(-1.)-30 -1856 y Fs(4.3)68 b(Elemen)l(tary)47 b(rational)f(functions)-30 -2062 y Fr(Eac)m(h)31 b(of)g(the)f(classes)i Fp(cl_RA)p -Fr(,)d Fp(cl_I)g Fr(de\014nes)g(the)i(follo)m(wing)h(op)s(erations:)-30 -2241 y Fp(cl_I)d(numerator)f(\(const)h Fl(t)m(yp)s(e)5 -b Fp(&)30 b(x\))450 2351 y Fr(Returns)f(the)i(n)m(umerator)e(of)i -Fp(x)p Fr(.)-30 2524 y Fp(cl_I)e(denominator)f(\(const)g -Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 2633 y Fr(Returns)e(the)i -(denominator)f(of)g Fp(x)p Fr(.)-30 2813 y(The)g(n)m(umerator)g(and)g -(denominator)g(of)h(a)g(rational)h(n)m(um)m(b)s(er)c(are)k(normalized)e -(in)g(suc)m(h)h(a)g(w)m(a)m(y)g(that)h(they)f(ha)m(v)m(e)-30 -2922 y(no)f(factor)i(in)e(common)e(and)i(the)h(denominator)e(is)i(p)s -(ositiv)m(e.)-30 3213 y Fs(4.4)68 b(Elemen)l(tary)47 -b(complex)e(functions)-30 3419 y Fr(The)30 b(class)h -Fp(cl_N)e Fr(de\014nes)h(the)g(follo)m(wing)i(op)s(eration:)-30 -3599 y Fp(cl_N)d(complex)g(\(const)f(cl_R&)h(a,)h(const)f(cl_R&)g(b\)) -450 3708 y Fr(Returns)35 b(the)h(complex)g(n)m(um)m(b)s(er)e -Fp(a+bi)p Fr(,)i(that)h(is,)g(the)f(complex)g(n)m(um)m(b)s(er)e(with)h -(real)i(part)f Fp(a)g Fr(and)450 3818 y(imaginary)30 -b(part)g Fp(b)p Fr(.)-30 3997 y(Eac)m(h)h(of)g(the)f(classes)i -Fp(cl_N)p Fr(,)d Fp(cl_R)g Fr(de\014nes)h(the)g(follo)m(wing)i(op)s -(erations:)-30 4177 y Fp(cl_R)d(realpart)f(\(const)h -Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 4286 y Fr(Returns)f(the)i(real)g -(part)f(of)h Fp(x)p Fr(.)-30 4459 y Fp(cl_R)e(imagpart)f(\(const)h -Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 4569 y Fr(Returns)f(the)i -(imaginary)f(part)g(of)h Fp(x)p Fr(.)-30 4742 y Fl(t)m(yp)s(e)36 -b Fp(conjugate)27 b(\(const)i Fl(t)m(yp)s(e)5 b Fp(&)31 -b(x\))450 4851 y Fr(Returns)e(the)i(complex)f(conjugate)i(of)e -Fp(x)p Fr(.)-30 5031 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(relations)150 -5179 y Fp(x)f(=)g(complex\(realpart\(x\),)25 b(imagpart\(x\)\))150 -5320 y(conjugate\(x\))i(=)j(complex\(realpart\(x\),)25 -b(-imagpart\(x\)\))p eop -%%Page: 13 15 -13 14 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2523 b(13)-30 299 y Fs(4.5)68 -b(Comparisons)-30 489 y Fr(Eac)m(h)27 b(of)f(the)h(classes)g -Fp(cl_N)p Fr(,)f Fp(cl_R)p Fr(,)g Fp(cl_RA)p Fr(,)g Fp(cl_I)p -Fr(,)g Fp(cl_F)p Fr(,)g Fp(cl_SF)p Fr(,)g Fp(cl_FF)p -Fr(,)g Fp(cl_DF)p Fr(,)g Fp(cl_LF)f Fr(de\014nes)g(the)h(follo)m(wing) --30 598 y(op)s(erations:)-30 752 y Fp(bool)j(operator)f(==)i(\(const)f -Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5 -b Fp(&\))-30 862 y(bool)29 b(operator)f(!=)i(\(const)f -Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5 -b Fp(&\))450 971 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p -Fp(++)p Fr(.)-30 1125 y Fp(uint32)g(equal_hashcode)d(\(const)j -Fl(t)m(yp)s(e)5 b Fp(&\))450 1235 y Fr(Returns)35 b(a)h(32-bit)h(hash)e -(co)s(de)h(that)g(is)g(the)g(same)f(for)h(an)m(y)g(t)m(w)m(o)h(n)m(um)m -(b)s(ers)c(whic)m(h)i(are)i(the)f(same)450 1345 y(according)24 -b(to)h Fp(==)p Fr(.)38 b(This)22 b(hash)h(co)s(de)h(dep)s(ends)e(on)h -(the)h(n)m(um)m(b)s(er's)d(v)-5 b(alue,)26 b(not)d(its)h(t)m(yp)s(e)g -(or)g(precision.)-30 1499 y Fp(cl_boolean)k(zerop)h(\(const)f -Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 1608 y Fr(Compare)e(against)j -(zero:)41 b Fp(x)30 b(==)g(0)-30 1762 y Fr(Eac)m(h)43 -b(of)g(the)f(classes)i Fp(cl_R)p Fr(,)h Fp(cl_RA)p Fr(,)f -Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f Fp(cl_SF)p Fr(,)g Fp(cl_FF)p -Fr(,)h Fp(cl_DF)p Fr(,)f Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing) --30 1872 y(op)s(erations:)-30 2026 y Fp(cl_signean)28 -b(compare)g(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f -Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 2135 y Fr(Compares)f -Fp(x)h Fr(and)g Fp(y)p Fr(.)40 b(Returns)30 b Fp(+)p -Fr(1)g(if)g Fp(x>y)p Fr(,)g(-1)h(if)g Fp(x=)i(\(const)f -Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5 -b Fp(&\))-30 2618 y(bool)29 b(operator)f(>)i(\(const)f -Fl(t)m(yp)s(e)5 b Fp(&,)30 b(const)f Fl(t)m(yp)s(e)5 -b Fp(&\))450 2727 y Fr(Comparison,)29 b(as)i(in)f(C)g(and)f(C)p -Fp(++)p Fr(.)-30 2881 y Fp(cl_boolean)f(minusp)g(\(const)h -Fl(t)m(yp)s(e)5 b Fp(&)30 b(x\))450 2991 y Fr(Compare)f(against)j -(zero:)41 b Fp(x)30 b(<)g(0)-30 3145 y(cl_boolean)e(plusp)h(\(const)f -Fl(t)m(yp)s(e)5 b Fp(&)31 b(x\))450 3255 y Fr(Compare)e(against)j -(zero:)41 b Fp(x)30 b(>)g(0)-30 3408 y Fl(t)m(yp)s(e)36 -b Fp(max)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f -Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 3518 y Fr(Return)g(the)g(maxim)m -(um)d(of)k Fp(x)f Fr(and)g Fp(y)p Fr(.)-30 3672 y Fl(t)m(yp)s(e)36 -b Fp(min)29 b(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f -Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 3782 y Fr(Return)g(the)g(minim)m -(um)d(of)j Fp(x)g Fr(and)g Fp(y)p Fr(.)-30 3936 y(When)35 -b(a)g(\015oating)h(p)s(oin)m(t)f(n)m(um)m(b)s(er)d(and)i(a)i(rational)g -(n)m(um)m(b)s(er)c(are)k(compared,)f(the)g(\015oat)h(is)f(\014rst)f -(con)m(v)m(erted)i(to)-30 4045 y(a)f(rational)h(n)m(um)m(b)s(er)c -(using)i(the)h(function)f Fp(rational)p Fr(.)51 b(Since)35 -b(a)g(\015oating)g(p)s(oin)m(t)g(n)m(um)m(b)s(er)d(actually)k(represen) -m(ts)-30 4155 y(an)h(in)m(terv)-5 b(al)39 b(of)e(real)h(n)m(um)m(b)s -(ers,)f(the)h(result)f(migh)m(t)g(b)s(e)g(surprising.)60 -b(F)-8 b(or)38 b(example,)h Fp(\(cl_F\)\(cl_R\)"1/3")26 -b(==)-30 4264 y(\(cl_R\)"1/3")h Fr(returns)i(false)i(b)s(ecause)g -(there)f(is)h(no)f(\015oating)h(p)s(oin)m(t)g(n)m(um)m(b)s(er)d(whose)i -(v)-5 b(alue)31 b(is)f(exactly)i Fp(1/3)p Fr(.)-30 4513 -y Fs(4.6)68 b(Rounding)45 b(functions)-30 4703 y Fr(When)40 -b(a)g(real)h(n)m(um)m(b)s(er)d(is)i(to)h(b)s(e)f(con)m(v)m(erted)h(to)g -(an)f(in)m(teger,)45 b(there)40 b(is)g(no)g(\\b)s(est")h(rounding.)69 -b(The)39 b(desired)-30 4813 y(rounding)27 b(function)g(dep)s(ends)f(on) -i(the)g(application.)41 b(The)28 b(Common)d(Lisp)i(and)g(ISO)g(Lisp)g -(standards)g(o\013er)i(four)-30 4923 y(rounding)g(functions:)-30 -5076 y Fp(floor\(x\))96 b Fr(This)30 b(is)g(the)h(largest)g(in)m(teger) -h Fp(<)p Fr(=)p Fp(x)p Fr(.)-30 5230 y Fp(ceiling\(x\))450 -5340 y Fr(This)e(is)g(the)h(smallest)f(in)m(teger)i Fp(>)p -Fr(=)p Fp(x)p Fr(.)p eop -%%Page: 14 16 -14 15 bop -30 -116 a Fr(Chapter)30 b(4:)41 b(F)-8 b(unctions)31 -b(on)f(n)m(um)m(b)s(ers)2523 b(14)-30 299 y Fp(truncate\(x\))450 -408 y Fr(Among)30 b(the)g(in)m(tegers)i(b)s(et)m(w)m(een)f(0)g(and)e -Fp(x)h Fr(\(inclusiv)m(e\))i(the)f(one)g(nearest)f(to)i -Fp(x)p Fr(.)-30 566 y Fp(round\(x\))96 b Fr(The)32 b(in)m(teger)i -(nearest)g(to)f Fp(x)p Fr(.)48 b(If)32 b Fp(x)g Fr(is)h(exactly)i -(halfw)m(a)m(y)e(b)s(et)m(w)m(een)h(t)m(w)m(o)g(in)m(tegers,)h(c)m(ho)s -(ose)e(the)g(ev)m(en)450 676 y(one.)-30 833 y(These)d(functions)g(ha)m -(v)m(e)i(di\013eren)m(t)e(adv)-5 b(an)m(tages:)-30 967 -y Fp(floor)44 b Fr(and)i Fp(ceiling)e Fr(are)i(translation)h(in)m(v)-5 -b(arian)m(t:)73 b Fp(floor\(x+n\))27 b(=)j(floor\(x\))e(+)i(n)46 -b Fr(and)f Fp(ceiling\(x+n\))27 b(=)-30 1076 y(ceiling\(x\))h(+)i(n)g -Fr(for)g(ev)m(ery)h Fp(x)f Fr(and)g(ev)m(ery)h(in)m(teger)h -Fp(n)p Fr(.)-30 1210 y(On)51 b(the)h(other)h(hand,)j -Fp(truncate)50 b Fr(and)h Fp(round)g Fr(are)h(symmetric:)83 -b Fp(truncate\(-x\))27 b(=)j(-truncate\(x\))49 b Fr(and)-30 -1319 y Fp(round\(-x\))28 b(=)i(-round\(x\))p Fr(,)44 -b(and)f(furthermore)e Fp(round)g Fr(is)j(un)m(biased:)65 -b(on)44 b(the)f(\\a)m(v)m(erage",)50 b(it)44 b(rounds)d(do)m(wn)-30 -1429 y(exactly)32 b(as)f(often)g(as)f(it)h(rounds)e(up.)-30 -1563 y(The)h(functions)g(are)h(related)g(lik)m(e)h(this:)150 -1696 y Fp(ceiling\(m/n\))27 b(=)j(floor\(\(m+n-1\)/n\))c(=)k -(floor\(\(m-1\)/n\)+1)f Fr(for)j(rational)j(n)m(um)m(b)s(ers)30 -b Fp(m/n)i Fr(\()p Fp(m)p Fr(,)i Fp(n)f Fr(in)m(te-)150 -1806 y(gers,)e Fp(n>)p Fr(0\),)g(and)150 1939 y Fp(truncate\(x\))c(=)j -(sign\(x\))f(*)h(floor\(abs\(x\)\))-30 2097 y Fr(Eac)m(h)f(of)f(the)g -(classes)h Fp(cl_R)p Fr(,)e Fp(cl_RA)p Fr(,)h Fp(cl_F)p -Fr(,)f Fp(cl_SF)p Fr(,)h Fp(cl_FF)p Fr(,)f Fp(cl_DF)p -Fr(,)g Fp(cl_LF)g Fr(de\014nes)g(the)h(follo)m(wing)h(op)s(erations:) --30 2254 y Fp(cl_I)g(floor1)g(\(const)g Fl(t)m(yp)s(e)5 -b Fp(&)30 b(x\))450 2364 y Fr(Returns)f Fp(floor\(x\))p -Fr(.)-30 2521 y Fp(cl_I)g(ceiling1)f(\(const)h Fl(t)m(yp)s(e)5 -b Fp(&)30 b(x\))450 2631 y Fr(Returns)f Fp(ceiling\(x\))p -Fr(.)-30 2789 y Fp(cl_I)g(truncate1)f(\(const)h Fl(t)m(yp)s(e)5 -b Fp(&)30 b(x\))450 2898 y Fr(Returns)f Fp(truncate\(x\))p -Fr(.)-30 3056 y Fp(cl_I)g(round1)g(\(const)g Fl(t)m(yp)s(e)5 -b Fp(&)30 b(x\))450 3165 y Fr(Returns)f Fp(round\(x\))p -Fr(.)-30 3323 y(Eac)m(h)43 b(of)g(the)f(classes)i Fp(cl_R)p -Fr(,)h Fp(cl_RA)p Fr(,)f Fp(cl_I)p Fr(,)h Fp(cl_F)p Fr(,)f -Fp(cl_SF)p Fr(,)g Fp(cl_FF)p Fr(,)h Fp(cl_DF)p Fr(,)f -Fp(cl_LF)d Fr(de\014nes)h(the)g(follo)m(wing)-30 3432 -y(op)s(erations:)-30 3590 y Fp(cl_I)29 b(floor1)g(\(const)g -Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5 -b Fp(&)30 b(y\))450 3700 y Fr(Returns)f Fp(floor\(x/y\))p -Fr(.)-30 3857 y Fp(cl_I)g(ceiling1)f(\(const)h Fl(t)m(yp)s(e)5 -b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 -3967 y Fr(Returns)f Fp(ceiling\(x/y\))p Fr(.)-30 4124 -y Fp(cl_I)g(truncate1)f(\(const)h Fl(t)m(yp)s(e)5 b Fp(&)30 -b(x,)g(const)f Fl(t)m(yp)s(e)5 b Fp(&)30 b(y\))450 4234 -y Fr(Returns)f Fp(truncate\(x/y\))p Fr(.)-30 4391 y Fp(cl_I)g(round1)g -(\(const)g Fl(t)m(yp)s(e)5 b Fp(&)30 b(x,)g(const)f Fl(t)m(yp)s(e)5 -b Fp(&)30 b(y\))450 4501 y Fr(Returns)f Fp(round\(x/y\))p -Fr(.)-30 4658 y(These)41 b(functions)g(are)h(called)h(`)p -Fp(floor1)p Fr(',)48 b(.)23 b(.)f(.)52 b(here)42 b(instead)f(of)h(`)p -Fp(floor)p Fr(',)49 b(.)22 b(.)h(.)11 b(,)44 b(b)s(ecause)e(on)f(some)g -(systems,)-30 4768 y(system)30 b(dep)s(enden)m(t)f(include)h(\014les)g -(de\014ne)g(`)p Fp(floor)p Fr(')f(and)h(`)p Fp(ceiling)p -Fr(')f(as)i(macros.)-30 4902 y(In)h(man)m(y)g(cases,)j(one)e(needs)g(b) -s(oth)f(the)h(quotien)m(t)i(and)d(the)h(remainder)e(of)j(a)f(division.) -49 b(It)33 b(is)g(more)f(e\016cien)m(t)i(to)-30 5011 -y(compute)h(b)s(oth)h(at)h(the)f(same)g(time)f(than)h(to)h(p)s(erform)d -(t)m(w)m(o)j(divisions,)h(one)e(for)g(quotien)m(t)i(and)d(the)h(next)h -(one)-30 5121 y(for)28 b(the)g(remainder.)38 b(The)27 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Fp(\(gdb\))46 -b(print)h(s)390 4925 y($7)g(=)h({)43 b(=)48 -b({)f(=)h({pointer)d(=)j(0x8055b60,)d(heappointer)f(=)k(0x8055b60,)485 -5029 y(word)f(=)h(134568800}},)c(})390 5132 y(\(gdb\))i(call)h -(s.debug_print\(\))390 5236 y(\(cl_string\))e("")390 -5340 y(\(gdb\))h(define)g(cprint)p eop -%%Page: 51 53 -51 52 bop -30 -116 a Fr(Chapter)30 b(11:)41 b(Using)31 -b(the)f(library)2681 b(51)390 299 y Fp(>call)46 b -(\($1\).debug_print\(\))390 403 y(>end)390 506 y(\(gdb\))g(cprint)g(s) -390 610 y(\(cl_string\))f("")150 745 y Fr(Unfortunately)-8 -b(,)31 b(this)g(feature)f(do)s(es)g(not)h(seem)f(to)h(w)m(ork)f(under)f -(all)i(circumstances.)p eop -%%Page: 52 54 -52 53 bop -30 -116 a Fr(Chapter)30 b(12:)41 b(Customizing)2863 -b(52)-30 299 y Fo(12)80 b(Customizing)-30 656 y Fs(12.1)68 -b(Error)46 b(handling)-30 848 y Fr(When)30 b(a)h(fatal)g(error)f(o)s -(ccurs,)g(an)g(error)g(message)g(is)g(output)g(to)h(the)f(standard)g -(error)g(output)f(stream,)h(and)g(the)-30 958 y(function)c -Fp(cl_abort)e 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Fd(mask_field)28 b(\(\))20 -b Fc(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g -(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.) -g(.)46 b Fb(22)-30 1147 y Fd(max)26 b(\(\))12 b Fc(.)h(.)f(.)h(.)f(.)g -(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.) -f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h -(.)f(.)g(.)38 b Fb(13)-30 1239 y Fd(min)26 b(\(\))12 -b Fc(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.) -f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h -(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fb(13)-30 1331 -y Fd(minus)27 b(\(\))9 b Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h -(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.) -h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b -Fb(33)-30 1423 y Fd(minus1)27 b(\(\))8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)g -(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.) 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-y(7.2)92 b(F)-8 b(unctions)31 b(on)f(mo)s(dular)e(in)m(tegers)16 -b Fe(.)h(.)e(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)46 b Fr(34)-30 2002 y Fs(8)135 b(Sym)l(b)t(olic)45 -b(data)g(t)l(yp)t(es)28 b Fa(.)21 b(.)e(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h -(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.) -73 b Fs(37)269 2139 y Fr(8.1)92 b(Strings)9 b Fe(.)14 -b(.)h(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)38 b Fr(37)269 2249 y(8.2)92 b(Sym)m(b)s(ols)24 -b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)55 b Fr(37)-30 2489 y Fs(9)135 b(Univ)-7 -b(ariate)46 b(p)t(olynomials)40 b Fa(.)19 b(.)h(.)f(.)g(.)h(.)f(.)h(.)f -(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)83 -b Fs(38)269 2626 y Fr(9.1)92 b(Univ)-5 b(ariate)32 b(p)s(olynomial)d -(rings)c Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)54 b Fr(38)269 2736 y(9.2)92 b(F)-8 -b(unctions)31 b(on)f(univ)-5 b(ariate)31 b(p)s(olynomials)d -Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)58 -b Fr(40)269 2845 y(9.3)92 b(Sp)s(ecial)31 b(p)s(olynomials)13 -b Fe(.)h(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)43 b Fr(42)-30 -3085 y Fs(10)135 b(In)l(ternals)20 b Fa(.)g(.)g(.)f(.)g(.)h(.)f(.)h(.)f -(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.) -f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)64 b -Fs(43)269 3222 y Fr(10.1)92 b(Wh)m(y)31 b(C)p Fp(++)f -Fr(?)d Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)57 b Fr(43)269 3332 y(10.2)92 b(Memory)30 b(e\016ciency)25 -b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)53 b Fr(43)269 -3442 y(10.3)92 b(Sp)s(eed)30 b(e\016ciency)e Fe(.)15 -b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)57 b Fr(43)269 -3551 y(10.4)92 b(Garbage)32 b(collection)14 b Fe(.)k(.)d(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)44 b Fr(44)-30 3791 y Fs(11)135 b(Using)46 -b(the)f(library)25 b Fa(.)20 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h -(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.) -h(.)69 b Fs(45)269 3928 y Fr(11.1)92 b(Compiler)30 b(options)21 -b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)50 b -Fr(45)269 4038 y(11.2)92 b(Compatibilit)m(y)31 b(to)g(old)g(CLN)f(v)m -(ersions)17 b Fe(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 -b Fr(45)269 4147 y(11.3)92 b(Include)30 b(\014les)18 -b Fe(.)d(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)47 -b Fr(45)269 4257 y(11.4)92 b(An)30 b(Example)d Fe(.)15 -b(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)57 -b Fr(49)269 4367 y(11.5)92 b(Debugging)32 b(supp)s(ort)23 -b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)54 b Fr(50)-30 -4607 y Fs(12)135 b(Customizing)39 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)h(.)f -(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.) -h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)83 b Fs(52)269 4744 -y Fr(12.1)92 b(Error)30 b(handling)24 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)h -(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f -(.)g(.)g(.)g(.)g(.)g(.)g(.)54 b Fr(52)269 4853 y(12.2)92 -b(Floating-p)s(oin)m(t)33 b(under\015o)m(w)19 b Fe(.)14 -b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)49 b Fr(52)269 4963 y(12.3)92 b(Customizing)30 -b(I/O)25 b Fe(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) -g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54 -b Fr(52)269 5072 y(12.4)92 b(Customizing)30 b(the)h(memory)d(allo)s -(cator)16 b Fe(.)i(.)d(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g -(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 -b Fr(52)-30 5313 y Fs(Index)30 b Fa(.)19 b(.)g(.)h(.)f(.)h(.)f(.)h(.)f -(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.) -h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f -(.)h(.)74 b Fs(53)p eop -%%Trailer -end -userdict /end-hook known{end-hook}if -%%EOF diff --git a/doc/cln_1.html b/doc/cln_1.html deleted file mode 100644 index 2b5c237..0000000 --- a/doc/cln_1.html +++ /dev/null @@ -1,175 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 1. Introduction - - -Go to the first, previous, next, last section, table of contents. -

- - -

1. Introduction

- -

-CLN is a library for computations with all kinds of numbers. -It has a rich set of number classes: - - - -

- -

-The subtypes of the complex numbers among these are exactly the -types of numbers known to the Common Lisp language. Therefore -CLN can be used for Common Lisp implementations, giving -`CLN' another meaning: it becomes an abbreviation of -"Common Lisp Numbers". - - -

-The CLN package implements - - - -

- -

-CLN is a C++ library. Using C++ as an implementation language provides - - - -

- -

-CLN is memory efficient: - - - -

- -

-CLN is speed efficient: - - - -

- -

-CLN aims at being easily integrated into larger software packages: - - - -

- -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_10.html b/doc/cln_10.html deleted file mode 100644 index f66fa70..0000000 --- a/doc/cln_10.html +++ /dev/null @@ -1,163 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 10. Internals - - -Go to the first, previous, next, last section, table of contents. -

- - -

10. Internals

- - - -

10.1 Why C++ ?

-

- - - -

-Using C++ as an implementation language provides - - - -

- -

-With these language features, there is no need for two separate languages, -one for the implementation of the library and one in which the library's users -can program. This means that a prototype implementation of an algorithm -can be integrated into the library immediately after it has been tested and -debugged. No need to rewrite it in a low-level language after having prototyped -in a high-level language. - - - - -

10.2 Memory efficiency

- -

-In order to save memory allocations, CLN implements: - - - -

- - - -

10.3 Speed efficiency

- -

-Speed efficiency is obtained by the combination of the following tricks -and algorithms: - - - -

- - - -

10.4 Garbage collection

-

- - - -

-All the number classes are reference count classes: They only contain a pointer -to an object in the heap. Upon construction, assignment and destruction of -number objects, only the objects' reference count are manipulated. - - -

-Memory occupied by number objects are automatically reclaimed as soon as -their reference count drops to zero. - - -

-For number rings, another strategy is implemented: There is a cache of, -for example, the modular integer rings. A modular integer ring is destroyed -only if its reference count dropped to zero and the cache is about to be -resized. The effect of this strategy is that recently used rings remain -cached, whereas undue memory consumption through cached rings is avoided. - - -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_11.html b/doc/cln_11.html deleted file mode 100644 index 2ff22a4..0000000 --- a/doc/cln_11.html +++ /dev/null @@ -1,476 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 11. Using the library - - -Go to the first, previous, next, last section, table of contents. -

- - -

11. Using the library

- -

-For the following discussion, we will assume that you have installed -the CLN source in $CLN_DIR and built it in $CLN_TARGETDIR. -For example, for me it's CLN_DIR="$HOME/cln" and -CLN_TARGETDIR="$HOME/cln/linuxelf". You might define these as -environment variables, or directly substitute the appropriate values. - - - - -

11.1 Compiler options

-

- - - -

-Until you have installed CLN in a public place, the following options are -needed: - - -

-When you compile CLN application code, add the flags - -

-   -I$CLN_DIR/include -I$CLN_TARGETDIR/include
-
- -

-to the C++ compiler's command line (make variable CFLAGS or CXXFLAGS). -When you link CLN application code to form an executable, add the flags - -

-   $CLN_TARGETDIR/src/libcln.a
-
- -

-to the C/C++ compiler's command line (make variable LIBS). - - -

-If you did a make install, the include files are installed in a -public directory (normally /usr/local/include), hence you don't -need special flags for compiling. The library has been installed to a -public directory as well (normally /usr/local/lib), hence when -linking a CLN application it is sufficient to give the flag -lcln. - - - - -

11.2 Compatibility to old CLN versions

-

- - - - -

-As of CLN version 1.1 all non-macro identifiers were hidden in namespace -cln in order to avoid potential name clashes with other C++ -libraries. If you have an old application, you will have to manually -port it to the new scheme. The following principles will help during -the transition: - -

- -

-When developing other libraries, please keep in mind not to import the -namespace cln in one of your public header files by saying -using namespace cln;. This would propagate to other applications -and can cause name clashes there. - - - - -

11.3 Include files

-

- - - - -

-Here is a summary of the include files and their contents. - - -

- -
<cln/object.h> -
-General definitions, reference counting, garbage collection. -
<cln/number.h> -
-The class cl_number. -
<cln/complex.h> -
-Functions for class cl_N, the complex numbers. -
<cln/real.h> -
-Functions for class cl_R, the real numbers. -
<cln/float.h> -
-Functions for class cl_F, the floats. -
<cln/sfloat.h> -
-Functions for class cl_SF, the short-floats. -
<cln/ffloat.h> -
-Functions for class cl_FF, the single-floats. -
<cln/dfloat.h> -
-Functions for class cl_DF, the double-floats. -
<cln/lfloat.h> -
-Functions for class cl_LF, the long-floats. -
<cln/rational.h> -
-Functions for class cl_RA, the rational numbers. -
<cln/integer.h> -
-Functions for class cl_I, the integers. -
<cln/io.h> -
-Input/Output. -
<cln/complex_io.h> -
-Input/Output for class cl_N, the complex numbers. -
<cln/real_io.h> -
-Input/Output for class cl_R, the real numbers. -
<cln/float_io.h> -
-Input/Output for class cl_F, the floats. -
<cln/sfloat_io.h> -
-Input/Output for class cl_SF, the short-floats. -
<cln/ffloat_io.h> -
-Input/Output for class cl_FF, the single-floats. -
<cln/dfloat_io.h> -
-Input/Output for class cl_DF, the double-floats. -
<cln/lfloat_io.h> -
-Input/Output for class cl_LF, the long-floats. -
<cln/rational_io.h> -
-Input/Output for class cl_RA, the rational numbers. -
<cln/integer_io.h> -
-Input/Output for class cl_I, the integers. -
<cln/input.h> -
-Flags for customizing input operations. -
<cln/output.h> -
-Flags for customizing output operations. -
<cln/malloc.h> -
-malloc_hook, free_hook. -
<cln/abort.h> -
-cl_abort. -
<cln/condition.h> -
-Conditions/exceptions. -
<cln/string.h> -
-Strings. -
<cln/symbol.h> -
-Symbols. -
<cln/proplist.h> -
-Property lists. -
<cln/ring.h> -
-General rings. -
<cln/null_ring.h> -
-The null ring. -
<cln/complex_ring.h> -
-The ring of complex numbers. -
<cln/real_ring.h> -
-The ring of real numbers. -
<cln/rational_ring.h> -
-The ring of rational numbers. -
<cln/integer_ring.h> -
-The ring of integers. -
<cln/numtheory.h> -
-Number threory functions. -
<cln/modinteger.h> -
-Modular integers. -
<cln/V.h> -
-Vectors. -
<cln/GV.h> -
-General vectors. -
<cln/GV_number.h> -
-General vectors over cl_number. -
<cln/GV_complex.h> -
-General vectors over cl_N. -
<cln/GV_real.h> -
-General vectors over cl_R. -
<cln/GV_rational.h> -
-General vectors over cl_RA. -
<cln/GV_integer.h> -
-General vectors over cl_I. -
<cln/GV_modinteger.h> -
-General vectors of modular integers. -
<cln/SV.h> -
-Simple vectors. -
<cln/SV_number.h> -
-Simple vectors over cl_number. -
<cln/SV_complex.h> -
-Simple vectors over cl_N. -
<cln/SV_real.h> -
-Simple vectors over cl_R. -
<cln/SV_rational.h> -
-Simple vectors over cl_RA. -
<cln/SV_integer.h> -
-Simple vectors over cl_I. -
<cln/SV_ringelt.h> -
-Simple vectors of general ring elements. -
<cln/univpoly.h> -
-Univariate polynomials. -
<cln/univpoly_integer.h> -
-Univariate polynomials over the integers. -
<cln/univpoly_rational.h> -
-Univariate polynomials over the rational numbers. -
<cln/univpoly_real.h> -
-Univariate polynomials over the real numbers. -
<cln/univpoly_complex.h> -
-Univariate polynomials over the complex numbers. -
<cln/univpoly_modint.h> -
-Univariate polynomials over modular integer rings. -
<cln/timing.h> -
-Timing facilities. -
<cln/cln.h> -
-Includes all of the above. -
- - - -

11.4 An Example

- -

-A function which computes the nth Fibonacci number can be written as follows. - - - - -

-#include <cln/integer.h>
-#include <cln/real.h>
-using namespace cln;
-
-// Returns F_n, computed as the nearest integer to
-// ((1+sqrt(5))/2)^n/sqrt(5). Assume n>=0.
-const cl_I fibonacci (int n)
-{
-        // Need a precision of ((1+sqrt(5))/2)^-n.
-        cl_float_format_t prec = cl_float_format((int)(0.208987641*n+5));
-        cl_R sqrt5 = sqrt(cl_float(5,prec));
-        cl_R phi = (1+sqrt5)/2;
-        return round1( expt(phi,n)/sqrt5 );
-}
-
- -

-Let's explain what is going on in detail. - - -

-The include file <cln/integer.h> is necessary because the type -cl_I is used in the function, and the include file <cln/real.h> -is needed for the type cl_R and the floating point number functions. -The order of the include files does not matter. In order not to write out -cln::foo we can safely import the whole namespace cln. - - -

-Then comes the function declaration. The argument is an int, the -result an integer. The return type is defined as `const cl_I', not -simply `cl_I', because that allows the compiler to detect typos like -`fibonacci(n) = 100'. It would be possible to declare the return -type as const cl_R (real number) or even const cl_N (complex -number). We use the most specialized possible return type because functions -which call `fibonacci' will be able to profit from the compiler's type -analysis: Adding two integers is slightly more efficient than adding the -same objects declared as complex numbers, because it needs less type -dispatch. Also, when linking to CLN as a non-shared library, this minimizes -the size of the resulting executable program. - - -

-The result will be computed as expt(phi,n)/sqrt(5), rounded to the nearest -integer. In order to get a correct result, the absolute error should be less -than 1/2, i.e. the relative error should be less than sqrt(5)/(2*expt(phi,n)). -To this end, the first line computes a floating point precision for sqrt(5) -and phi. - - -

-Then sqrt(5) is computed by first converting the integer 5 to a floating point -number and than taking the square root. The converse, first taking the square -root of 5, and then converting to the desired precision, would not work in -CLN: The square root would be computed to a default precision (normally -single-float precision), and the following conversion could not help about -the lacking accuracy. This is because CLN is not a symbolic computer algebra -system and does not represent sqrt(5) in a non-numeric way. - - -

-The type cl_R for sqrt5 and, in the following line, phi is the only -possible choice. You cannot write cl_F because the C++ compiler can -only infer that cl_float(5,prec) is a real number. You cannot write -cl_N because a `round1' does not exist for general complex -numbers. - - -

-When the function returns, all the local variables in the function are -automatically reclaimed (garbage collected). Only the result survives and -gets passed to the caller. - - -

-The file fibonacci.cc in the subdirectory examples -contains this implementation together with an even faster algorithm. - - - - -

11.5 Debugging support

-

- - - -

-When debugging a CLN application with GNU gdb, two facilities are -available from the library: - - - -

- -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_12.html b/doc/cln_12.html deleted file mode 100644 index 5c86ada..0000000 --- a/doc/cln_12.html +++ /dev/null @@ -1,108 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 12. Customizing - - -Go to the first, previous, next, last section, table of contents. -

- - -

12. Customizing

-

- - - - - -

12.1 Error handling

- -

-When a fatal error occurs, an error message is output to the standard error -output stream, and the function cl_abort is called. The default -version of this function (provided in the library) terminates the application. -To catch such a fatal error, you need to define the function cl_abort -yourself, with the prototype - -

-#include <cln/abort.h>
-void cl_abort (void);
-
- -

- -This function must not return control to its caller. - - - - -

12.2 Floating-point underflow

-

- - - -

-Floating point underflow denotes the situation when a floating-point number -is to be created which is so close to 0 that its exponent is too -low to be represented internally. By default, this causes a fatal error. -If you set the global variable - -

-cl_boolean cl_inhibit_floating_point_underflow
-
- -

-to cl_true, the error will be inhibited, and a floating-point zero -will be generated instead. The default value of -cl_inhibit_floating_point_underflow is cl_false. - - - - -

12.3 Customizing I/O

- -

-The output of the function fprint may be customized by changing the -value of the global variable default_print_flags. - - - - - -

12.4 Customizing the memory allocator

- -

-Every memory allocation of CLN is done through the function pointer -malloc_hook. Freeing of this memory is done through the function -pointer free_hook. The default versions of these functions, -provided in the library, call malloc and free and check -the malloc result against NULL. -If you want to provide another memory allocator, you need to define -the variables malloc_hook and free_hook yourself, -like this: - -

-#include <cln/malloc.h>
-namespace cln {
-        void* (*malloc_hook) (size_t size) = ...;
-        void (*free_hook) (void* ptr)      = ...;
-}
-
- -

- - -The cl_malloc_hook function must not return a NULL pointer. - - -

-It is not possible to change the memory allocator at runtime, because -it is already called at program startup by the constructors of some -global variables. - - -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_13.html b/doc/cln_13.html deleted file mode 100644 index d485564..0000000 --- a/doc/cln_13.html +++ /dev/null @@ -1,22 +0,0 @@ - - - - -CLN, a Class Library for Numbers - Index - - -Go to the first, previous, next, last section, table of contents. -

- - -

Index

- -

-Jump to: -

- - -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_2.html b/doc/cln_2.html deleted file mode 100644 index 16bfa17..0000000 --- a/doc/cln_2.html +++ /dev/null @@ -1,279 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 2. Installation - - -Go to the first, previous, next, last section, table of contents. -

- - -

2. Installation

- -

-This section describes how to install the CLN package on your system. - - - - -

2.1 Prerequisites

- - - -

2.1.1 C++ compiler

- -

-To build CLN, you need a C++ compiler. -Actually, you need GNU g++ 2.90 or newer, the EGCS compilers will -do. -I recommend GNU g++ 2.95 or newer. - - -

-The following C++ features are used: -classes, member functions, overloading of functions and operators, -constructors and destructors, inline, const, multiple inheritance, -templates and namespaces. - - -

-The following C++ features are not used: -new, delete, virtual inheritance, exceptions. - - -

-CLN relies on semi-automatic ordering of initializations -of static and global variables, a feature which I could -implement for GNU g++ only. - - - - -

2.1.2 Make utility

-

- - - -

-To build CLN, you also need to have GNU make installed. - - - - -

2.1.3 Sed utility

-

- - - -

-To build CLN on HP-UX, you also need to have GNU sed installed. -This is because the libtool script, which creates the CLN library, relies -on sed, and the vendor's sed utility on these systems is too -limited. - - - - -

2.2 Building the library

- -

-As with any autoconfiguring GNU software, installation is as easy as this: - - - -

-$ ./configure
-$ make
-$ make check
-
- -

-If on your system, `make' is not GNU make, you have to use -`gmake' instead of `make' above. - - -

-The configure command checks out some features of your system and -C++ compiler and builds the Makefiles. The make command -builds the library. This step may take 4 hours on an average workstation. -The make check runs some test to check that no important subroutine -has been miscompiled. - - -

-The configure command accepts options. To get a summary of them, try - - - -

-$ ./configure --help
-
- -

-Some of the options are explained in detail in the `INSTALL.generic' file. - - -

-You can specify the C compiler, the C++ compiler and their options through -the following environment variables when running configure: - - -

- -
CC -
-Specifies the C compiler. - -
CFLAGS -
-Flags to be given to the C compiler when compiling programs (not when linking). - -
CXX -
-Specifies the C++ compiler. - -
CXXFLAGS -
-Flags to be given to the C++ compiler when compiling programs (not when linking). -
- -

-Examples: - - - -

-$ CC="gcc" CFLAGS="-O" CXX="g++" CXXFLAGS="-O" ./configure
-$ CC="gcc -V egcs-2.91.60" CFLAGS="-O -g" \
-  CXX="g++ -V egcs-2.91.60" CXXFLAGS="-O -g" ./configure
-$ CC="gcc -V 2.95.2" CFLAGS="-O2 -fno-exceptions" \
-  CXX="g++ -V 2.95.2" CFLAGS="-O2 -fno-exceptions" ./configure
-
- -

-Note that for these environment variables to take effect, you have to set -them (assuming a Bourne-compatible shell) on the same line as the -configure command. If you made the settings in earlier shell -commands, you have to export the environment variables before -calling configure. In a csh shell, you have to use the -`setenv' command for setting each of the environment variables. - - -

-Currently CLN works only with the GNU g++ compiler, and only in -optimizing mode. So you should specify at least -O in the CXXFLAGS, -or no CXXFLAGS at all. (If CXXFLAGS is not set, CLN will use -O.) - - -

-If you use g++ version 2.8.x or egcs-2.91.x (a.k.a. egcs-1.1) or -gcc-2.95.x, I recommend adding `-fno-exceptions' to the CXXFLAGS. -This will likely generate better code. - - -

-If you use g++ version egcs-2.91.x (egcs-1.1) or gcc-2.95.x on Sparc, -add either `-O', `-O1' or `-O2 -fno-schedule-insns' to the -CXXFLAGS. With full `-O2', g++ miscompiles the division routines. -Also, if you have g++ version egcs-1.1.1 or older on Sparc, you must -specify `--disable-shared' because g++ would miscompile parts of -the library. - - -

-By default, both a shared and a static library are built. You can build -CLN as a static (or shared) library only, by calling configure with -the option `--disable-shared' (or `--disable-static'). While -shared libraries are usually more convenient to use, they may not work -on all architectures. Try disabling them if you run into linker -problems. Also, they are generally somewhat slower than static -libraries so runtime-critical applications should be linked statically. - - - - -

2.2.1 Using the GNU MP Library

-

- - - -

-Starting with version 1.1, CLN may be configured to make use of a -preinstalled gmp library. Please make sure that you have at -least gmp version 3.0 installed since earlier versions are -unsupported and likely not to work. Enabling this feature by calling -configure with the option `--with-gmp' is known to be quite -a boost for CLN's performance. - - -

-If you have installed the gmp library and its header file in -some place where your compiler cannot find it by default, you must help -configure by setting CPPFLAGS and LDFLAGS. Here is -an example: - - - -

-$ CC="gcc" CFLAGS="-O2" CXX="g++" CXXFLAGS="-O2 -fno-exceptions" \
-  CPPFLAGS="-I/opt/gmp/include" LDFLAGS="-L/opt/gmp/lib" ./configure --with-gmp
-
- - - -

2.3 Installing the library

-

- - - -

-As with any autoconfiguring GNU software, installation is as easy as this: - - - -

-$ make install
-
- -

-The `make install' command installs the library and the include files -into public places (`/usr/local/lib/' and `/usr/local/include/', -if you haven't specified a --prefix option to configure). -This step may require superuser privileges. - - -

-If you have already built the library and wish to install it, but didn't -specify --prefix=... at configure time, just re-run -configure, giving it the same options as the first time, plus -the --prefix=... option. - - - - -

2.4 Cleaning up

- -

-You can remove system-dependent files generated by make through - - - -

-$ make clean
-
- -

-You can remove all files generated by make, thus reverting to a -virgin distribution of CLN, through - - - -

-$ make distclean
-
- -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_3.html b/doc/cln_3.html deleted file mode 100644 index 154d8d6..0000000 --- a/doc/cln_3.html +++ /dev/null @@ -1,401 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 3. Ordinary number types - - -Go to the first, previous, next, last section, table of contents. -

- - -

3. Ordinary number types

- -

-CLN implements the following class hierarchy: - - - -

-                        Number
-                      cl_number
-                    <cln/number.h>
-                          |
-                          |
-                 Real or complex number
-                        cl_N
-                    <cln/complex.h>
-                          |
-                          |
-                     Real number
-                        cl_R
-                     <cln/real.h>
-                          |
-      +-------------------+-------------------+
-      |                                       |
-Rational number                     Floating-point number
-    cl_RA                                   cl_F
-<cln/rational.h>                         <cln/float.h>
-      |                                       |
-      |                +--------------+--------------+--------------+
-   Integer             |              |              |              |
-    cl_I          Short-Float    Single-Float   Double-Float    Long-Float
-<cln/integer.h>      cl_SF          cl_FF          cl_DF          cl_LF
-                 <cln/sfloat.h> <cln/ffloat.h> <cln/dfloat.h> <cln/lfloat.h>
-
- -

- - -The base class cl_number is an abstract base class. -It is not useful to declare a variable of this type except if you want -to completely disable compile-time type checking and use run-time type -checking instead. - - -

- - - -The class cl_N comprises real and complex numbers. There is -no special class for complex numbers since complex numbers with imaginary -part 0 are automatically converted to real numbers. - - -

- -The class cl_R comprises real numbers of different kinds. It is an -abstract class. - - -

- - - -The class cl_RA comprises exact real numbers: rational numbers, including -integers. There is no special class for non-integral rational numbers -since rational numbers with denominator 1 are automatically converted -to integers. - - -

- -The class cl_F implements floating-point approximations to real numbers. -It is an abstract class. - - - - -

3.1 Exact numbers

-

- - - -

-Some numbers are represented as exact numbers: there is no loss of information -when such a number is converted from its mathematical value to its internal -representation. On exact numbers, the elementary operations (+, --, *, /, comparisons, ...) compute the completely -correct result. - - -

-In CLN, the exact numbers are: - - - -

- -

-Rational numbers are always normalized to the form -numerator/denominator where the numerator and denominator -are coprime integers and the denominator is positive. If the resulting -denominator is 1, the rational number is converted to an integer. - - -

-Small integers (typically in the range -2^30...2^30-1, -for 32-bit machines) are especially efficient, because they consume no heap -allocation. Otherwise the distinction between these immediate integers -(called "fixnums") and heap allocated integers (called "bignums") -is completely transparent. - - - - -

3.2 Floating-point numbers

-

- - - -

-Not all real numbers can be represented exactly. (There is an easy mathematical -proof for this: Only a countable set of numbers can be stored exactly in -a computer, even if one assumes that it has unlimited storage. But there -are uncountably many real numbers.) So some approximation is needed. -CLN implements ordinary floating-point numbers, with mantissa and exponent. - - -

- -The elementary operations (+, -, *, /, ...) -only return approximate results. For example, the value of the expression -(cl_F) 0.3 + (cl_F) 0.4 prints as `0.70000005', not as -`0.7'. Rounding errors like this one are inevitable when computing -with floating-point numbers. - - -

-Nevertheless, CLN rounds the floating-point results of the operations +, --, *, /, sqrt according to the "round-to-even" -rule: It first computes the exact mathematical result and then returns the -floating-point number which is nearest to this. If two floating-point numbers -are equally distant from the ideal result, the one with a 0 in its least -significant mantissa bit is chosen. - - -

-Similarly, testing floating point numbers for equality `x == y' -is gambling with random errors. Better check for `abs(x - y) < epsilon' -for some well-chosen epsilon. - - -

-Floating point numbers come in four flavors: - - - -

- -

-Of course, computations with long floats are more expensive than those -with smaller floating-point formats. - - -

-CLN does not implement features like NaNs, denormalized numbers and -gradual underflow. If the exponent range of some floating-point type -is too limited for your application, choose another floating-point type -with larger exponent range. - - -

- -As a user of CLN, you can forget about the differences between the -four floating-point types and just declare all your floating-point -variables as being of type cl_F. This has the advantage that -when you change the precision of some computation (say, from cl_DF -to cl_LF), you don't have to change the code, only the precision -of the initial values. Also, many transcendental functions have been -declared as returning a cl_F when the argument is a cl_F, -but such declarations are missing for the types cl_SF, cl_FF, -cl_DF, cl_LF. (Such declarations would be wrong if -the floating point contagion rule happened to change in the future.) - - - - -

3.3 Complex numbers

-

- - - -

-Complex numbers, as implemented by the class cl_N, have a real -part and an imaginary part, both real numbers. A complex number whose -imaginary part is the exact number 0 is automatically converted -to a real number. - - -

-Complex numbers can arise from real numbers alone, for example -through application of sqrt or transcendental functions. - - - - -

3.4 Conversions

-

- - - -

-Conversions from any class to any its superclasses ("base classes" in -C++ terminology) is done automatically. - - -

-Conversions from the C built-in types `long' and `unsigned long' -are provided for the classes cl_I, cl_RA, cl_R, -cl_N and cl_number. - - -

-Conversions from the C built-in types `int' and `unsigned int' -are provided for the classes cl_I, cl_RA, cl_R, -cl_N and cl_number. However, these conversions emphasize -efficiency. Their range is therefore limited: - - - -

- -

-In a declaration like `cl_I x = 10;' the C++ compiler is able to -do the conversion of 10 from `int' to `cl_I' at compile time -already. On the other hand, code like `cl_I x = 1000000000;' is -in error. -So, if you want to be sure that an `int' whose magnitude is not guaranteed -to be < 2^29 is correctly converted to a `cl_I', first convert it to a -`long'. Similarly, if a large `unsigned int' is to be converted to a -`cl_I', first convert it to an `unsigned long'. - - -

-Conversions from the C built-in type `float' are provided for the classes -cl_FF, cl_F, cl_R, cl_N and cl_number. - - -

-Conversions from the C built-in type `double' are provided for the classes -cl_DF, cl_F, cl_R, cl_N and cl_number. - - -

-Conversions from `const char *' are provided for the classes -cl_I, cl_RA, -cl_SF, cl_FF, cl_DF, cl_LF, cl_F, -cl_R, cl_N. -The easiest way to specify a value which is outside of the range of the -C++ built-in types is therefore to specify it as a string, like this: - - -

-   cl_I order_of_rubiks_cube_group = "43252003274489856000";
-
- -

-Note that this conversion is done at runtime, not at compile-time. - - -

-Conversions from cl_I to the C built-in types `int', -`unsigned int', `long', `unsigned long' are provided through -the functions - - -

- -
int cl_I_to_int (const cl_I& x) -
- -
unsigned int cl_I_to_uint (const cl_I& x) -
- -
long cl_I_to_long (const cl_I& x) -
- -
unsigned long cl_I_to_ulong (const cl_I& x) -
- -Returns x as element of the C type ctype. If x is not -representable in the range of ctype, a runtime error occurs. -
- -

-Conversions from the classes cl_I, cl_RA, -cl_SF, cl_FF, cl_DF, cl_LF, cl_F and -cl_R -to the C built-in types `float' and `double' are provided through -the functions - - -

- -
float float_approx (const type& x) -
- -
double double_approx (const type& x) -
- -Returns an approximation of x of C type ctype. -If abs(x) is too close to 0 (underflow), 0 is returned. -If abs(x) is too large (overflow), an IEEE infinity is returned. -
- -

-Conversions from any class to any of its subclasses ("derived classes" in -C++ terminology) are not provided. Instead, you can assert and check -that a value belongs to a certain subclass, and return it as element of that -class, using the `As' and `The' macros. - -As(type)(value) checks that value belongs to -type and returns it as such. - -The(type)(value) assumes that value belongs to -type and returns it as such. It is your responsibility to ensure -that this assumption is valid. -Example: - - - -

-   cl_I x = ...;
-   if (!(x >= 0)) abort();
-   cl_I ten_x = The(cl_I)(expt(10,x)); // If x >= 0, 10^x is an integer.
-                // In general, it would be a rational number.
-
- -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_4.html b/doc/cln_4.html deleted file mode 100644 index 43ff034..0000000 --- a/doc/cln_4.html +++ /dev/null @@ -1,2018 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 4. Functions on numbers - - -Go to the first, previous, next, last section, table of contents. -

- - -

4. Functions on numbers

- -

-Each of the number classes declares its mathematical operations in the -corresponding include file. For example, if your code operates with -objects of type cl_I, it should #include <cln/integer.h>. - - - - -

4.1 Constructing numbers

- -

-Here is how to create number objects "from nothing". - - - - -

4.1.1 Constructing integers

- -

-cl_I objects are most easily constructed from C integers and from -strings. See section 3.4 Conversions. - - - - -

4.1.2 Constructing rational numbers

- -

-cl_RA objects can be constructed from strings. The syntax -for rational numbers is described in section 5.1 Internal and printed representation. -Another standard way to produce a rational number is through application -of `operator /' or `recip' on integers. - - - - -

4.1.3 Constructing floating-point numbers

- -

-cl_F objects with low precision are most easily constructed from -C `float' and `double'. See section 3.4 Conversions. - - -

-To construct a cl_F with high precision, you can use the conversion -from `const char *', but you have to specify the desired precision -within the string. (See section 5.1 Internal and printed representation.) -Example: - -

-   cl_F e = "0.271828182845904523536028747135266249775724709369996e+1_40";
-
- -

-will set `e' to the given value, with a precision of 40 decimal digits. - - -

-The programmatic way to construct a cl_F with high precision is -through the cl_float conversion function, see -section 4.11.1 Conversion to floating-point numbers. For example, to compute -e to 40 decimal places, first construct 1.0 to 40 decimal places -and then apply the exponential function: - -

-   cl_float_format_t precision = cl_float_format(40);
-   cl_F e = exp(cl_float(1,precision));
-
- - - -

4.1.4 Constructing complex numbers

- -

-Non-real cl_N objects are normally constructed through the function - -

-   cl_N complex (const cl_R& realpart, const cl_R& imagpart)
-
- -

-See section 4.4 Elementary complex functions. - - - - -

4.2 Elementary functions

- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
type operator + (const type&, const type&) -
- -Addition. - -
type operator - (const type&, const type&) -
- -Subtraction. - -
type operator - (const type&) -
-Returns the negative of the argument. - -
type plus1 (const type& x) -
- -Returns x + 1. - -
type minus1 (const type& x) -
- -Returns x - 1. - -
type operator * (const type&, const type&) -
- -Multiplication. - -
type square (const type& x) -
- -Returns x * x. -
- -

-Each of the classes cl_N, cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
type operator / (const type&, const type&) -
- -Division. - -
type recip (const type&) -
- -Returns the reciprocal of the argument. -
- -

-The class cl_I doesn't define a `/' operation because -in the C/C++ language this operator, applied to integral types, -denotes the `floor' or `truncate' operation (which one of these, -is implementation dependent). (See section 4.6 Rounding functions.) -Instead, cl_I defines an "exact quotient" function: - - -

- -
cl_I exquo (const cl_I& x, const cl_I& y) -
- -Checks that y divides x, and returns the quotient x/y. -
- -

-The following exponentiation functions are defined: - - -

- -
cl_I expt_pos (const cl_I& x, const cl_I& y) -
- -
cl_RA expt_pos (const cl_RA& x, const cl_I& y) -
-y must be > 0. Returns x^y. - -
cl_RA expt (const cl_RA& x, const cl_I& y) -
- -
cl_R expt (const cl_R& x, const cl_I& y) -
-
cl_N expt (const cl_N& x, const cl_I& y) -
-Returns x^y. -
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operation: - - -

- -
type abs (const type& x) -
- -Returns the absolute value of x. -This is x if x >= 0, and -x if x <= 0. -
- -

-The class cl_N implements this as follows: - - -

- -
cl_R abs (const cl_N x) -
-Returns the absolute value of x. -
- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operation: - - -

- -
type signum (const type& x) -
- -Returns the sign of x, in the same number format as x. -This is defined as x / abs(x) if x is non-zero, and -x if x is zero. If x is real, the value is either -0 or 1 or -1. -
- - - -

4.3 Elementary rational functions

- -

-Each of the classes cl_RA, cl_I defines the following operations: - - -

- -
cl_I numerator (const type& x) -
- -Returns the numerator of x. - -
cl_I denominator (const type& x) -
- -Returns the denominator of x. -
- -

-The numerator and denominator of a rational number are normalized in such -a way that they have no factor in common and the denominator is positive. - - - - -

4.4 Elementary complex functions

- -

-The class cl_N defines the following operation: - - -

- -
cl_N complex (const cl_R& a, const cl_R& b) -
- -Returns the complex number a+bi, that is, the complex number with -real part a and imaginary part b. -
- -

-Each of the classes cl_N, cl_R defines the following operations: - - -

- -
cl_R realpart (const type& x) -
- -Returns the real part of x. - -
cl_R imagpart (const type& x) -
- -Returns the imaginary part of x. - -
type conjugate (const type& x) -
- -Returns the complex conjugate of x. -
- -

-We have the relations - - - -

- - - -

4.5 Comparisons

-

- - - -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
bool operator == (const type&, const type&) -
- -
bool operator != (const type&, const type&) -
- -Comparison, as in C and C++. - -
uint32 equal_hashcode (const type&) -
- -Returns a 32-bit hash code that is the same for any two numbers which are -the same according to ==. This hash code depends on the number's value, -not its type or precision. - -
cl_boolean zerop (const type& x) -
- -Compare against zero: x == 0 -
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
cl_signean compare (const type& x, const type& y) -
- -Compares x and y. Returns +1 if x>y, --1 if x<y, 0 if x=y. - -
bool operator <= (const type&, const type&) -
- -
bool operator < (const type&, const type&) -
- -
bool operator >= (const type&, const type&) -
- -
bool operator > (const type&, const type&) -
- -Comparison, as in C and C++. - -
cl_boolean minusp (const type& x) -
- -Compare against zero: x < 0 - -
cl_boolean plusp (const type& x) -
- -Compare against zero: x > 0 - -
type max (const type& x, const type& y) -
- -Return the maximum of x and y. - -
type min (const type& x, const type& y) -
- -Return the minimum of x and y. -
- -

-When a floating point number and a rational number are compared, the float -is first converted to a rational number using the function rational. -Since a floating point number actually represents an interval of real numbers, -the result might be surprising. -For example, (cl_F)(cl_R)"1/3" == (cl_R)"1/3" returns false because -there is no floating point number whose value is exactly 1/3. - - - - -

4.6 Rounding functions

-

- - - -

-When a real number is to be converted to an integer, there is no "best" -rounding. The desired rounding function depends on the application. -The Common Lisp and ISO Lisp standards offer four rounding functions: - - -

- -
floor(x) -
-This is the largest integer <=x. - -
ceiling(x) -
-This is the smallest integer >=x. - -
truncate(x) -
-Among the integers between 0 and x (inclusive) the one nearest to x. - -
round(x) -
-The integer nearest to x. If x is exactly halfway between two -integers, choose the even one. -
- -

-These functions have different advantages: - - -

-floor and ceiling are translation invariant: -floor(x+n) = floor(x) + n and ceiling(x+n) = ceiling(x) + n -for every x and every integer n. - - -

-On the other hand, truncate and round are symmetric: -truncate(-x) = -truncate(x) and round(-x) = -round(x), -and furthermore round is unbiased: on the "average", it rounds -down exactly as often as it rounds up. - - -

-The functions are related like this: - - - -

- -

-Each of the classes cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
cl_I floor1 (const type& x) -
- -Returns floor(x). -
cl_I ceiling1 (const type& x) -
- -Returns ceiling(x). -
cl_I truncate1 (const type& x) -
- -Returns truncate(x). -
cl_I round1 (const type& x) -
- -Returns round(x). -
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
cl_I floor1 (const type& x, const type& y) -
-Returns floor(x/y). -
cl_I ceiling1 (const type& x, const type& y) -
-Returns ceiling(x/y). -
cl_I truncate1 (const type& x, const type& y) -
-Returns truncate(x/y). -
cl_I round1 (const type& x, const type& y) -
-Returns round(x/y). -
- -

-These functions are called `floor1', ... here instead of -`floor', ..., because on some systems, system dependent include -files define `floor' and `ceiling' as macros. - - -

-In many cases, one needs both the quotient and the remainder of a division. -It is more efficient to compute both at the same time than to perform -two divisions, one for quotient and the next one for the remainder. -The following functions therefore return a structure containing both -the quotient and the remainder. The suffix `2' indicates the number -of "return values". The remainder is defined as follows: - - - -

- -

-and similarly for the other three operations. - - -

-Each of the classes cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
struct type_div_t { cl_I quotient; type remainder; }; -
-
type_div_t floor2 (const type& x) -
-
type_div_t ceiling2 (const type& x) -
-
type_div_t truncate2 (const type& x) -
-
type_div_t round2 (const type& x) -
-
- -

-Each of the classes cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
struct type_div_t { cl_I quotient; type remainder; }; -
-
type_div_t floor2 (const type& x, const type& y) -
- -
type_div_t ceiling2 (const type& x, const type& y) -
- -
type_div_t truncate2 (const type& x, const type& y) -
- -
type_div_t round2 (const type& x, const type& y) -
- -
- -

-Sometimes, one wants the quotient as a floating-point number (of the -same format as the argument, if the argument is a float) instead of as -an integer. The prefix `f' indicates this. - - -

-Each of the classes -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - -

- -
type ffloor (const type& x) -
- -
type fceiling (const type& x) -
- -
type ftruncate (const type& x) -
- -
type fround (const type& x) -
- -
- -

-and similarly for class cl_R, but with return type cl_F. - - -

-The class cl_R defines the following operations: - - -

- -
cl_F ffloor (const type& x, const type& y) -
-
cl_F fceiling (const type& x, const type& y) -
-
cl_F ftruncate (const type& x, const type& y) -
-
cl_F fround (const type& x, const type& y) -
-
- -

-These functions also exist in versions which return both the quotient -and the remainder. The suffix `2' indicates this. - - -

-Each of the classes -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations: - - - - - - - -

- -
struct type_fdiv_t { type quotient; type remainder; }; -
-
type_fdiv_t ffloor2 (const type& x) -
- -
type_fdiv_t fceiling2 (const type& x) -
- -
type_fdiv_t ftruncate2 (const type& x) -
- -
type_fdiv_t fround2 (const type& x) -
- -
-

-and similarly for class cl_R, but with quotient type cl_F. - - - -

-The class cl_R defines the following operations: - - -

- -
struct type_fdiv_t { cl_F quotient; cl_R remainder; }; -
-
type_fdiv_t ffloor2 (const type& x, const type& y) -
-
type_fdiv_t fceiling2 (const type& x, const type& y) -
-
type_fdiv_t ftruncate2 (const type& x, const type& y) -
-
type_fdiv_t fround2 (const type& x, const type& y) -
-
- -

-Other applications need only the remainder of a division. -The remainder of `floor' and `ffloor' is called `mod' -(abbreviation of "modulo"). The remainder `truncate' and -`ftruncate' is called `rem' (abbreviation of "remainder"). - - - -

- -

-If x and y are both >= 0, mod(x,y) = rem(x,y) >= 0. -In general, mod(x,y) has the sign of y or is zero, -and rem(x,y) has the sign of x or is zero. - - -

-The classes cl_R, cl_I define the following operations: - - -

- -
type mod (const type& x, const type& y) -
- -
type rem (const type& x, const type& y) -
- -
- - - -

4.7 Roots

- -

-Each of the classes cl_R, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operation: - - -

- -
type sqrt (const type& x) -
- -x must be >= 0. This function returns the square root of x, -normalized to be >= 0. If x is the square of a rational number, -sqrt(x) will be a rational number, else it will return a -floating-point approximation. -
- -

-The classes cl_RA, cl_I define the following operation: - - -

- -
cl_boolean sqrtp (const type& x, type* root) -
- -This tests whether x is a perfect square. If so, it returns true -and the exact square root in *root, else it returns false. -
- -

-Furthermore, for integers, similarly: - - -

- -
cl_boolean isqrt (const type& x, type* root) -
- -x should be >= 0. This function sets *root to -floor(sqrt(x)) and returns the same value as sqrtp: -the boolean value (expt(*root,2) == x). -
- -

-For nth roots, the classes cl_RA, cl_I -define the following operation: - - -

- -
cl_boolean rootp (const type& x, const cl_I& n, type* root) -
- -x must be >= 0. n must be > 0. -This tests whether x is an nth power of a rational number. -If so, it returns true and the exact root in *root, else it returns -false. -
- -

-The only square root function which accepts negative numbers is the one -for class cl_N: - - -

- -
cl_N sqrt (const cl_N& z) -
- -Returns the square root of z, as defined by the formula -sqrt(z) = exp(log(z)/2). Conversion to a floating-point type -or to a complex number are done if necessary. The range of the result is the -right half plane realpart(sqrt(z)) >= 0 -including the positive imaginary axis and 0, but excluding -the negative imaginary axis. -The result is an exact number only if z is an exact number. -
- - - -

4.8 Transcendental functions

-

- - - -

-The transcendental functions return an exact result if the argument -is exact and the result is exact as well. Otherwise they must return -inexact numbers even if the argument is exact. -For example, cos(0) = 1 returns the rational number 1. - - - - -

4.8.1 Exponential and logarithmic functions

- -
- -
cl_R exp (const cl_R& x) -
- -
cl_N exp (const cl_N& x) -
-Returns the exponential function of x. This is e^x where -e is the base of the natural logarithms. The range of the result -is the entire complex plane excluding 0. - -
cl_R ln (const cl_R& x) -
- -x must be > 0. Returns the (natural) logarithm of x. - -
cl_N log (const cl_N& x) -
- -Returns the (natural) logarithm of x. If x is real and positive, -this is ln(x). In general, log(x) = log(abs(x)) + i*phase(x). -The range of the result is the strip in the complex plane --pi < imagpart(log(x)) <= pi. - -
cl_R phase (const cl_N& x) -
- -Returns the angle part of x in its polar representation as a -complex number. That is, phase(x) = atan(realpart(x),imagpart(x)). -This is also the imaginary part of log(x). -The range of the result is the interval -pi < phase(x) <= pi. -The result will be an exact number only if zerop(x) or -if x is real and positive. - -
cl_R log (const cl_R& a, const cl_R& b) -
-a and b must be > 0. Returns the logarithm of a with -respect to base b. log(a,b) = ln(a)/ln(b). -The result can be exact only if a = 1 or if a and b -are both rational. - -
cl_N log (const cl_N& a, const cl_N& b) -
-Returns the logarithm of a with respect to base b. -log(a,b) = log(a)/log(b). - -
cl_N expt (const cl_N& x, const cl_N& y) -
- -Exponentiation: Returns x^y = exp(y*log(x)). -
- -

-The constant e = exp(1) = 2.71828... is returned by the following functions: - - -

- -
cl_F exp1 (cl_float_format_t f) -
- -Returns e as a float of format f. - -
cl_F exp1 (const cl_F& y) -
-Returns e in the float format of y. - -
cl_F exp1 (void) -
-Returns e as a float of format default_float_format. -
- - - -

4.8.2 Trigonometric functions

- -
- -
cl_R sin (const cl_R& x) -
- -Returns sin(x). The range of the result is the interval --1 <= sin(x) <= 1. - -
cl_N sin (const cl_N& z) -
-Returns sin(z). The range of the result is the entire complex plane. - -
cl_R cos (const cl_R& x) -
- -Returns cos(x). The range of the result is the interval --1 <= cos(x) <= 1. - -
cl_N cos (const cl_N& x) -
-Returns cos(z). The range of the result is the entire complex plane. - -
struct cos_sin_t { cl_R cos; cl_R sin; }; -
- -
cos_sin_t cos_sin (const cl_R& x) -
-Returns both sin(x) and cos(x). This is more efficient than - -computing them separately. The relation cos^2 + sin^2 = 1 will -hold only approximately. - -
cl_R tan (const cl_R& x) -
- -
cl_N tan (const cl_N& x) -
-Returns tan(x) = sin(x)/cos(x). - -
cl_N cis (const cl_R& x) -
- -
cl_N cis (const cl_N& x) -
-Returns exp(i*x). The name `cis' means "cos + i sin", because -e^(i*x) = cos(x) + i*sin(x). - - - -
cl_N asin (const cl_N& z) -
-Returns arcsin(z). This is defined as -arcsin(z) = log(iz+sqrt(1-z^2))/i and satisfies -arcsin(-z) = -arcsin(z). -The range of the result is the strip in the complex domain --pi/2 <= realpart(arcsin(z)) <= pi/2, excluding the numbers -with realpart = -pi/2 and imagpart < 0 and the numbers -with realpart = pi/2 and imagpart > 0. - -
cl_N acos (const cl_N& z) -
- -Returns arccos(z). This is defined as -arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i -and satisfies arccos(-z) = pi - arccos(z). -The range of the result is the strip in the complex domain -0 <= realpart(arcsin(z)) <= pi, excluding the numbers -with realpart = 0 and imagpart < 0 and the numbers -with realpart = pi and imagpart > 0. - - - -
cl_R atan (const cl_R& x, const cl_R& y) -
-Returns the angle of the polar representation of the complex number -x+iy. This is atan(y/x) if x>0. The range of -the result is the interval -pi < atan(x,y) <= pi. The result will -be an exact number only if x > 0 and y is the exact 0. -WARNING: In Common Lisp, this function is called as (atan y x), -with reversed order of arguments. - -
cl_R atan (const cl_R& x) -
-Returns arctan(x). This is the same as atan(1,x). The range -of the result is the interval -pi/2 < atan(x) < pi/2. The result -will be an exact number only if x is the exact 0. - -
cl_N atan (const cl_N& z) -
-Returns arctan(z). This is defined as -arctan(z) = (log(1+iz)-log(1-iz)) / 2i and satisfies -arctan(-z) = -arctan(z). The range of the result is -the strip in the complex domain --pi/2 <= realpart(arctan(z)) <= pi/2, excluding the numbers -with realpart = -pi/2 and imagpart >= 0 and the numbers -with realpart = pi/2 and imagpart <= 0. - -
- -

- - -Archimedes' constant pi = 3.14... is returned by the following functions: - - -

- -
cl_F pi (cl_float_format_t f) -
- -Returns pi as a float of format f. - -
cl_F pi (const cl_F& y) -
-Returns pi in the float format of y. - -
cl_F pi (void) -
-Returns pi as a float of format default_float_format. -
- - - -

4.8.3 Hyperbolic functions

- -
- -
cl_R sinh (const cl_R& x) -
- -Returns sinh(x). - -
cl_N sinh (const cl_N& z) -
-Returns sinh(z). The range of the result is the entire complex plane. - -
cl_R cosh (const cl_R& x) -
- -Returns cosh(x). The range of the result is the interval -cosh(x) >= 1. - -
cl_N cosh (const cl_N& z) -
-Returns cosh(z). The range of the result is the entire complex plane. - -
struct cosh_sinh_t { cl_R cosh; cl_R sinh; }; -
- -
cosh_sinh_t cosh_sinh (const cl_R& x) -
- -Returns both sinh(x) and cosh(x). This is more efficient than -computing them separately. The relation cosh^2 - sinh^2 = 1 will -hold only approximately. - -
cl_R tanh (const cl_R& x) -
- -
cl_N tanh (const cl_N& x) -
-Returns tanh(x) = sinh(x)/cosh(x). - -
cl_N asinh (const cl_N& z) -
- -Returns arsinh(z). This is defined as -arsinh(z) = log(z+sqrt(1+z^2)) and satisfies -arsinh(-z) = -arsinh(z). -The range of the result is the strip in the complex domain --pi/2 <= imagpart(arsinh(z)) <= pi/2, excluding the numbers -with imagpart = -pi/2 and realpart > 0 and the numbers -with imagpart = pi/2 and realpart < 0. - -
cl_N acosh (const cl_N& z) -
- -Returns arcosh(z). This is defined as -arcosh(z) = 2*log(sqrt((z+1)/2)+sqrt((z-1)/2)). -The range of the result is the half-strip in the complex domain --pi < imagpart(arcosh(z)) <= pi, realpart(arcosh(z)) >= 0, -excluding the numbers with realpart = 0 and -pi < imagpart < 0. - -
cl_N atanh (const cl_N& z) -
- -Returns artanh(z). This is defined as -artanh(z) = (log(1+z)-log(1-z)) / 2 and satisfies -artanh(-z) = -artanh(z). The range of the result is -the strip in the complex domain --pi/2 <= imagpart(artanh(z)) <= pi/2, excluding the numbers -with imagpart = -pi/2 and realpart <= 0 and the numbers -with imagpart = pi/2 and realpart >= 0. -
- - - -

4.8.4 Euler gamma

-

- - - -

-Euler's constant C = 0.577... is returned by the following functions: - - -

- -
cl_F eulerconst (cl_float_format_t f) -
- -Returns Euler's constant as a float of format f. - -
cl_F eulerconst (const cl_F& y) -
-Returns Euler's constant in the float format of y. - -
cl_F eulerconst (void) -
-Returns Euler's constant as a float of format default_float_format. -
- -

-Catalan's constant G = 0.915... is returned by the following functions: - - - -

- -
cl_F catalanconst (cl_float_format_t f) -
- -Returns Catalan's constant as a float of format f. - -
cl_F catalanconst (const cl_F& y) -
-Returns Catalan's constant in the float format of y. - -
cl_F catalanconst (void) -
-Returns Catalan's constant as a float of format default_float_format. -
- - - -

4.8.5 Riemann zeta

-

- - - -

-Riemann's zeta function at an integral point s>1 is returned by the -following functions: - - -

- -
cl_F zeta (int s, cl_float_format_t f) -
- -Returns Riemann's zeta function at s as a float of format f. - -
cl_F zeta (int s, const cl_F& y) -
-Returns Riemann's zeta function at s in the float format of y. - -
cl_F zeta (int s) -
-Returns Riemann's zeta function at s as a float of format -default_float_format. -
- - - -

4.9 Functions on integers

- - - -

4.9.1 Logical functions

- -

-Integers, when viewed as in two's complement notation, can be thought as -infinite bit strings where the bits' values eventually are constant. -For example, - -

-    17 = ......00010001
-    -6 = ......11111010
-
- -

-The logical operations view integers as such bit strings and operate -on each of the bit positions in parallel. - - -

- -
cl_I lognot (const cl_I& x) -
- -
cl_I operator ~ (const cl_I& x) -
- -Logical not, like ~x in C. This is the same as -1-x. - -
cl_I logand (const cl_I& x, const cl_I& y) -
- -
cl_I operator & (const cl_I& x, const cl_I& y) -
- -Logical and, like x & y in C. - -
cl_I logior (const cl_I& x, const cl_I& y) -
- -
cl_I operator | (const cl_I& x, const cl_I& y) -
- -Logical (inclusive) or, like x | y in C. - -
cl_I logxor (const cl_I& x, const cl_I& y) -
- -
cl_I operator ^ (const cl_I& x, const cl_I& y) -
- -Exclusive or, like x ^ y in C. - -
cl_I logeqv (const cl_I& x, const cl_I& y) -
- -Bitwise equivalence, like ~(x ^ y) in C. - -
cl_I lognand (const cl_I& x, const cl_I& y) -
- -Bitwise not and, like ~(x & y) in C. - -
cl_I lognor (const cl_I& x, const cl_I& y) -
- -Bitwise not or, like ~(x | y) in C. - -
cl_I logandc1 (const cl_I& x, const cl_I& y) -
- -Logical and, complementing the first argument, like ~x & y in C. - -
cl_I logandc2 (const cl_I& x, const cl_I& y) -
- -Logical and, complementing the second argument, like x & ~y in C. - -
cl_I logorc1 (const cl_I& x, const cl_I& y) -
- -Logical or, complementing the first argument, like ~x | y in C. - -
cl_I logorc2 (const cl_I& x, const cl_I& y) -
- -Logical or, complementing the second argument, like x | ~y in C. -
- -

-These operations are all available though the function -

- -
cl_I boole (cl_boole op, const cl_I& x, const cl_I& y) -
- -
-

-where op must have one of the 16 values (each one stands for a function -which combines two bits into one bit): boole_clr, boole_set, -boole_1, boole_2, boole_c1, boole_c2, -boole_and, boole_ior, boole_xor, boole_eqv, -boole_nand, boole_nor, boole_andc1, boole_andc2, -boole_orc1, boole_orc2. - - - - - - - - - - - - - - - - - -

-Other functions that view integers as bit strings: - - -

- -
cl_boolean logtest (const cl_I& x, const cl_I& y) -
- -Returns true if some bit is set in both x and y, i.e. if -logand(x,y) != 0. - -
cl_boolean logbitp (const cl_I& n, const cl_I& x) -
- -Returns true if the nth bit (from the right) of x is set. -Bit 0 is the least significant bit. - -
uintL logcount (const cl_I& x) -
- -Returns the number of one bits in x, if x >= 0, or -the number of zero bits in x, if x < 0. -
- -

-The following functions operate on intervals of bits in integers. -The type - -

-struct cl_byte { uintL size; uintL position; };
-
- -

- -represents the bit interval containing the bits -position...position+size-1 of an integer. -The constructor cl_byte(size,position) constructs a cl_byte. - - -

- -
cl_I ldb (const cl_I& n, const cl_byte& b) -
- -extracts the bits of n described by the bit interval b -and returns them as a nonnegative integer with b.size bits. - -
cl_boolean ldb_test (const cl_I& n, const cl_byte& b) -
- -Returns true if some bit described by the bit interval b is set in -n. - -
cl_I dpb (const cl_I& newbyte, const cl_I& n, const cl_byte& b) -
- -Returns n, with the bits described by the bit interval b -replaced by newbyte. Only the lowest b.size bits of -newbyte are relevant. -
- -

-The functions ldb and dpb implicitly shift. The following -functions are their counterparts without shifting: - - -

- -
cl_I mask_field (const cl_I& n, const cl_byte& b) -
- -returns an integer with the bits described by the bit interval b -copied from the corresponding bits in n, the other bits zero. - -
cl_I deposit_field (const cl_I& newbyte, const cl_I& n, const cl_byte& b) -
- -returns an integer where the bits described by the bit interval b -come from newbyte and the other bits come from n. -
- -

-The following relations hold: - - - -

- -

-The following operations on integers as bit strings are efficient shortcuts -for common arithmetic operations: - - -

- -
cl_boolean oddp (const cl_I& x) -
- -Returns true if the least significant bit of x is 1. Equivalent to -mod(x,2) != 0. - -
cl_boolean evenp (const cl_I& x) -
- -Returns true if the least significant bit of x is 0. Equivalent to -mod(x,2) == 0. - -
cl_I operator << (const cl_I& x, const cl_I& n) -
- -Shifts x by n bits to the left. n should be >=0. -Equivalent to x * expt(2,n). - -
cl_I operator >> (const cl_I& x, const cl_I& n) -
- -Shifts x by n bits to the right. n should be >=0. -Bits shifted out to the right are thrown away. -Equivalent to floor(x / expt(2,n)). - -
cl_I ash (const cl_I& x, const cl_I& y) -
- -Shifts x by y bits to the left (if y>=0) or -by -y bits to the right (if y<=0). In other words, this -returns floor(x * expt(2,y)). - -
uintL integer_length (const cl_I& x) -
- -Returns the number of bits (excluding the sign bit) needed to represent x -in two's complement notation. This is the smallest n >= 0 such that --2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that -2^(n-1) <= x < 2^n. - -
uintL ord2 (const cl_I& x) -
- -x must be non-zero. This function returns the number of 0 bits at the -right of x in two's complement notation. This is the largest n >= 0 -such that 2^n divides x. - -
uintL power2p (const cl_I& x) -
- -x must be > 0. This function checks whether x is a power of 2. -If x = 2^(n-1), it returns n. Else it returns 0. -(See also the function logp.) -
- - - -

4.9.2 Number theoretic functions

- -
- -
uint32 gcd (uint32 a, uint32 b) -
- -
cl_I gcd (const cl_I& a, const cl_I& b) -
-This function returns the greatest common divisor of a and b, -normalized to be >= 0. - -
cl_I xgcd (const cl_I& a, const cl_I& b, cl_I* u, cl_I* v) -
- -This function ("extended gcd") returns the greatest common divisor g of -a and b and at the same time the representation of g -as an integral linear combination of a and b: -u and v with u*a+v*b = g, g >= 0. -u and v will be normalized to be of smallest possible absolute -value, in the following sense: If a and b are non-zero, and -abs(a) != abs(b), u and v will satisfy the inequalities -abs(u) <= abs(b)/(2*g), abs(v) <= abs(a)/(2*g). - -
cl_I lcm (const cl_I& a, const cl_I& b) -
- -This function returns the least common multiple of a and b, -normalized to be >= 0. - -
cl_boolean logp (const cl_I& a, const cl_I& b, cl_RA* l) -
- -
cl_boolean logp (const cl_RA& a, const cl_RA& b, cl_RA* l) -
-a must be > 0. b must be >0 and != 1. If log(a,b) is -rational number, this function returns true and sets *l = log(a,b), else -it returns false. -
- - - -

4.9.3 Combinatorial functions

- -
- -
cl_I factorial (uintL n) -
- -n must be a small integer >= 0. This function returns the factorial -n! = 1*2*...*n. - -
cl_I doublefactorial (uintL n) -
- -n must be a small integer >= 0. This function returns the -doublefactorial n!! = 1*3*...*n or -n!! = 2*4*...*n, respectively. - -
cl_I binomial (uintL n, uintL k) -
- -n and k must be small integers >= 0. This function returns the -binomial coefficient -(n choose k) = n! / k! (n-k)! -for 0 <= k <= n, 0 else. -
- - - -

4.10 Functions on floating-point numbers

- -

-Recall that a floating-point number consists of a sign s, an -exponent e and a mantissa m. The value of the number is -(-1)^s * 2^e * m. - - -

-Each of the classes -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines the following operations. - - -

- -
type scale_float (const type& x, sintL delta) -
- -
type scale_float (const type& x, const cl_I& delta) -
-Returns x*2^delta. This is more efficient than an explicit multiplication -because it copies x and modifies the exponent. -
- -

-The following functions provide an abstract interface to the underlying -representation of floating-point numbers. - - -

- -
sintL float_exponent (const type& x) -
- -Returns the exponent e of x. -For x = 0.0, this is 0. For x non-zero, this is the unique -integer with 2^(e-1) <= abs(x) < 2^e. - -
sintL float_radix (const type& x) -
- -Returns the base of the floating-point representation. This is always 2. - -
type float_sign (const type& x) -
- -Returns the sign s of x as a float. The value is 1 for -x >= 0, -1 for x < 0. - -
uintL float_digits (const type& x) -
- -Returns the number of mantissa bits in the floating-point representation -of x, including the hidden bit. The value only depends on the type -of x, not on its value. - -
uintL float_precision (const type& x) -
- -Returns the number of significant mantissa bits in the floating-point -representation of x. Since denormalized numbers are not supported, -this is the same as float_digits(x) if x is non-zero, and -0 if x = 0. -
- -

-The complete internal representation of a float is encoded in the type - - - - - -decoded_float (or decoded_sfloat, decoded_ffloat, -decoded_dfloat, decoded_lfloat, respectively), defined by - -

-struct decoded_typefloat {
-        type mantissa; cl_I exponent; type sign;
-};
-
- -

-and returned by the function - - -

- -
decoded_typefloat decode_float (const type& x) -
- -For x non-zero, this returns (-1)^s, e, m with -x = (-1)^s * 2^e * m and 0.5 <= m < 1.0. For x = 0, -it returns (-1)^s=1, e=0, m=0. -e is the same as returned by the function float_exponent. -
- -

-A complete decoding in terms of integers is provided as type - - -

-struct cl_idecoded_float {
-        cl_I mantissa; cl_I exponent; cl_I sign;
-};
-
- -

-by the following function: - - -

- -
cl_idecoded_float integer_decode_float (const type& x) -
- -For x non-zero, this returns (-1)^s, e, m with -x = (-1)^s * 2^e * m and m an integer with float_digits(x) -bits. For x = 0, it returns (-1)^s=1, e=0, m=0. -WARNING: The exponent e is not the same as the one returned by -the functions decode_float and float_exponent. -
- -

-Some other function, implemented only for class cl_F: - - -

- -
cl_F float_sign (const cl_F& x, const cl_F& y) -
- -This returns a floating point number whose precision and absolute value -is that of y and whose sign is that of x. If x is -zero, it is treated as positive. Same for y. -
- - - -

4.11 Conversion functions

-

- - - - - -

4.11.1 Conversion to floating-point numbers

- -

-The type cl_float_format_t describes a floating-point format. - - - -

- -
cl_float_format_t cl_float_format (uintL n) -
- -Returns the smallest float format which guarantees at least n -decimal digits in the mantissa (after the decimal point). - -
cl_float_format_t cl_float_format (const cl_F& x) -
-Returns the floating point format of x. - -
cl_float_format_t default_float_format -
- -Global variable: the default float format used when converting rational numbers -to floats. -
- -

-To convert a real number to a float, each of the types -cl_R, cl_F, cl_I, cl_RA, -int, unsigned int, float, double -defines the following operations: - - -

- -
cl_F cl_float (const type&x, cl_float_format_t f) -
- -Returns x as a float of format f. -
cl_F cl_float (const type&x, const cl_F& y) -
-Returns x in the float format of y. -
cl_F cl_float (const type&x) -
-Returns x as a float of format default_float_format if -it is an exact number, or x itself if it is already a float. -
- -

-Of course, converting a number to a float can lose precision. - - -

-Every floating-point format has some characteristic numbers: - - -

- -
cl_F most_positive_float (cl_float_format_t f) -
- -Returns the largest (most positive) floating point number in float format f. - -
cl_F most_negative_float (cl_float_format_t f) -
- -Returns the smallest (most negative) floating point number in float format f. - -
cl_F least_positive_float (cl_float_format_t f) -
- -Returns the least positive floating point number (i.e. > 0 but closest to 0) -in float format f. - -
cl_F least_negative_float (cl_float_format_t f) -
- -Returns the least negative floating point number (i.e. < 0 but closest to 0) -in float format f. - -
cl_F float_epsilon (cl_float_format_t f) -
- -Returns the smallest floating point number e > 0 such that 1+e != 1. - -
cl_F float_negative_epsilon (cl_float_format_t f) -
- -Returns the smallest floating point number e > 0 such that 1-e != 1. -
- - - -

4.11.2 Conversion to rational numbers

- -

-Each of the classes cl_R, cl_RA, cl_F -defines the following operation: - - -

- -
cl_RA rational (const type& x) -
- -Returns the value of x as an exact number. If x is already -an exact number, this is x. If x is a floating-point number, -the value is a rational number whose denominator is a power of 2. -
- -

-In order to convert back, say, (cl_F)(cl_R)"1/3" to 1/3, there is -the function - - -

- -
cl_RA rationalize (const cl_R& x) -
- -If x is a floating-point number, it actually represents an interval -of real numbers, and this function returns the rational number with -smallest denominator (and smallest numerator, in magnitude) -which lies in this interval. -If x is already an exact number, this function returns x. -
- -

-If x is any float, one has - - - -

- - - -

4.12 Random number generators

- -

-A random generator is a machine which produces (pseudo-)random numbers. -The include file <cln/random.h> defines a class random_state -which contains the state of a random generator. If you make a copy -of the random number generator, the original one and the copy will produce -the same sequence of random numbers. - - -

-The following functions return (pseudo-)random numbers in different formats. -Calling one of these modifies the state of the random number generator in -a complicated but deterministic way. - - -

-The global variable - - - -

-random_state default_random_state
-
- -

-contains a default random number generator. It is used when the functions -below are called without random_state argument. - - -

- -
uint32 random32 (random_state& randomstate) -
-
uint32 random32 () -
- -Returns a random unsigned 32-bit number. All bits are equally random. - -
cl_I random_I (random_state& randomstate, const cl_I& n) -
-
cl_I random_I (const cl_I& n) -
- -n must be an integer > 0. This function returns a random integer x -in the range 0 <= x < n. - -
cl_F random_F (random_state& randomstate, const cl_F& n) -
-
cl_F random_F (const cl_F& n) -
- -n must be a float > 0. This function returns a random floating-point -number of the same format as n in the range 0 <= x < n. - -
cl_R random_R (random_state& randomstate, const cl_R& n) -
-
cl_R random_R (const cl_R& n) -
- -Behaves like random_I if n is an integer and like random_F -if n is a float. -
- - - -

4.13 Obfuscating operators

-

- - - -

-The modifying C/C++ operators +=, -=, *=, /=, -&=, |=, ^=, <<=, >>= -are not available by default because their -use tends to make programs unreadable. It is trivial to get away without -them. However, if you feel that you absolutely need these operators -to get happy, then add - -

-#define WANT_OBFUSCATING_OPERATORS
-
- -

- -to the beginning of your source files, before the inclusion of any CLN -include files. This flag will enable the following operators: - - -

-For the classes cl_N, cl_R, cl_RA, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF: - - -

- -
type& operator += (type&, const type&) -
- -
type& operator -= (type&, const type&) -
- -
type& operator *= (type&, const type&) -
- -
type& operator /= (type&, const type&) -
- -
- -

-For the class cl_I: - - -

- -
type& operator += (type&, const type&) -
-
type& operator -= (type&, const type&) -
-
type& operator *= (type&, const type&) -
-
type& operator &= (type&, const type&) -
- -
type& operator |= (type&, const type&) -
- -
type& operator ^= (type&, const type&) -
- -
type& operator <<= (type&, const type&) -
- -
type& operator >>= (type&, const type&) -
- -
- -

-For the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF: - - -

- -
type& operator ++ (type& x) -
- -The prefix operator ++x. - -
void operator ++ (type& x, int) -
-The postfix operator x++. - -
type& operator -- (type& x) -
- -The prefix operator --x. - -
void operator -- (type& x, int) -
-The postfix operator x--. -
- -

-Note that by using these obfuscating operators, you wouldn't gain efficiency: -In CLN `x += y;' is exactly the same as `x = x+y;', not more -efficient. - - -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_5.html b/doc/cln_5.html deleted file mode 100644 index 42ea42d..0000000 --- a/doc/cln_5.html +++ /dev/null @@ -1,424 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 5. Input/Output - - -Go to the first, previous, next, last section, table of contents. -

- - -

5. Input/Output

-

- - - - - -

5.1 Internal and printed representation

-

- - - -

-All computations deal with the internal representations of the numbers. - - -

-Every number has an external representation as a sequence of ASCII characters. -Several external representations may denote the same number, for example, -"20.0" and "20.000". - - -

-Converting an internal to an external representation is called "printing", - -converting an external to an internal representation is called "reading". - -In CLN, it is always true that conversion of an internal to an external -representation and then back to an internal representation will yield the -same internal representation. Symbolically: read(print(x)) == x. -This is called "print-read consistency". - - -

-Different types of numbers have different external representations (case -is insignificant): - - -

- -
Integers -
-External representation: sign{digit}+. The reader also accepts the -Common Lisp syntaxes sign{digit}+. with a trailing dot -for decimal integers -and the #nR, #b, #o, #x prefixes. - -
Rational numbers -
-External representation: sign{digit}+/{digit}+. -The #nR, #b, #o, #x prefixes are allowed -here as well. - -
Floating-point numbers -
-External representation: sign{digit}*exponent or -sign{digit}*.{digit}*exponent or -sign{digit}*.{digit}+. A precision specifier -of the form _prec may be appended. There must be at least -one digit in the non-exponent part. The exponent has the syntax -expmarker expsign {digit}+. -The exponent marker is - - -
    -
  • - -`s' for short-floats, -
  • - -`f' for single-floats, -
  • - -`d' for double-floats, -
  • - -`L' for long-floats, -
- -or `e', which denotes a default float format. The precision specifying -suffix has the syntax _prec where prec denotes the number of -valid mantissa digits (in decimal, excluding leading zeroes), cf. also -function `cl_float_format'. - -
Complex numbers -
-External representation: - -
    -
  • - -In algebraic notation: realpart+imagparti. Of course, -if imagpart is negative, its printed representation begins with -a `-', and the `+' between realpart and imagpart -may be omitted. Note that this notation cannot be used when the imagpart -is rational and the rational number's base is >18, because the `i' -is then read as a digit. -
  • - -In Common Lisp notation: #C(realpart imagpart). -
- -
- - - -

5.2 Input functions

- -

-Including <cln/io.h> defines a type cl_istream, which is -the type of the first argument to all input functions. cl_istream -is the same as std::istream&. - - -

-The variable - -

- -

-contains the standard input stream. - - -

-These are the simple input functions: - - -

- -
int freadchar (cl_istream stream) -
-Reads a character from stream. Returns cl_EOF (not a `char'!) -if the end of stream was encountered or an error occurred. - -
int funreadchar (cl_istream stream, int c) -
-Puts back c onto stream. c must be the result of the -last freadchar operation on stream. -
- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines, in <cln/type_io.h>, the following input function: - - -

- -
cl_istream operator>> (cl_istream stream, type& result) -
-Reads a number from stream and stores it in the result. -
- -

-The most flexible input functions, defined in <cln/type_io.h>, -are the following: - - -

- -
cl_N read_complex (cl_istream stream, const cl_read_flags& flags) -
-
cl_R read_real (cl_istream stream, const cl_read_flags& flags) -
-
cl_F read_float (cl_istream stream, const cl_read_flags& flags) -
-
cl_RA read_rational (cl_istream stream, const cl_read_flags& flags) -
-
cl_I read_integer (cl_istream stream, const cl_read_flags& flags) -
-Reads a number from stream. The flags are parameters which -affect the input syntax. Whitespace before the number is silently skipped. - -
cl_N read_complex (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_R read_real (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_F read_float (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_RA read_rational (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-
cl_I read_integer (const cl_read_flags& flags, const char * string, const char * string_limit, const char * * end_of_parse) -
-Reads a number from a string in memory. The flags are parameters which -affect the input syntax. The string starts at string and ends at -string_limit (exclusive limit). string_limit may also be -NULL, denoting the entire string, i.e. equivalent to -string_limit = string + strlen(string). If end_of_parse is -NULL, the string in memory must contain exactly one number and nothing -more, else a fatal error will be signalled. If end_of_parse -is not NULL, *end_of_parse will be assigned a pointer past -the last parsed character (i.e. string_limit if nothing came after -the number). Whitespace is not allowed. -
- -

-The structure cl_read_flags contains the following fields: - - -

- -
cl_read_syntax_t syntax -
-The possible results of the read operation. Possible values are -syntax_number, syntax_real, syntax_rational, -syntax_integer, syntax_float, syntax_sfloat, -syntax_ffloat, syntax_dfloat, syntax_lfloat. - -
cl_read_lsyntax_t lsyntax -
-Specifies the language-dependent syntax variant for the read operation. -Possible values are - -
- -
lsyntax_standard -
-accept standard algebraic notation only, no complex numbers, -
lsyntax_algebraic -
-accept the algebraic notation x+yi for complex numbers, -
lsyntax_commonlisp -
-accept the #b, #o, #x syntaxes for binary, octal, -hexadecimal numbers, -#baseR for rational numbers in a given base, -#c(realpart imagpart) for complex numbers, -
lsyntax_all -
-accept all of these extensions. -
- -
unsigned int rational_base -
-The base in which rational numbers are read. - -
cl_float_format_t float_flags.default_float_format -
-The float format used when reading floats with exponent marker `e'. - -
cl_float_format_t float_flags.default_lfloat_format -
-The float format used when reading floats with exponent marker `l'. - -
cl_boolean float_flags.mantissa_dependent_float_format -
-When this flag is true, floats specified with more digits than corresponding -to the exponent marker they contain, but without _nnn suffix, will get a -precision corresponding to their number of significant digits. -
- - - -

5.3 Output functions

- -

-Including <cln/io.h> defines a type cl_ostream, which is -the type of the first argument to all output functions. cl_ostream -is the same as std::ostream&. - - -

-The variable - -

- -

-contains the standard output stream. - - -

-The variable - -

- -

-contains the standard error output stream. - - -

-These are the simple output functions: - - -

- -
void fprintchar (cl_ostream stream, char c) -
-Prints the character x literally on the stream. - -
void fprint (cl_ostream stream, const char * string) -
-Prints the string literally on the stream. - -
void fprintdecimal (cl_ostream stream, int x) -
-
void fprintdecimal (cl_ostream stream, const cl_I& x) -
-Prints the integer x in decimal on the stream. - -
void fprintbinary (cl_ostream stream, const cl_I& x) -
-Prints the integer x in binary (base 2, without prefix) -on the stream. - -
void fprintoctal (cl_ostream stream, const cl_I& x) -
-Prints the integer x in octal (base 8, without prefix) -on the stream. - -
void fprinthexadecimal (cl_ostream stream, const cl_I& x) -
-Prints the integer x in hexadecimal (base 16, without prefix) -on the stream. -
- -

-Each of the classes cl_N, cl_R, cl_RA, cl_I, -cl_F, cl_SF, cl_FF, cl_DF, cl_LF -defines, in <cln/type_io.h>, the following output functions: - - -

- -
void fprint (cl_ostream stream, const type& x) -
-
cl_ostream operator<< (cl_ostream stream, const type& x) -
-Prints the number x on the stream. The output may depend -on the global printer settings in the variable default_print_flags. -The ostream flags and settings (flags, width and locale) are -ignored. -
- -

-The most flexible output function, defined in <cln/type_io.h>, -are the following: - -

-void print_complex  (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_N& z);
-void print_real     (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_R& z);
-void print_float    (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_F& z);
-void print_rational (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_RA& z);
-void print_integer  (cl_ostream stream, const cl_print_flags& flags,
-                     const cl_I& z);
-
- -

-Prints the number x on the stream. The flags are -parameters which affect the output. - - -

-The structure type cl_print_flags contains the following fields: - - -

- -
unsigned int rational_base -
-The base in which rational numbers are printed. Default is 10. - -
cl_boolean rational_readably -
-If this flag is true, rational numbers are printed with radix specifiers in -Common Lisp syntax (#nR or #b or #o or #x -prefixes, trailing dot). Default is false. - -
cl_boolean float_readably -
-If this flag is true, type specific exponent markers have precedence over 'E'. -Default is false. - -
cl_float_format_t default_float_format -
-Floating point numbers of this format will be printed using the 'E' exponent -marker. Default is cl_float_format_ffloat. - -
cl_boolean complex_readably -
-If this flag is true, complex numbers will be printed using the Common Lisp -syntax #C(realpart imagpart). Default is false. - -
cl_string univpoly_varname -
-Univariate polynomials with no explicit indeterminate name will be printed -using this variable name. Default is "x". -
- -

-The global variable default_print_flags contains the default values, -used by the function fprint. - - -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_6.html b/doc/cln_6.html deleted file mode 100644 index efab2fe..0000000 --- a/doc/cln_6.html +++ /dev/null @@ -1,125 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 6. Rings - - -Go to the first, previous, next, last section, table of contents. -

- - -

6. Rings

- -

-CLN has a class of abstract rings. - - - -

-                         Ring
-                       cl_ring
-                     <cln/ring.h>
-
- -

-Rings can be compared for equality: - - -

- -
bool operator== (const cl_ring&, const cl_ring&) -
-
bool operator!= (const cl_ring&, const cl_ring&) -
-These compare two rings for equality. -
- -

-Given a ring R, the following members can be used. - - -

- -
void R->fprint (cl_ostream stream, const cl_ring_element& x) -
- -
cl_boolean R->equal (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->zero () -
- -
cl_boolean R->zerop (const cl_ring_element& x) -
- -
cl_ring_element R->plus (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->minus (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->uminus (const cl_ring_element& x) -
- -
cl_ring_element R->one () -
- -
cl_ring_element R->canonhom (const cl_I& x) -
- -
cl_ring_element R->mul (const cl_ring_element& x, const cl_ring_element& y) -
- -
cl_ring_element R->square (const cl_ring_element& x) -
- -
cl_ring_element R->expt_pos (const cl_ring_element& x, const cl_I& y) -
- -
- -

-The following rings are built-in. - - -

- -
cl_null_ring cl_0_ring -
-The null ring, containing only zero. - -
cl_complex_ring cl_C_ring -
-The ring of complex numbers. This corresponds to the type cl_N. - -
cl_real_ring cl_R_ring -
-The ring of real numbers. This corresponds to the type cl_R. - -
cl_rational_ring cl_RA_ring -
-The ring of rational numbers. This corresponds to the type cl_RA. - -
cl_integer_ring cl_I_ring -
-The ring of integers. This corresponds to the type cl_I. -
- -

-Type tests can be performed for any of cl_C_ring, cl_R_ring, -cl_RA_ring, cl_I_ring: - - -

- -
cl_boolean instanceof (const cl_number& x, const cl_number_ring& R) -
- -Tests whether the given number is an element of the number ring R. -
- -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_7.html b/doc/cln_7.html deleted file mode 100644 index 50d7716..0000000 --- a/doc/cln_7.html +++ /dev/null @@ -1,251 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 7. Modular integers - - -Go to the first, previous, next, last section, table of contents. -

- - -

7. Modular integers

-

- - - - - -

7.1 Modular integer rings

-

- - - -

-CLN implements modular integers, i.e. integers modulo a fixed integer N. -The modulus is explicitly part of every modular integer. CLN doesn't -allow you to (accidentally) mix elements of different modular rings, -e.g. (3 mod 4) + (2 mod 5) will result in a runtime error. -(Ideally one would imagine a generic data type cl_MI(N), but C++ -doesn't have generic types. So one has to live with runtime checks.) - - -

-The class of modular integer rings is - - - -

-                         Ring
-                       cl_ring
-                     <cln/ring.h>
-                          |
-                          |
-                 Modular integer ring
-                    cl_modint_ring
-                  <cln/modinteger.h>
-
- -

- - - -

-and the class of all modular integers (elements of modular integer rings) is - - - -

-                    Modular integer
-                         cl_MI
-                   <cln/modinteger.h>
-
- -

-Modular integer rings are constructed using the function - - -

- -
cl_modint_ring find_modint_ring (const cl_I& N) -
- -This function returns the modular ring `Z/NZ'. It takes care -of finding out about special cases of N, like powers of two -and odd numbers for which Montgomery multiplication will be a win, - -and precomputes any necessary auxiliary data for computing modulo N. -There is a cache table of rings, indexed by N (or, more precisely, -by abs(N)). This ensures that the precomputation costs are reduced -to a minimum. -
- -

-Modular integer rings can be compared for equality: - - -

- -
bool operator== (const cl_modint_ring&, const cl_modint_ring&) -
- -
bool operator!= (const cl_modint_ring&, const cl_modint_ring&) -
- -These compare two modular integer rings for equality. Two different calls -to find_modint_ring with the same argument necessarily return the -same ring because it is memoized in the cache table. -
- - - -

7.2 Functions on modular integers

- -

-Given a modular integer ring R, the following members can be used. - - -

- -
cl_I R->modulus -
- -This is the ring's modulus, normalized to be nonnegative: abs(N). - -
cl_MI R->zero() -
- -This returns 0 mod N. - -
cl_MI R->one() -
- -This returns 1 mod N. - -
cl_MI R->canonhom (const cl_I& x) -
- -This returns x mod N. - -
cl_I R->retract (const cl_MI& x) -
- -This is a partial inverse function to R->canonhom. It returns the -standard representative (>=0, <N) of x. - -
cl_MI R->random(random_state& randomstate) -
-
cl_MI R->random() -
- -This returns a random integer modulo N. -
- -

-The following operations are defined on modular integers. - - -

- -
cl_modint_ring x.ring () -
- -Returns the ring to which the modular integer x belongs. - -
cl_MI operator+ (const cl_MI&, const cl_MI&) -
- -Returns the sum of two modular integers. One of the arguments may also -be a plain integer. - -
cl_MI operator- (const cl_MI&, const cl_MI&) -
- -Returns the difference of two modular integers. One of the arguments may also -be a plain integer. - -
cl_MI operator- (const cl_MI&) -
-Returns the negative of a modular integer. - -
cl_MI operator* (const cl_MI&, const cl_MI&) -
- -Returns the product of two modular integers. One of the arguments may also -be a plain integer. - -
cl_MI square (const cl_MI&) -
- -Returns the square of a modular integer. - -
cl_MI recip (const cl_MI& x) -
- -Returns the reciprocal x^-1 of a modular integer x. x -must be coprime to the modulus, otherwise an error message is issued. - -
cl_MI div (const cl_MI& x, const cl_MI& y) -
- -Returns the quotient x*y^-1 of two modular integers x, y. -y must be coprime to the modulus, otherwise an error message is issued. - -
cl_MI expt_pos (const cl_MI& x, const cl_I& y) -
- -y must be > 0. Returns x^y. - -
cl_MI expt (const cl_MI& x, const cl_I& y) -
- -Returns x^y. If y is negative, x must be coprime to the -modulus, else an error message is issued. - -
cl_MI operator<< (const cl_MI& x, const cl_I& y) -
- -Returns x*2^y. - -
cl_MI operator>> (const cl_MI& x, const cl_I& y) -
- -Returns x*2^-y. When y is positive, the modulus must be odd, -or an error message is issued. - -
bool operator== (const cl_MI&, const cl_MI&) -
- -
bool operator!= (const cl_MI&, const cl_MI&) -
- -Compares two modular integers, belonging to the same modular integer ring, -for equality. - -
cl_boolean zerop (const cl_MI& x) -
- -Returns true if x is 0 mod N. -
- -

-The following output functions are defined (see also the chapter on -input/output). - - -

- -
void fprint (cl_ostream stream, const cl_MI& x) -
- -
cl_ostream operator<< (cl_ostream stream, const cl_MI& x) -
- -Prints the modular integer x on the stream. The output may depend -on the global printer settings in the variable default_print_flags. -
- -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_8.html b/doc/cln_8.html deleted file mode 100644 index 22c6343..0000000 --- a/doc/cln_8.html +++ /dev/null @@ -1,139 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 8. Symbolic data types - - -Go to the first, previous, next, last section, table of contents. -

- - -

8. Symbolic data types

-

- - - -

-CLN implements two symbolic (non-numeric) data types: strings and symbols. - - - - -

8.1 Strings

-

- - - - -

-The class - - - -

-                      String
-                     cl_string
-                   <cln/string.h>
-
- -

-implements immutable strings. - - -

-Strings are constructed through the following constructors: - - -

- -
cl_string (const char * s) -
-Returns an immutable copy of the (zero-terminated) C string s. - -
cl_string (const char * ptr, unsigned long len) -
-Returns an immutable copy of the len characters at -ptr[0], ..., ptr[len-1]. NUL characters are allowed. -
- -

-The following functions are available on strings: - - -

- -
operator = -
-Assignment from cl_string and const char *. - -
s.length() -
- -
strlen(s) -
- -Returns the length of the string s. - -
s[i] -
- -Returns the ith character of the string s. -i must be in the range 0 <= i < s.length(). - -
bool equal (const cl_string& s1, const cl_string& s2) -
- -Compares two strings for equality. One of the arguments may also be a -plain const char *. -
- - - -

8.2 Symbols

-

- - - - -

-Symbols are uniquified strings: all symbols with the same name are shared. -This means that comparison of two symbols is fast (effectively just a pointer -comparison), whereas comparison of two strings must in the worst case walk -both strings until their end. -Symbols are used, for example, as tags for properties, as names of variables -in polynomial rings, etc. - - -

-Symbols are constructed through the following constructor: - - -

- -
cl_symbol (const cl_string& s) -
-Looks up or creates a new symbol with a given name. -
- -

-The following operations are available on symbols: - - -

- -
cl_string (const cl_symbol& sym) -
-Conversion to cl_string: Returns the string which names the symbol -sym. - -
bool equal (const cl_symbol& sym1, const cl_symbol& sym2) -
- -Compares two symbols for equality. This is very fast. -
- -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_9.html b/doc/cln_9.html deleted file mode 100644 index 4e5ae21..0000000 --- a/doc/cln_9.html +++ /dev/null @@ -1,359 +0,0 @@ - - - - -CLN, a Class Library for Numbers - 9. Univariate polynomials - - -Go to the first, previous, next, last section, table of contents. -

- - -

9. Univariate polynomials

-

- - - - - - -

9.1 Univariate polynomial rings

- -

-CLN implements univariate polynomials (polynomials in one variable) over an -arbitrary ring. The indeterminate variable may be either unnamed (and will be -printed according to default_print_flags.univpoly_varname, which -defaults to `x') or carry a given name. The base ring and the -indeterminate are explicitly part of every polynomial. CLN doesn't allow you to -(accidentally) mix elements of different polynomial rings, e.g. -(a^2+1) * (b^3-1) will result in a runtime error. (Ideally this should -return a multivariate polynomial, but they are not yet implemented in CLN.) - - -

-The classes of univariate polynomial rings are - - - -

-                           Ring
-                         cl_ring
-                       <cln/ring.h>
-                            |
-                            |
-                 Univariate polynomial ring
-                      cl_univpoly_ring
-                      <cln/univpoly.h>
-                            |
-           +----------------+-------------------+
-           |                |                   |
- Complex polynomial ring    |    Modular integer polynomial ring
- cl_univpoly_complex_ring   |        cl_univpoly_modint_ring
- <cln/univpoly_complex.h>   |        <cln/univpoly_modint.h>
-                            |
-           +----------------+
-           |                |
-   Real polynomial ring     |
-   cl_univpoly_real_ring    |
-   <cln/univpoly_real.h>    |
-                            |
-           +----------------+
-           |                |
- Rational polynomial ring   |
- cl_univpoly_rational_ring  |
- <cln/univpoly_rational.h>  |
-                            |
-           +----------------+
-           |
- Integer polynomial ring
- cl_univpoly_integer_ring
- <cln/univpoly_integer.h>
-
- -

-and the corresponding classes of univariate polynomials are - - - -

-                   Univariate polynomial
-                          cl_UP
-                      <cln/univpoly.h>
-                            |
-           +----------------+-------------------+
-           |                |                   |
-   Complex polynomial       |      Modular integer polynomial
-        cl_UP_N             |                cl_UP_MI
- <cln/univpoly_complex.h>   |        <cln/univpoly_modint.h>
-                            |
-           +----------------+
-           |                |
-     Real polynomial        |
-        cl_UP_R             |
-  <cln/univpoly_real.h>     |
-                            |
-           +----------------+
-           |                |
-   Rational polynomial      |
-        cl_UP_RA            |
- <cln/univpoly_rational.h>  |
-                            |
-           +----------------+
-           |
-   Integer polynomial
-        cl_UP_I
- <cln/univpoly_integer.h>
-
- -

-Univariate polynomial rings are constructed using the functions - - -

- -
cl_univpoly_ring find_univpoly_ring (const cl_ring& R) -
-
cl_univpoly_ring find_univpoly_ring (const cl_ring& R, const cl_symbol& varname) -
-This function returns the polynomial ring `R[X]', unnamed or named. -R may be an arbitrary ring. This function takes care of finding out -about special cases of R, such as the rings of complex numbers, -real numbers, rational numbers, integers, or modular integer rings. -There is a cache table of rings, indexed by R and varname. -This ensures that two calls of this function with the same arguments will -return the same polynomial ring. - -
cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R) -
- -
cl_univpoly_complex_ring find_univpoly_ring (const cl_complex_ring& R, const cl_symbol& varname) -
-
cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R) -
-
cl_univpoly_real_ring find_univpoly_ring (const cl_real_ring& R, const cl_symbol& varname) -
-
cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R) -
-
cl_univpoly_rational_ring find_univpoly_ring (const cl_rational_ring& R, const cl_symbol& varname) -
-
cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R) -
-
cl_univpoly_integer_ring find_univpoly_ring (const cl_integer_ring& R, const cl_symbol& varname) -
-
cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R) -
-
cl_univpoly_modint_ring find_univpoly_ring (const cl_modint_ring& R, const cl_symbol& varname) -
-These functions are equivalent to the general find_univpoly_ring, -only the return type is more specific, according to the base ring's type. -
- - - -

9.2 Functions on univariate polynomials

- -

-Given a univariate polynomial ring R, the following members can be used. - - -

- -
cl_ring R->basering() -
- -This returns the base ring, as passed to `find_univpoly_ring'. - -
cl_UP R->zero() -
- -This returns 0 in R, a polynomial of degree -1. - -
cl_UP R->one() -
- -This returns 1 in R, a polynomial of degree <= 0. - -
cl_UP R->canonhom (const cl_I& x) -
- -This returns x in R, a polynomial of degree <= 0. - -
cl_UP R->monomial (const cl_ring_element& x, uintL e) -
- -This returns a sparse polynomial: x * X^e, where X is the -indeterminate. - -
cl_UP R->create (sintL degree) -
- -Creates a new polynomial with a given degree. The zero polynomial has degree --1. After creating the polynomial, you should put in the coefficients, -using the set_coeff member function, and then call the finalize -member function. -
- -

-The following are the only destructive operations on univariate polynomials. - - -

- -
void set_coeff (cl_UP& x, uintL index, const cl_ring_element& y) -
- -This changes the coefficient of X^index in x to be y. -After changing a polynomial and before applying any "normal" operation on it, -you should call its finalize member function. - -
void finalize (cl_UP& x) -
- -This function marks the endpoint of destructive modifications of a polynomial. -It normalizes the internal representation so that subsequent computations have -less overhead. Doing normal computations on unnormalized polynomials may -produce wrong results or crash the program. -
- -

-The following operations are defined on univariate polynomials. - - -

- -
cl_univpoly_ring x.ring () -
- -Returns the ring to which the univariate polynomial x belongs. - -
cl_UP operator+ (const cl_UP&, const cl_UP&) -
- -Returns the sum of two univariate polynomials. - -
cl_UP operator- (const cl_UP&, const cl_UP&) -
- -Returns the difference of two univariate polynomials. - -
cl_UP operator- (const cl_UP&) -
-Returns the negative of a univariate polynomial. - -
cl_UP operator* (const cl_UP&, const cl_UP&) -
- -Returns the product of two univariate polynomials. One of the arguments may -also be a plain integer or an element of the base ring. - -
cl_UP square (const cl_UP&) -
- -Returns the square of a univariate polynomial. - -
cl_UP expt_pos (const cl_UP& x, const cl_I& y) -
- -y must be > 0. Returns x^y. - -
bool operator== (const cl_UP&, const cl_UP&) -
- -
bool operator!= (const cl_UP&, const cl_UP&) -
- -Compares two univariate polynomials, belonging to the same univariate -polynomial ring, for equality. - -
cl_boolean zerop (const cl_UP& x) -
- -Returns true if x is 0 in R. - -
sintL degree (const cl_UP& x) -
- -Returns the degree of the polynomial. The zero polynomial has degree -1. - -
cl_ring_element coeff (const cl_UP& x, uintL index) -
- -Returns the coefficient of X^index in the polynomial x. - -
cl_ring_element x (const cl_ring_element& y) -
- -Evaluation: If x is a polynomial and y belongs to the base ring, -then `x(y)' returns the value of the substitution of y into -x. - -
cl_UP deriv (const cl_UP& x) -
- -Returns the derivative of the polynomial x with respect to the -indeterminate X. -
- -

-The following output functions are defined (see also the chapter on -input/output). - - -

- -
void fprint (cl_ostream stream, const cl_UP& x) -
- -
cl_ostream operator<< (cl_ostream stream, const cl_UP& x) -
- -Prints the univariate polynomial x on the stream. The output may -depend on the global printer settings in the variable -default_print_flags. -
- - - -

9.3 Special polynomials

- -

-The following functions return special polynomials. - - -

- -
cl_UP_I tschebychev (sintL n) -
- - -Returns the n-th Chebyshev polynomial (n >= 0). - -
cl_UP_I hermite (sintL n) -
- - -Returns the n-th Hermite polynomial (n >= 0). - -
cl_UP_RA legendre (sintL n) -
- - -Returns the n-th Legendre polynomial (n >= 0). - -
cl_UP_I laguerre (sintL n) -
- - -Returns the n-th Laguerre polynomial (n >= 0). -
- -

-Information how to derive the differential equation satisfied by each -of these polynomials from their definition can be found in the -doc/polynomial/ directory. - - -

-Go to the first, previous, next, last section, table of contents. - - diff --git a/doc/cln_toc.html b/doc/cln_toc.html deleted file mode 100644 index 8bd5aca..0000000 --- a/doc/cln_toc.html +++ /dev/null @@ -1,125 +0,0 @@ - - - - -CLN, a Class Library for Numbers - Table of Contents - - -

CLN, a Class Library for Numbers

-
by Bruno Haible
-

-

-

-

-This document was generated on 28 August 2000 using -texi2html 1.56k. - -