This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
@page
@vskip 0pt plus 1filll
-Copyright @copyright{} 1999-2001 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2003 Johannes Gutenberg University Mainz, Germany
@sp 2
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
@section License
The GiNaC framework for symbolic computation within the C++ programming
-language is Copyright @copyright{} 1999-2001 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2003 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
@menu
* Expressions:: The fundamental GiNaC class.
-* The Class Hierarchy:: Overview of GiNaC's classes.
+* Automatic evaluation:: Evaluation and canonicalization.
* Error handling:: How the library reports errors.
+* The Class Hierarchy:: Overview of GiNaC's classes.
* Symbols:: Symbolic objects.
* Numbers:: Numerical objects.
* Constants:: Pre-defined constants.
-* Fundamental containers:: The power, add and mul classes.
+* Fundamental containers:: Sums, products and powers.
* Lists:: Lists of expressions.
* Mathematical functions:: Mathematical functions.
* Relations:: Equality, Inequality and all that.
@end menu
-@node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
+@node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
@c node-name, next, previous, up
@section Expressions
@cindex expression (class @code{ex})
@code{ex}.
-@node The Class Hierarchy, Error handling, Expressions, Basic Concepts
+@node Automatic evaluation, Error handling, Expressions, Basic Concepts
+@c node-name, next, previous, up
+@section Automatic evaluation and canonicalization of expressions
+@cindex evaluation
+
+GiNaC performs some automatic transformations on expressions, to simplify
+them and put them into a canonical form. Some examples:
+
+@example
+ex MyEx1 = 2*x - 1 + x; // 3*x-1
+ex MyEx2 = x - x; // 0
+ex MyEx3 = cos(2*Pi); // 1
+ex MyEx4 = x*y/x; // y
+@end example
+
+This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
+evaluation}. GiNaC only performs transformations that are
+
+@itemize @bullet
+@item
+at most of complexity @math{O(n log n)}
+@item
+algebraically correct, possibly except for a set of measure zero (e.g.
+@math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
+@end itemize
+
+There are two types of automatic transformations in GiNaC that may not
+behave in an entirely obvious way at first glance:
+
+@itemize
+@item
+The terms of sums and products (and some other things like the arguments of
+symmetric functions, the indices of symmetric tensors etc.) are re-ordered
+into a canonical form that is deterministic, but not lexicographical or in
+any other way easily guessable (it almost always depends on the number and
+order of the symbols you define). However, constructing the same expression
+twice, either implicitly or explicitly, will always result in the same
+canonical form.
+@item
+Expressions of the form 'number times sum' are automatically expanded (this
+has to do with GiNaC's internal representation of sums and products). For
+example
+@example
+ex MyEx5 = 2*(x + y); // 2*x+2*y
+ex MyEx6 = z*(x + y); // z*(x+y)
+@end example
+@end itemize
+
+The general rule is that when you construct expressions, GiNaC automatically
+creates them in canonical form, which might differ from the form you typed in
+your program. This may create some awkward looking output (@samp{-y+x} instead
+of @samp{y-x}) but allows for more efficient operation and usually yields
+some immediate simplifications.
+
+@cindex @code{eval()}
+Internally, the anonymous evaluator in GiNaC is implemented by the methods
+
+@example
+ex ex::eval(int level = 0) const;
+ex basic::eval(int level = 0) const;
+@end example
+
+but unless you are extending GiNaC with your own classes or functions, there
+should never be any reason to call them explicitly. All GiNaC methods that
+transform expressions, like @code{subs()} or @code{normal()}, automatically
+re-evaluate their results.
+
+
+@node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
+@c node-name, next, previous, up
+@section Error handling
+@cindex exceptions
+@cindex @code{pole_error} (class)
+
+GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
+generated by GiNaC are subclassed from the standard @code{exception} class
+defined in the @file{<stdexcept>} header. In addition to the predefined
+@code{logic_error}, @code{domain_error}, @code{out_of_range},
+@code{invalid_argument}, @code{runtime_error}, @code{range_error} and
+@code{overflow_error} types, GiNaC also defines a @code{pole_error}
+exception that gets thrown when trying to evaluate a mathematical function
+at a singularity.
+
+The @code{pole_error} class has a member function
+
+@example
+int pole_error::degree(void) const;
+@end example
+
+that returns the order of the singularity (or 0 when the pole is
+logarithmic or the order is undefined).
+
+When using GiNaC it is useful to arrange for exceptions to be catched in
+the main program even if you don't want to do any special error handling.
+Otherwise whenever an error occurs in GiNaC, it will be delegated to the
+default exception handler of your C++ compiler's run-time system which
+usually only aborts the program without giving any information what went
+wrong.
+
+Here is an example for a @code{main()} function that catches and prints
+exceptions generated by GiNaC:
+
+@example
+#include <iostream>
+#include <stdexcept>
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main(void)
+@{
+ try @{
+ ...
+ // code using GiNaC
+ ...
+ @} catch (exception &p) @{
+ cerr << p.what() << endl;
+ return 1;
+ @}
+ return 0;
+@}
+@end example
+
+
+@node The Class Hierarchy, Symbols, Error handling, Basic Concepts
@c node-name, next, previous, up
@section The Class Hierarchy
@cindex container
@cindex atom
-To get an idea about what kinds of symbolic composits may be built we
+To get an idea about what kinds of symbolic composites may be built we
have a look at the most important classes in the class hierarchy and
some of the relations among the classes:
@end cartouche
-@node Error handling, Symbols, The Class Hierarchy, Basic Concepts
-@c node-name, next, previous, up
-@section Error handling
-@cindex exceptions
-@cindex @code{pole_error} (class)
-
-GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
-generated by GiNaC are subclassed from the standard @code{exception} class
-defined in the @file{<stdexcept>} header. In addition to the predefined
-@code{logic_error}, @code{domain_error}, @code{out_of_range},
-@code{invalid_argument}, @code{runtime_error}, @code{range_error} and
-@code{overflow_error} types, GiNaC also defines a @code{pole_error}
-exception that gets thrown when trying to evaluate a mathematical function
-at a singularity.
-
-The @code{pole_error} class has a member function
-
-@example
-int pole_error::degree(void) const;
-@end example
-
-that returns the order of the singularity (or 0 when the pole is
-logarithmic or the order is undefined).
-
-When using GiNaC it is useful to arrange for exceptions to be catched in
-the main program even if you don't want to do any special error handling.
-Otherwise whenever an error occurs in GiNaC, it will be delegated to the
-default exception handler of your C++ compiler's run-time system which
-usually only aborts the program without giving any information what went
-wrong.
-
-Here is an example for a @code{main()} function that catches and prints
-exceptions generated by GiNaC:
-
-@example
-#include <iostream>
-#include <stdexcept>
-#include <ginac/ginac.h>
-using namespace std;
-using namespace GiNaC;
-
-int main(void)
-@{
- try @{
- ...
- // code using GiNaC
- ...
- @} catch (exception &p) @{
- cerr << p.what() << endl;
- return 1;
- @}
- return 0;
-@}
-@end example
-
-
-@node Symbols, Numbers, Error handling, Basic Concepts
+@node Symbols, Numbers, The Class Hierarchy, Basic Concepts
@c node-name, next, previous, up
@section Symbols
@cindex @code{symbol} (class)
For storing numerical things, GiNaC uses Bruno Haible's library CLN.
The classes therein serve as foundation classes for GiNaC. CLN stands
for Class Library for Numbers or alternatively for Common Lisp Numbers.
-In order to find out more about CLN's internals the reader is refered to
+In order to find out more about CLN's internals, the reader is referred to
the documentation of that library. @inforef{Introduction, , cln}, for
more information. Suffice to say that it is by itself build on top of
another library, the GNU Multiple Precision library GMP, which is an
numeric trott("1.0841015122311136151E-2");
std::cout << two*p << std::endl; // floating point 6.283...
+ ...
+@end example
+
+@cindex @code{I}
+@cindex complex numbers
+The imaginary unit in GiNaC is a predefined @code{numeric} object with the
+name @code{I}:
+
+@example
+ ...
+ numeric z1 = 2-3*I; // exact complex number 2-3i
+ numeric z2 = 5.9+1.6*I; // complex floating point number
@}
@end example
-It may be tempting to construct numbers writing @code{numeric r(3/2)}.
+It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
This would, however, call C's built-in operator @code{/} for integers
first and result in a numeric holding a plain integer 1. @strong{Never
use the operator @code{/} on integers} unless you know exactly what you
that in both cases you got a couple of extra digits. This is because
numbers are internally stored by CLN as chunks of binary digits in order
to match your machine's word size and to not waste precision. Thus, on
-architectures with differnt word size, the above output might even
+architectures with different word size, the above output might even
differ with regard to actually computed digits.
It should be clear that objects of class @code{numeric} should be used
@node Fundamental containers, Lists, Constants, Basic Concepts
@c node-name, next, previous, up
-@section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
+@section Sums, products and powers
@cindex polynomial
@cindex @code{add}
@cindex @code{mul}
@cindex @code{power}
-Simple polynomial expressions are written down in GiNaC pretty much like
+Simple rational expressions are written down in GiNaC pretty much like
in other CAS or like expressions involving numerical variables in C.
The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
been overloaded to achieve this goal. When you run the following
and safe simplifications are carried out like transforming
@code{3*x+4-x} to @code{2*x+4}.
-The general rule is that when you construct such objects, GiNaC
-automatically creates them in canonical form, which might differ from
-the form you typed in your program. This allows for rapid comparison of
-expressions, since after all @code{a-a} is simply zero. Note, that the
-canonical form is not necessarily lexicographical ordering or in any way
-easily guessable. It is only guaranteed that constructing the same
-expression twice, either implicitly or explicitly, results in the same
-canonical form.
-
@node Lists, Mathematical functions, Fundamental containers, Basic Concepts
@c node-name, next, previous, up
There are a couple of ways to construct matrices, with or without preset
elements:
+@cindex @code{lst_to_matrix()}
+@cindex @code{diag_matrix()}
+@cindex @code{unit_matrix()}
+@cindex @code{symbolic_matrix()}
@example
matrix::matrix(unsigned r, unsigned c);
matrix::matrix(unsigned r, unsigned c, const lst & l);
ex lst_to_matrix(const lst & l);
ex diag_matrix(const lst & l);
+ex unit_matrix(unsigned x);
+ex unit_matrix(unsigned r, unsigned c);
+ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
+ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
@end example
The first two functions are @code{matrix} constructors which create a matrix
with @samp{r} rows and @samp{c} columns. The matrix elements can be
initialized from a (flat) list of expressions @samp{l}. Otherwise they are
all set to zero. The @code{lst_to_matrix()} function constructs a matrix
-from a list of lists, each list representing a matrix row. Finally,
-@code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
-elements. Note that the last two functions return expressions, not matrix
-objects.
+from a list of lists, each list representing a matrix row. @code{diag_matrix()}
+constructs a diagonal matrix given the list of diagonal elements.
+@code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
+unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
+with newly generated symbols made of the specified base name and the
+position of each element in the matrix.
Matrix elements can be accessed and set using the parenthesis (function call)
operator:
the @code{op()} method. But C++-style subscripting with square brackets
@samp{[]} is not available.
-Here are a couple of examples that all construct the same 2x2 diagonal
-matrix:
+Here are a couple of examples of constructing matrices:
@example
@{
symbol a("a"), b("b");
- ex e;
matrix M(2, 2);
M(0, 0) = a;
M(1, 1) = b;
- e = M;
-
- e = matrix(2, 2, lst(a, 0, 0, b));
+ cout << M << endl;
+ // -> [[a,0],[0,b]]
- e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
+ cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
+ // -> [[a,0],[0,b]]
- e = diag_matrix(lst(a, b));
+ cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
+ // -> [[a,0],[0,b]]
- cout << e << endl;
+ cout << diag_matrix(lst(a, b)) << endl;
// -> [[a,0],[0,b]]
+
+ cout << unit_matrix(3) << endl;
+ // -> [[1,0,0],[0,1,0],[0,0,1]]
+
+ cout << symbolic_matrix(2, 3, "x") << endl;
+ // -> [[x00,x01,x02],[x10,x11,x12]]
@}
@end example
The @code{matrix} class provides a couple of additional methods for
computing determinants, traces, and characteristic polynomials:
+@cindex @code{determinant()}
+@cindex @code{trace()}
+@cindex @code{charpoly()}
@example
ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
ex matrix::trace(void) const;
// -> 2*A.j.i
cout << indexed(B, sy_anti(), i, j)
+ indexed(B, sy_anti(), j, i) << endl;
- // -> -B.j.i
+ // -> 0
cout << indexed(B, sy_anti(), i, j, k)
- + indexed(B, sy_anti(), j, i, k) << endl;
+ - indexed(B, sy_anti(), j, k, i) << endl;
// -> 0
...
@end example
dummy nor free indices.
To be recognized as a dummy index pair, the two indices must be of the same
-class and dimension and their value must be the same single symbol (an index
-like @samp{2*n+1} is never a dummy index). If the indices are of class
+class and their value must be the same single symbol (an index like
+@samp{2*n+1} is never a dummy index). If the indices are of class
@code{varidx} they must also be of opposite variance; if they are of class
@code{spinidx} they must be both dotted or both undotted.
multiples of the unity element, even though it's customary to omit it.
E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
-GiNaC may produce incorrect results.
+GiNaC will complain and/or produce incorrect results.
@cindex @code{dirac_gamma5()}
-There's a special element @samp{gamma5} that commutes with all other
-gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
-provided by
+There is a special element @samp{gamma5} that commutes with all other
+gammas, has a unit square, and in 4 dimensions equals
+@samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
@example
ex dirac_gamma5(unsigned char rl = 0);
@end example
-@cindex @code{dirac_gamma6()}
-@cindex @code{dirac_gamma7()}
-The two additional functions
+@cindex @code{dirac_gammaL()}
+@cindex @code{dirac_gammaR()}
+The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
+objects, constructed by
@example
-ex dirac_gamma6(unsigned char rl = 0);
-ex dirac_gamma7(unsigned char rl = 0);
+ex dirac_gammaL(unsigned char rl = 0);
+ex dirac_gammaR(unsigned char rl = 0);
@end example
-return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
-respectively.
+They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
+and @samp{gammaL gammaR = gammaR gammaL = 0}.
@cindex @code{dirac_slash()}
Finally, the function
Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
In products of dirac gammas, superfluous unity elements are automatically
-removed, squares are replaced by their values and @samp{gamma5} is
-anticommuted to the front. The @code{simplify_indexed()} function performs
-contractions in gamma strings, for example
+removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
+and @samp{gammaR} are moved to the front.
+
+The @code{simplify_indexed()} function performs contractions in gamma strings,
+for example
@example
@{
are also valid patterns.
@end itemize
+@subsection Matching expressions
@cindex @code{match()}
The most basic application of patterns is to check whether an expression
matches a given pattern. This is done by the function
@{$0==x^2@}
@end example
+@subsection Matching parts of expressions
@cindex @code{has()}
A more general way to look for patterns in expressions is provided by the
member function
@end example
works a bit like @code{has()} but it doesn't stop upon finding the first
-match. Instead, it appends all found matches to the specified list. If there
-are multiple occurrences of the same expression, it is entered only once to
-the list. @code{find()} returns false if no matches were found (in
+match. Instead, it inserts all found matches into the specified list. If
+there are multiple occurrences of the same expression, it is entered only
+once to the list. @code{find()} returns false if no matches were found (in
@command{ginsh}, it returns an empty list):
@example
@{sin(y),sin(x)@}
@end example
+@subsection Substituting expressions
@cindex @code{subs()}
Probably the most useful application of patterns is to use them for
substituting expressions with the @code{subs()} method. Wildcards can be
b^3+a^3+(x+y)^3
> subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
b^4+a^4+(x+y)^4
-> subs((a+b+c)^2,a+b=x);
+> subs((a+b+c)^2,a+b==x);
(a+b+c)^2
> subs((a+b+c)^2,a+b+$1==x+$1);
(x+c)^2
-> subs(a+2*b,a+b=x);
+> subs(a+2*b,a+b==x);
a+2*b
> subs(4*x^3-2*x^2+5*x-1,x==a);
-1+5*a-2*a^2+4*a^3
@subsection Expanding and collecting
@cindex @code{expand()}
@cindex @code{collect()}
+@cindex @code{collect_common_factors()}
A polynomial in one or more variables has many equivalent
representations. Some useful ones serve a specific purpose. Consider
(1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)
@end example
+Polynomials can often be brought into a more compact form by collecting
+common factors from the terms of sums. This is accomplished by the function
+
+@example
+ex collect_common_factors(const ex & e);
+@end example
+
+This function doesn't perform a full factorization but only looks for
+factors which are already explicitly present:
+
+@example
+> collect_common_factors(a*x+a*y);
+(x+y)*a
+> collect_common_factors(a*x^2+2*a*x*y+a*y^2);
+a*(2*x*y+y^2+x^2)
+> collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
+(c+a)*a*(x*y+y^2+x)*b
+@end example
+
@subsection Degree and coefficients
@cindex @code{degree()}
@cindex @code{ldegree()}
int ex::ldegree(const ex & s);
@end example
-which also work reliably on non-expanded input polynomials (they even work
-on rational functions, returning the asymptotic degree). To extract
-a coefficient with a certain power from an expanded polynomial you use
+These functions only work reliably if the input polynomial is collected in
+terms of the object @samp{s}. Otherwise, they are only guaranteed to return
+the upper/lower bounds of the exponents. If you need accurate results, you
+have to call @code{expand()} and/or @code{collect()} on the input polynomial.
+For example
+
+@example
+> a=(x+1)^2-x^2;
+(1+x)^2-x^2;
+> degree(a,x);
+2
+> degree(expand(a),x);
+1
+@end example
+
+@code{degree()} also works on rational functions, returning the asymptotic
+degree:
+
+@example
+> degree((x+1)/(x^3+1),x);
+-2
+@end example
+
+If the input is not a polynomial or rational function in the variable @samp{s},
+the behavior of @code{degree()} and @code{ldegree()} is undefined.
+
+To extract a coefficient with a certain power from an expanded
+polynomial you use
@example
ex ex::coeff(const ex & s, int n);
@math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
series raised to the power @math{-2}.
-@cindex M@'echain's formula
+@cindex Machin's formula
As another instructive application, let us calculate the numerical
value of Archimedes' constant
@tex
$\pi$
@end tex
(for which there already exists the built-in constant @code{Pi})
-using M@'echain's amazing formula
+using Machin's amazing formula
@tex
$\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
@end tex
#include <ginac/ginac.h>
using namespace GiNaC;
-ex mechain_pi(int degr)
+ex machin_pi(int degr)
@{
symbol x;
ex pi_expansion = series_to_poly(atan(x).series(x,degr));
using std::endl; // ...dealing with this namespace std.
ex pi_frac;
for (int i=2; i<12; i+=2) @{
- pi_frac = mechain_pi(i);
+ pi_frac = machin_pi(i);
cout << i << ":\t" << pi_frac << endl
<< "\t" << pi_frac.evalf() << endl;
@}
The above example will produce (note the @code{x^2} being converted to @code{x*x}):
@example
-float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
-double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
-cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
+float f = (3.0/2.0)*(x*x)+4.500000e+00;
+double d = (3.0/2.0)*(x*x)+4.5000000000000000e+00;
+cl_N n = cln::cl_RA("3/2")*(x*x)+cln::cl_F("4.5_17");
@end example
The @code{print_context} type @code{print_tree} provides a dump of the
desired symbols to the @code{>>} stream input operator.
Instead, GiNaC lets you construct an expression from a string, specifying the
-list of symbols to be used:
+list of symbols and indices to be used:
@example
@{
- symbol x("x"), y("y");
- ex e("2*x+sin(y)", lst(x, y));
+ symbol x("x"), y("y"), p("p");
+ idx i(symbol("i"), 3);
+ ex e("2*x+sin(y)+p.i", lst(x, y, p, i));
@}
@end example
The input syntax is the same as that used by @command{ginsh} and the stream
-output operator @code{<<}. The symbols in the string are matched by name to
-the symbols in the list and if GiNaC encounters a symbol not specified in
-the list it will throw an exception.
+output operator @code{<<}. The symbols and indices in the string are matched
+by name to the symbols and indices in the list and if GiNaC encounters a
+symbol or index not specified in the list it will throw an exception. Only
+indices whose values are single symbols can be used (i.e. numeric indices
+or compound indices as in "A.(2*n+1)" are not allowed).
With this constructor, it's also easy to implement interactive GiNaC programs:
@subsection Automatic evaluation
-@cindex @code{hold()}
-@cindex @code{eval()}
@cindex evaluation
+@cindex @code{eval()}
+@cindex @code{hold()}
When dealing with objects that are just a little more complicated than the
simple string objects we have implemented, chances are that you will want to
have some automatic simplifications or canonicalizations performed on them.
@end example
This @file{Makefile.am}, says that we are building a single executable,
-from a single sourcefile @file{simple.cpp}. Since every program
+from a single source file @file{simple.cpp}. Since every program
we are building uses GiNaC we simply added the GiNaC options
to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
want to specify them on a per-program basis: for instance by
git://www.ginac.de / ginac.git / blobdiff