This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2001 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{} 1999-2000 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2001 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{} 1999-2000 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2001 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
ex HermitePoly(const symbol & x, int n)
* Lists:: Lists of expressions.
* Mathematical functions:: Mathematical functions.
* Relations:: Equality, Inequality and all that.
+* Indexed objects:: Handling indexed quantities.
@end menu
@item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
@item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
@item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
-@item @code{color}, @code{coloridx} @tab Element and index of the @math{SU(3)} Lie-algebra
-@item @code{isospin} @tab Element of the @math{SU(2)} Lie-algebra
-@item @code{idx} @tab Index of a general tensor object
+@item @code{indexed} @tab Indexed object like @math{A_ij}
+@item @code{tensor} @tab Special tensor like the delta and metric tensors
+@item @code{idx} @tab Index of an indexed object
+@item @code{varidx} @tab Index with variance
@end multitable
@end cartouche
Although symbols can be assigned expressions for internal reasons, you
should not do it (and we are not going to tell you how it is done). If
you want to replace a symbol with something else in an expression, you
-can use the expression's @code{.subs()} method (@xref{Substituting Symbols},
+can use the expression's @code{.subs()} method (@xref{Substituting Expressions},
for more information).
// Trott's constant in scientific notation:
numeric trott("1.0841015122311136151E-2");
- cout << two*p << endl; // floating point 6.283...
+ std::cout << two*p << std::endl; // floating point 6.283...
@}
@end example
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
void foo()
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
// some very important constants:
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
-program, the constructor for an object of type @code{mul} is
+code snippet, the constructor for an object of type @code{mul} is
automatically called to hold the product of @code{a} and @code{b} and
then the constructor for an object of type @code{add} is called to hold
the sum of that @code{mul} object and the number one:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
+ ...
symbol a("a"), b("b");
ex MyTerm = 1+a*b;
- // ...
-@}
+ ...
@end example
@cindex @code{pow()}
statement @code{pow(x,2);} to represent @code{x} squared. This direct
construction is necessary since we cannot safely overload the constructor
@code{^} in C++ to construct a @code{power} object. If we did, it would
-have several counterintuitive effects:
+have several counterintuitive and undesired effects:
@itemize @bullet
@item
instance, all trigonometric and hyperbolic functions are implemented
(@xref{Built-in Functions}, for a complete list).
-These functions are all objects of class @code{function}. They accept one
-or more expressions as arguments and return one expression. If the arguments
-are not numerical, the evaluation of the function may be halted, as it
-does in the next example:
+These functions are all objects of class @code{function}. They accept
+one or more expressions as arguments and return one expression. If the
+arguments are not numerical, the evaluation of the function may be
+halted, as it does in the next example, showing how a function returns
+itself twice and finally an expression that may be really useful:
@cindex Gamma function
@cindex @code{subs()}
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- symbol x("x"), y("y");
-
+ ...
+ symbol x("x"), y("y");
ex foo = x+y/2;
- cout << "tgamma(" << foo << ") -> " << tgamma(foo) << endl;
+ cout << tgamma(foo) << endl;
+ // -> tgamma(x+(1/2)*y)
ex bar = foo.subs(y==1);
- cout << "tgamma(" << bar << ") -> " << tgamma(bar) << endl;
+ cout << tgamma(bar) << endl;
+ // -> tgamma(x+1/2)
ex foobar = bar.subs(x==7);
- cout << "tgamma(" << foobar << ") -> " << tgamma(foobar) << endl;
-@}
-@end example
-
-This program shows how the function returns itself twice and finally an
-expression that may be really useful:
-
-@example
-tgamma(x+(1/2)*y) -> tgamma(x+(1/2)*y)
-tgamma(x+1/2) -> tgamma(x+1/2)
-tgamma(15/2) -> (135135/128)*Pi^(1/2)
+ cout << tgamma(foobar) << endl;
+ // -> (135135/128)*Pi^(1/2)
+ ...
@end example
Besides evaluation most of these functions allow differentiation, series
this.
-@node Relations, Methods and Functions, Mathematical functions, Basic Concepts
+@node Relations, Indexed objects, Mathematical functions, Basic Concepts
@c node-name, next, previous, up
@section Relations
@cindex @code{relational} (class)
@code{expand()} must be called explicitly.
-@node Methods and Functions, Information About Expressions, Relations, Top
+@node Indexed objects, Methods and Functions, Relations, Basic Concepts
+@c node-name, next, previous, up
+@section Indexed objects
+
+GiNaC allows you to handle expressions containing general indexed objects in
+arbitrary spaces. It is also able to canonicalize and simplify such
+expressions and perform symbolic dummy index summations. There are a number
+of predefined indexed objects provided, like delta and metric tensors.
+
+There are few restrictions placed on indexed objects and their indices and
+it is easy to construct nonsense expressions, but our intention is to
+provide a general framework that allows you to implement algorithms with
+indexed quantities, getting in the way as little as possible.
+
+@cindex @code{idx} (class)
+@cindex @code{indexed} (class)
+@subsection Indexed quantities and their indices
+
+Indexed expressions in GiNaC are constructed of two special types of objects,
+@dfn{index objects} and @dfn{indexed objects}.
+
+@itemize @bullet
+
+@cindex contravariant
+@cindex covariant
+@cindex variance
+@item Index objects are of class @code{idx} or a subclass. Every index has
+a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
+the index lives in) which can both be arbitrary expressions but are usually
+a number or a simple symbol. In addition, indices of class @code{varidx} have
+a @dfn{variance} (they can be co- or contravariant).
+
+@item Indexed objects are of class @code{indexed} or a subclass. They
+contain a @dfn{base expression} (which is the expression being indexed), and
+one or more indices.
+
+@end itemize
+
+@strong{Note:} when printing expressions, covariant indices and indices
+without variance are denoted @samp{.i} while contravariant indices are denoted
+@samp{~i}. In the following, we are going to use that notation in the text
+so instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions
+are not visible in the output.
+
+A simple example shall illustrate the concepts:
+
+@example
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ symbol i_sym("i"), j_sym("j");
+ idx i(i_sym, 3), j(j_sym, 3);
+
+ symbol A("A");
+ cout << indexed(A, i, j) << endl;
+ // -> A.i.j
+ ...
+@end example
+
+The @code{idx} constructor takes two arguments, the index value and the
+index dimension. First we define two index objects, @code{i} and @code{j},
+both with the numeric dimension 3. The value of the index @code{i} is the
+symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
+@code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
+construct an expression containing one indexed object, @samp{A.i.j}. It has
+the symbol @code{A} as its base expression and the two indices @code{i} and
+@code{j}.
+
+Note the difference between the indices @code{i} and @code{j} which are of
+class @code{idx}, and the index values which are the sybols @code{i_sym}
+and @code{j_sym}. The indices of indexed objects cannot directly be symbols
+or numbers but must be index objects. For example, the following is not
+correct and will raise an exception:
+
+@example
+symbol i("i"), j("j");
+e = indexed(A, i, j); // ERROR: indices must be of type idx
+@end example
+
+You can have multiple indexed objects in an expression, index values can
+be numeric, and index dimensions symbolic:
+
+@example
+ ...
+ symbol B("B"), dim("dim");
+ cout << 4 * indexed(A, i)
+ + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
+ // -> B.j.2.i+4*A.i
+ ...
+@end example
+
+@code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
+index of value 2, and a symbolic index @samp{i} with the symbolic dimension
+@samp{dim}. Note that GiNaC doesn't automatically notify you that the free
+indices of @samp{A} and @samp{B} in the sum don't match (you have to call
+@code{simplify_indexed()} for that, see below).
+
+In fact, base expressions, index values and index dimensions can be
+arbitrary expressions:
+
+@example
+ ...
+ cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
+ // -> (B+A).(1+2*i)
+ ...
+@end example
+
+It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
+get an error message from this but you will probably not be able to do
+anything useful with it.
+
+@cindex @code{get_value()}
+@cindex @code{get_dimension()}
+The methods
+
+@example
+ex idx::get_value(void);
+ex idx::get_dimension(void);
+@end example
+
+return the value and dimension of an @code{idx} object. If you have an index
+in an expression, such as returned by calling @code{.op()} on an indexed
+object, you can get a reference to the @code{idx} object with the function
+@code{ex_to_idx()} on the expression.
+
+There are also the methods
+
+@example
+bool idx::is_numeric(void);
+bool idx::is_symbolic(void);
+bool idx::is_dim_numeric(void);
+bool idx::is_dim_symbolic(void);
+@end example
+
+for checking whether the value and dimension are numeric or symbolic
+(non-numeric). Using the @code{info()} method of an index (see @ref{Information
+About Expressions}) returns information about the index value.
+
+@cindex @code{varidx} (class)
+If you need co- and contravariant indices, use the @code{varidx} class:
+
+@example
+ ...
+ symbol mu_sym("mu"), nu_sym("nu");
+ varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
+ varidx mu_co(mu_sym, 4, true); // covariant index .mu
+
+ cout << indexed(A, mu, nu) << endl;
+ // -> A~mu~nu
+ cout << indexed(A, mu_co, nu) << endl;
+ // -> A.mu~nu
+ cout << indexed(A, mu.toggle_variance(), nu) << endl;
+ // -> A.mu~nu
+ ...
+@end example
+
+A @code{varidx} is an @code{idx} with an additional flag that marks it as
+co- or contravariant. The default is a contravariant (upper) index, but
+this can be overridden by supplying a third argument to the @code{varidx}
+constructor. The two methods
+
+@example
+bool varidx::is_covariant(void);
+bool varidx::is_contravariant(void);
+@end example
+
+allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
+to get the object reference from an expression). There's also the very useful
+method
+
+@example
+ex varidx::toggle_variance(void);
+@end example
+
+which makes a new index with the same value and dimension but the opposite
+variance. By using it you only have to define the index once.
+
+@subsection Substituting indices
+
+@cindex @code{subs()}
+Sometimes you will want to substitute one symbolic index with another
+symbolic or numeric index, for example when calculating one specific element
+of a tensor expression. This is done with the @code{.subs()} method, as it
+is done for symbols (see @ref{Substituting Expressions}).
+
+You have two possibilities here. You can either substitute the whole index
+by another index or expression:
+
+@example
+ ...
+ ex e = indexed(A, mu_co);
+ cout << e << " becomes " << e.subs(mu_co == nu) << endl;
+ // -> A.mu becomes A~nu
+ cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
+ // -> A.mu becomes A~0
+ cout << e << " becomes " << e.subs(mu_co == 0) << endl;
+ // -> A.mu becomes A.0
+ ...
+@end example
+
+The third example shows that trying to replace an index with something that
+is not an index will substitute the index value instead.
+
+Alternatively, you can substitute the @emph{symbol} of a symbolic index by
+another expression:
+
+@example
+ ...
+ ex e = indexed(A, mu_co);
+ cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
+ // -> A.mu becomes A.nu
+ cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
+ // -> A.mu becomes A.0
+ ...
+@end example
+
+As you see, with the second method only the value of the index will get
+substituted. Its other properties, including its dimension, remain unchanged.
+If you want to change the dimension of an index you have to substitute the
+whole index by another one with the new dimension.
+
+Finally, substituting the base expression of an indexed object works as
+expected:
+
+@example
+ ...
+ ex e = indexed(A, mu_co);
+ cout << e << " becomes " << e.subs(A == A+B) << endl;
+ // -> A.mu becomes (B+A).mu
+ ...
+@end example
+
+@subsection Symmetries
+
+Indexed objects can be declared as being totally symmetric or antisymmetric
+with respect to their indices. In this case, GiNaC will automatically bring
+the indices into a canonical order which allows for some immediate
+simplifications:
+
+@example
+ ...
+ cout << indexed(A, indexed::symmetric, i, j)
+ + indexed(A, indexed::symmetric, j, i) << endl;
+ // -> 2*A.j.i
+ cout << indexed(B, indexed::antisymmetric, i, j)
+ + indexed(B, indexed::antisymmetric, j, j) << endl;
+ // -> -B.j.i
+ cout << indexed(B, indexed::antisymmetric, i, j)
+ + indexed(B, indexed::antisymmetric, j, i) << endl;
+ // -> 0
+ ...
+@end example
+
+@cindex @code{get_free_indices()}
+@cindex Dummy index
+@subsection Dummy indices
+
+GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
+that a summation over the index range is implied. Symbolic indices which are
+not dummy indices are called @dfn{free indices}. Numeric indices are neither
+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
+@code{varidx}, they must also be of opposite variance.
+
+The method @code{.get_free_indices()} returns a vector containing the free
+indices of an expression. It also checks that the free indices of the terms
+of a sum are consistent:
+
+@example
+@{
+ symbol A("A"), B("B"), C("C");
+
+ symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
+ idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
+
+ ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
+ cout << exprseq(e.get_free_indices()) << endl;
+ // -> (.i,.k)
+ // 'j' and 'l' are dummy indices
+
+ symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
+ varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
+
+ e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
+ + indexed(C, mu, sigma, rho, sigma.toggle_variance());
+ cout << exprseq(e.get_free_indices()) << endl;
+ // -> (~mu,~rho)
+ // 'nu' is a dummy index, but 'sigma' is not
+
+ e = indexed(A, mu, mu);
+ cout << exprseq(e.get_free_indices()) << endl;
+ // -> (~mu)
+ // 'mu' is not a dummy index because it appears twice with the same
+ // variance
+
+ e = indexed(A, mu, nu) + 42;
+ cout << exprseq(e.get_free_indices()) << endl; // ERROR
+ // this will throw an exception:
+ // "add::get_free_indices: inconsistent indices in sum"
+@}
+@end example
+
+@cindex @code{simplify_indexed()}
+@subsection Simplifying indexed expressions
+
+In addition to the few automatic simplifications that GiNaC performs on
+indexed expressions (such as re-ordering the indices of symmetric tensors
+and calculating traces and convolutions of matrices and predefined tensors)
+there is the method
+
+@example
+ex ex::simplify_indexed(void);
+ex ex::simplify_indexed(const scalar_products & sp);
+@end example
+
+that performs some more expensive operations:
+
+@itemize
+@item it checks the consistency of free indices in sums in the same way
+ @code{get_free_indices()} does
+@item it (symbolically) calculates all possible dummy index summations/contractions
+ with the predefined tensors (this will be explained in more detail in the
+ next section)
+@item as a special case of dummy index summation, it can replace scalar products
+ of two tensors with a user-defined value
+@end itemize
+
+The last point is done with the help of the @code{scalar_products} class
+which is used to store scalar products with known values (this is not an
+arithmetic class, you just pass it to @code{simplify_indexed()}):
+
+@example
+@{
+ symbol A("A"), B("B"), C("C"), i_sym("i");
+ idx i(i_sym, 3);
+
+ scalar_products sp;
+ sp.add(A, B, 0); // A and B are orthogonal
+ sp.add(A, C, 0); // A and C are orthogonal
+ sp.add(A, A, 4); // A^2 = 4 (A has length 2)
+
+ e = indexed(A + B, i) * indexed(A + C, i);
+ cout << e << endl;
+ // -> (B+A).i*(A+C).i
+
+ cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
+ << endl;
+ // -> 4+C.i*B.i
+@}
+@end example
+
+The @code{scalar_products} object @code{sp} acts as a storage for the
+scalar products added to it with the @code{.add()} method. This method
+takes three arguments: the two expressions of which the scalar product is
+taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
+@code{simplify_indexed()} will replace all scalar products of indexed
+objects that have the symbols @code{A} and @code{B} as base expressions
+with the single value 0. The number, type and dimension of the indices
+doesn't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
+
+@cindex @code{expand()}
+The example above also illustrates a feature of the @code{expand()} method:
+if passed the @code{expand_indexed} option it will distribute indices
+over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
+
+@cindex @code{tensor} (class)
+@subsection Predefined tensors
+
+Some frequently used special tensors such as the delta, epsilon and metric
+tensors are predefined in GiNaC. They have special properties when
+contracted with other tensor expressions and some of them have constant
+matrix representations (they will evaluate to a number when numeric
+indices are specified).
+
+@cindex @code{delta_tensor()}
+@subsubsection Delta tensor
+
+The delta tensor takes two indices, is symmetric and has the matrix
+representation @code{diag(1,1,1,...)}. It is constructed by the function
+@code{delta_tensor()}:
+
+@example
+@{
+ symbol A("A"), B("B");
+
+ idx i(symbol("i"), 3), j(symbol("j"), 3),
+ k(symbol("k"), 3), l(symbol("l"), 3);
+
+ ex e = indexed(A, i, j) * indexed(B, k, l)
+ * delta_tensor(i, k) * delta_tensor(j, l) << endl;
+ cout << e.simplify_indexed() << endl;
+ // -> B.i.j*A.i.j
+
+ cout << delta_tensor(i, i) << endl;
+ // -> 3
+@}
+@end example
+
+@cindex @code{metric_tensor()}
+@subsubsection General metric tensor
+
+The function @code{metric_tensor()} creates a general symmetric metric
+tensor with two indices that can be used to raise/lower tensor indices. The
+metric tensor is denoted as @samp{g} in the output and if its indices are of
+mixed variance it is automatically replaced by a delta tensor:
+
+@example
+@{
+ symbol A("A");
+
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
+
+ ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
+ cout << e.simplify_indexed() << endl;
+ // -> A~mu~rho
+
+ e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
+ cout << e.simplify_indexed() << endl;
+ // -> g~mu~rho
+
+ e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
+ * metric_tensor(nu, rho);
+ cout << e.simplify_indexed() << endl;
+ // -> delta.mu~rho
+
+ e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
+ * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
+ + indexed(A, mu.toggle_variance(), rho));
+ cout << e.simplify_indexed() << endl;
+ // -> 4+A.rho~rho
+@}
+@end example
+
+@cindex @code{lorentz_g()}
+@subsubsection Minkowski metric tensor
+
+The Minkowski metric tensor is a special metric tensor with a constant
+matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
+signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
+It is created with the function @code{lorentz_g()} (although it is output as
+@samp{eta}):
+
+@example
+@{
+ varidx mu(symbol("mu"), 4);
+
+ e = delta_tensor(varidx(0, 4), mu.toggle_variance())
+ * lorentz_g(mu, varidx(0, 4)); // negative signature
+ cout << e.simplify_indexed() << endl;
+ // -> 1
+
+ e = delta_tensor(varidx(0, 4), mu.toggle_variance())
+ * lorentz_g(mu, varidx(0, 4), true); // positive signature
+ cout << e.simplify_indexed() << endl;
+ // -> -1
+@}
+@end example
+
+@subsubsection Epsilon tensor
+
+The epsilon tensor is totally antisymmetric, its number of indices is equal
+to the dimension of the index space (the indices must all be of the same
+numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
+defined to be 1. Its behaviour with indices that have a variance also
+depends on the signature of the metric. Epsilon tensors are output as
+@samp{eps}.
+
+There are three functions defined to create epsilon tensors in 2, 3 and 4
+dimensions:
+
+@example
+ex epsilon_tensor(const ex & i1, const ex & i2);
+ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
+ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
+@end example
+
+The first two functions create an epsilon tensor in 2 or 3 Euclidean
+dimensions, the last function creates an epsilon tensor in a 4-dimensional
+Minkowski space (the last @code{bool} argument specifies whether the metric
+has negative or positive signature, as in the case of the Minkowski metric
+tensor).
+
+@subsection Linear algebra
+
+The @code{matrix} class can be used with indices to do some simple linear
+algebra (linear combinations and products of vectors and matrices, traces
+and scalar products):
+
+@example
+@{
+ idx i(symbol("i"), 2), j(symbol("j"), 2);
+ symbol x("x"), y("y");
+
+ matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
+
+ cout << indexed(A, i, i) << endl;
+ // -> 5
+
+ ex e = indexed(A, i, j) * indexed(X, j);
+ cout << e.simplify_indexed() << endl;
+ // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
+
+ e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
+ cout << e.simplify_indexed() << endl;
+ // -> [[ [[3*y+3*x,6*y+2*x]] ]].j
+@}
+@end example
+
+You can of course obtain the same results with the @code{matrix::add()},
+@code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
+don't have to worry about transposing matrices.
+
+Matrix indices always start at 0 and their dimension must match the number
+of rows/columns of the matrix. Matrices with one row or one column are
+vectors and can have one or two indices (it doesn't matter whether it's a
+row or a column vector). Other matrices must have two indices.
+
+You should be careful when using indices with variance on matrices. GiNaC
+doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
+@samp{F.mu.nu} are different matrices. In this case you should use only
+one form for @samp{F} and explicitly multiply it with a matrix representation
+of the metric tensor.
+
+
+@node Methods and Functions, Information About Expressions, Indexed objects, Top
@c node-name, next, previous, up
@chapter Methods and Functions
@cindex polynomial
example:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- ex x = numeric(1.0);
-
- cout << "As method: " << sin(x).evalf() << endl;
- cout << "As function: " << evalf(sin(x)) << endl;
-@}
+ ...
+ cout << "As method: " << sin(1).evalf() << endl;
+ cout << "As function: " << evalf(sin(1)) << endl;
+ ...
@end example
@cindex @code{subs()}
@menu
* Information About Expressions::
-* Substituting Symbols::
+* Substituting Expressions::
* Polynomial Arithmetic:: Working with polynomials.
* Rational Expressions:: Working with rational functions.
* Symbolic Differentiation::
@end menu
-@node Information About Expressions, Substituting Symbols, Methods and Functions, Methods and Functions
+@node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
@c node-name, next, previous, up
@section Getting information about expressions
@subsection Checking expression types
@cindex @code{is_ex_of_type()}
+@cindex @code{ex_to_numeric()}
+@cindex @code{ex_to_@dots{}}
+@cindex @code{Converting ex to other classes}
@cindex @code{info()}
Sometimes it's useful to check whether a given expression is a plain number,
bool ex::info(unsigned flag);
@end example
+When the test made by @code{is_ex_of_type()} returns true, it is safe to
+call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
+one of the class names (@xref{The Class Hierarchy}, for a list of all
+classes). For example, assuming @code{e} is an @code{ex}:
+
+@example
+@{
+ @dots{}
+ if (is_ex_of_type(e, numeric))
+ numeric n = ex_to_numeric(e);
+ @dots{}
+@}
+@end example
+
@code{is_ex_of_type()} allows you to check whether the top-level object of
an expression @samp{e} is an instance of the GiNaC class @samp{t}
(@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
determines the number of subexpressions (@samp{operands}) contained, while
@code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
In the case of a @code{power} object, @code{op(0)} will return the basis
-and @code{op(1)} the exponent.
+and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
+is the base expression and @code{op(i)}, @math{i>0} are the indices.
The left-hand and right-hand side expressions of objects of class
@code{relational} (and only of these) can also be accessed with the methods
expressions will give very surprising results.
-@node Substituting Symbols, Polynomial Arithmetic, Information About Expressions, Methods and Functions
+@node Substituting Expressions, Polynomial Arithmetic, Information About Expressions, Methods and Functions
@c node-name, next, previous, up
-@section Substituting symbols
+@section Substituting expressions
@cindex @code{subs()}
-Symbols can be replaced with expressions via the @code{.subs()} method:
+Algebraic objects inside expressions can be replaced with arbitrary
+expressions via the @code{.subs()} method:
@example
ex ex::subs(const ex & e);
@end example
In the first form, @code{subs()} accepts a relational of the form
-@samp{symbol == expression} or a @code{lst} of such relationals. E.g.
+@samp{object == expression} or a @code{lst} of such relationals:
@example
@{
symbol x("x"), y("y");
+
ex e1 = 2*x^2-4*x+3;
cout << "e1(7) = " << e1.subs(x == 7) << endl;
+ // -> 73
+
ex e2 = x*y + x;
cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
+ // -> -10
@}
@end example
-will print @samp{73} and @samp{-10}, respectively.
+@code{subs()} performs syntactic substitution of any complete algebraic
+object; it does not try to match sub-expressions as is demonstrated by the
+following example:
+
+@example
+@{
+ symbol x("x"), y("y"), z("z");
+
+ ex e1 = pow(x+y, 2);
+ cout << e1.subs(x+y == 4) << endl;
+ // -> 16
+
+ ex e2 = sin(x)*cos(x);
+ cout << e2.subs(sin(x) == cos(x)) << endl;
+ // -> cos(x)^2
+
+ ex e3 = x+y+z;
+ cout << e3.subs(x+y == 4) << endl;
+ // -> x+y+z
+ // (and not 4+z as one might expect)
+@}
+@end example
If you specify multiple substitutions, they are performed in parallel, so e.g.
@code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
-The second form of @code{subs()} takes two lists, one for the symbols and
-one for the expressions to be substituted (both lists must contain the same
-number of elements). Using this form, you would write @code{subs(lst(x, y), lst(y, x))}
-to exchange @samp{x} and @samp{y}.
+The second form of @code{subs()} takes two lists, one for the objects to be
+replaced and one for the expressions to be substituted (both lists must
+contain the same number of elements). Using this form, you would write
+@code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
-@node Polynomial Arithmetic, Rational Expressions, Substituting Symbols, Methods and Functions
+@node Polynomial Arithmetic, Rational Expressions, Substituting Expressions, Methods and Functions
@c node-name, next, previous, up
@section Polynomial arithmetic
@code{collect()} accomplishes this task:
@example
-ex ex::collect(const symbol & s);
+ex ex::collect(const ex & s);
@end example
Note that the original polynomial needs to be in expanded form in order
methods
@example
-int ex::degree(const symbol & s);
-int ex::ldegree(const symbol & s);
+int ex::degree(const ex & s);
+int ex::ldegree(const ex & s);
@end example
which also work reliably on non-expanded input polynomials (they even work
a coefficient with a certain power from an expanded polynomial you use
@example
-ex ex::coeff(const symbol & s, int n);
+ex ex::coeff(const ex & s, int n);
@end example
You can also obtain the leading and trailing coefficients with the methods
@example
-ex ex::lcoeff(const symbol & s);
-ex ex::tcoeff(const symbol & s);
+ex ex::lcoeff(const ex & s);
+ex ex::tcoeff(const ex & s);
@end example
which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
or even from run to run since the internal canonical ordering is not
within the user's sphere of influence.
+@code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
+@code{tcoeff()} and @code{collect()} can also be used to a certain degree
+with non-polynomial expressions as they not only work with symbols but with
+constants, functions and indexed objects as well:
+
+@example
+@{
+ symbol a("a"), b("b"), c("c");
+ idx i(symbol("i"), 3);
+
+ ex e = pow(sin(x) - cos(x), 4);
+ cout << e.degree(cos(x)) << endl;
+ // -> 4
+ cout << e.expand().coeff(sin(x), 3) << endl;
+ // -> -4*cos(x)
+
+ e = indexed(a+b, i) * indexed(b+c, i);
+ e = e.expand(expand_options::expand_indexed);
+ cout << e.collect(indexed(b, i)) << endl;
+ // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
+@}
+@end example
+
@subsection Polynomial division
@cindex polynomial division
@end example
+@subsection Square-free decomposition
+@cindex square-free decomposition
+@cindex factorization
+@cindex @code{sqrfree()}
+
+GiNaC still lacks proper factorization support. Some form of
+factorization is, however, easily implemented by noting that factors
+appearing in a polynomial with power two or more also appear in the
+derivative and hence can easily be found by computing the GCD of the
+original polynomial and its derivatives. Any system has an interface
+for this so called square-free factorization. So we provide one, too:
+@example
+ex sqrfree(const ex & a, const lst & l = lst());
+@end example
+Here is an example that by the way illustrates how the result may depend
+on the order of differentiation:
+@example
+ ...
+ symbol x("x"), y("y");
+ ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
+
+ cout << sqrfree(BiVarPol, lst(x,y)) << endl;
+ // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
+
+ cout << sqrfree(BiVarPol, lst(y,x)) << endl;
+ // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
+
+ cout << sqrfree(BiVarPol) << endl;
+ // -> depending on luck, any of the above
+ ...
+@end example
+
+
@node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
@c node-name, next, previous, up
@section Rational expressions
@cindex simplification
@cindex temporary replacement
-Some basic from of simplification of expressions is called for frequently.
+Some basic form of simplification of expressions is called for frequently.
GiNaC provides the method @code{.normal()}, which converts a rational function
into an equivalent rational function of the form @samp{numerator/denominator}
where numerator and denominator are coprime. If the input expression is already
symbol x("x");
ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
- cout << "t1 is " << t1.normal() << endl;
- cout << "t2 is " << t2.normal() << endl;
+ std::cout << "t1 is " << t1.normal() << std::endl;
+ std::cout << "t2 is " << t2.normal() << std::endl;
@}
@end example
int main()
@{
for (unsigned i=0; i<11; i+=2)
- cout << EulerNumber(i) << endl;
+ std::cout << EulerNumber(i) << std::endl;
return 0;
@}
@end example
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
int main()
@{
+ using std::cout; // just for fun, another way of...
+ using std::endl; // ...dealing with this namespace std.
ex pi_frac;
for (int i=2; i<12; i+=2) @{
pi_frac = mechain_pi(i);
negative real axis where the points on the axis itself belong to the
upper part (i.e. continuous with quadrant II). The inverse
trigonometric and hyperbolic functions are not defined for complex
-arguments by the C++ standard, however. Here, we follow the conventions
-used by CLN, which in turn follow the carefully designed definitions
-in the Common Lisp standard. Hopefully, future revisions of the C++
+arguments by the C++ standard, however. In GiNaC we follow the
+conventions used by CLN, which in turn follow the carefully designed
+definitions in the Common Lisp standard. It should be noted that this
+convention is identical to the one used by the C99 standard and by most
+serious CAS. It is to be expected that future revisions of the C++
standard incorporate these functions in the complex domain in a manner
-compatible with Common Lisp.
+compatible with C99.
@node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
@{
symbol x("x");
ex e = 4.5+pow(x,2)*3/2;
- cout << e << endl; // prints '4.5+3/2*x^2'
+ cout << e << endl; // prints '(4.5)+3/2*x^2'
// ...
@end example
into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
is printed as @samp{x^2}).
-To print an expression in a way that can be directly used in a C or C++
-program, you use the method
+It is possible to print expressions in a number of different formats with
+the method
@example
-void ex::printcsrc(ostream & os, unsigned type, const char *name);
+void ex::print(const print_context & c, unsigned level = 0);
@end example
-This outputs a line in the form of a variable definition @code{<type> <name> = <expression>}.
-The possible types are defined in @file{ginac/flags.h} (@code{csrc_types})
-and mostly affect the way in which floating point numbers are written:
+The type of @code{print_context} object passed in determines the format
+of the output. The possible types are defined in @file{ginac/print.h}.
+All constructors of @code{print_context} and derived classes take an
+@code{ostream &} as their first argument.
+
+To print an expression in a way that can be directly used in a C or C++
+program, you pass a @code{print_csrc} object like this:
@example
// ...
- e.printcsrc(cout, csrc_types::ctype_float, "f");
- e.printcsrc(cout, csrc_types::ctype_double, "d");
- e.printcsrc(cout, csrc_types::ctype_cl_N, "n");
+ cout << "float f = ";
+ e.print(print_csrc_float(cout));
+ cout << ";\n";
+
+ cout << "double d = ";
+ e.print(print_csrc_double(cout));
+ cout << ";\n";
+
+ cout << "cl_N n = ";
+ e.print(print_csrc_cl_N(cout));
+ cout << ";\n";
// ...
@end example
+The three possible types mostly affect the way in which floating point
+numbers are written.
+
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 = (cl_F("3.0")/cl_F("2.0"))*(x*x)+cl_F("4.5");
+cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
@end example
-Finally, there are the two methods @code{printraw()} and @code{printtree()} intended for GiNaC
-developers, that provide a dump of the internal structure of an expression for
-debugging purposes:
+The @code{print_context} type @code{print_tree} provdes a dump of the
+internal structure of an expression for debugging purposes:
@example
// ...
- e.printraw(cout); cout << endl << endl;
- e.printtree(cout);
+ e.print(print_tree(cout));
@}
@end example
produces
@example
-ex(+((power(ex(symbol(name=x,serial=1,hash=150875740,flags=11)),ex(numeric(2)),hash=2,flags=3),numeric(3/2)),,hash=0,flags=3))
-
-type=Q25GiNaC3add, hash=0 (0x0), flags=3, nops=2
- power: hash=2 (0x2), flags=3
- x (symbol): serial=1, hash=150875740 (0x8fe2e5c), flags=11
- 2 (numeric): hash=2147483714 (0x80000042), flags=11
- 3/2 (numeric): hash=2147483745 (0x80000061), flags=11
+add, hash=0x0, flags=0x3, nops=2
+ power, hash=0x9, flags=0x3, nops=2
+ x (symbol), serial=3, hash=0x44a113a6, flags=0xf
+ 2 (numeric), hash=0x80000042, flags=0xf
+ 3/2 (numeric), hash=0x80000061, flags=0xf
-----
overall_coeff
- 4.5L0 (numeric): hash=2147483723 (0x8000004b), flags=11
+ 4.5L0 (numeric), hash=0x8000004b, flags=0xf
=====
@end example
-The @code{printtree()} method is also available in @command{ginsh} as the
-@code{print()} function.
+This kind of output is also available in @command{ginsh} as the @code{print()}
+function.
+
+If you need any fancy special output format, e.g. for interfacing GiNaC
+with other algebra systems or for producing code for different
+programming languages, you can always traverse the expression tree yourself:
+
+@example
+static void my_print(const ex & e)
+@{
+ if (is_ex_of_type(e, function))
+ cout << ex_to_function(e).get_name();
+ else
+ cout << e.bp->class_name();
+ cout << "(";
+ unsigned n = e.nops();
+ if (n)
+ for (unsigned i=0; i<n; i++) @{
+ my_print(e.op(i));
+ if (i != n-1)
+ cout << ",";
+ @}
+ else
+ cout << e;
+ cout << ")";
+@}
+
+int main(void)
+@{
+ my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
+ return 0;
+@}
+@end example
+
+This will produce
+
+@example
+add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
+symbol(y))),numeric(-2)))
+@end example
+
+If you need an output format that makes it possible to accurately
+reconstruct an expression by feeding the output to a suitable parser or
+object factory, you should consider storing the expression in an
+@code{archive} object and reading the object properties from there.
+See the section on archiving for more information.
@subsection Expression input
#include <string>
#include <stdexcept>
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
expression a unique name:
@example
-#include <ginac/ginac.h>
#include <fstream>
+using namespace std;
+#include <ginac/ginac.h>
using namespace GiNaC;
int main()
different symbol than the @code{x} which was defined at the beginning of
the program, altough both would appear as @samp{x} when printed.
+You can also use the information stored in an @code{archive} object to
+output expressions in a format suitable for exact reconstruction. The
+@code{archive} and @code{archive_node} classes have a couple of member
+functions that let you access the stored properties:
+
+@example
+static void my_print2(const archive_node & n)
+@{
+ string class_name;
+ n.find_string("class", class_name);
+ cout << class_name << "(";
+
+ archive_node::propinfovector p;
+ n.get_properties(p);
+
+ unsigned num = p.size();
+ for (unsigned i=0; i<num; i++) @{
+ const string &name = p[i].name;
+ if (name == "class")
+ continue;
+ cout << name << "=";
+
+ unsigned count = p[i].count;
+ if (count > 1)
+ cout << "@{";
+
+ for (unsigned j=0; j<count; j++) @{
+ switch (p[i].type) @{
+ case archive_node::PTYPE_BOOL: @{
+ bool x;
+ n.find_bool(name, x);
+ cout << (x ? "true" : "false");
+ break;
+ @}
+ case archive_node::PTYPE_UNSIGNED: @{
+ unsigned x;
+ n.find_unsigned(name, x);
+ cout << x;
+ break;
+ @}
+ case archive_node::PTYPE_STRING: @{
+ string x;
+ n.find_string(name, x);
+ cout << '\"' << x << '\"';
+ break;
+ @}
+ case archive_node::PTYPE_NODE: @{
+ const archive_node &x = n.find_ex_node(name, j);
+ my_print2(x);
+ break;
+ @}
+ @}
+
+ if (j != count-1)
+ cout << ",";
+ @}
+
+ if (count > 1)
+ cout << "@}";
+
+ if (i != num-1)
+ cout << ",";
+ @}
+
+ cout << ")";
+@}
+
+int main(void)
+@{
+ ex e = pow(2, x) - y;
+ archive ar(e, "e");
+ my_print2(ar.get_top_node(0)); cout << endl;
+ return 0;
+@}
+@end example
+
+This will produce:
+
+@example
+add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
+symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
+overall_coeff=numeric(number="0"))
+@end example
+
+Be warned, however, that the set of properties and their meaning for each
+class may change between GiNaC versions.
@node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
@menu
* What does not belong into GiNaC:: What to avoid.
* Symbolic functions:: Implementing symbolic functions.
+* Adding classes:: Defining new algebraic classes.
@end menu
provided by @acronym{CLN} are much better suited.
-@node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
+@node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
@c node-name, next, previous, up
@section Symbolic functions
implement does have a pole somewhere in the complex plane, you need to
write another method for Laurent expansion around that point.
-Now that all the ingrediences for @code{cos} have been set up, we need
+Now that all the ingredients for @code{cos} have been set up, we need
to tell the system about it. This is done by a macro and we are not
going to descibe how it expands, please consult your preprocessor if you
are curious:
assure you that functions are GiNaC's most macro-intense classes. We
have done our best to avoid macros where we can.)
+
+@node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
+@c node-name, next, previous, up
+@section Adding classes
+
+If you are doing some very specialized things with GiNaC you may find that
+you have to implement your own algebraic classes to fit your needs. This
+section will explain how to do this by giving the example of a simple
+'string' class. After reading this section you will know how to properly
+declare a GiNaC class and what the minimum required member functions are
+that you have to implement. We only cover the implementation of a 'leaf'
+class here (i.e. one that doesn't contain subexpressions). Creating a
+container class like, for example, a class representing tensor products is
+more involved but this section should give you enough information so you can
+consult the source to GiNaC's predefined classes if you want to implement
+something more complicated.
+
+@subsection GiNaC's run-time type information system
+
+@cindex hierarchy of classes
+@cindex RTTI
+All algebraic classes (that is, all classes that can appear in expressions)
+in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
+@code{basic *} (which is essentially what an @code{ex} is) represents a
+generic pointer to an algebraic class. Occasionally it is necessary to find
+out what the class of an object pointed to by a @code{basic *} really is.
+Also, for the unarchiving of expressions it must be possible to find the
+@code{unarchive()} function of a class given the class name (as a string). A
+system that provides this kind of information is called a run-time type
+information (RTTI) system. The C++ language provides such a thing (see the
+standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
+implements its own, simpler RTTI.
+
+The RTTI in GiNaC is based on two mechanisms:
+
+@itemize @bullet
+
+@item
+The @code{basic} class declares a member variable @code{tinfo_key} which
+holds an unsigned integer that identifies the object's class. These numbers
+are defined in the @file{tinfos.h} header file for the built-in GiNaC
+classes. They all start with @code{TINFO_}.
+
+@item
+By means of some clever tricks with static members, GiNaC maintains a list
+of information for all classes derived from @code{basic}. The information
+available includes the class names, the @code{tinfo_key}s, and pointers
+to the unarchiving functions. This class registry is defined in the
+@file{registrar.h} header file.
+
+@end itemize
+
+The disadvantage of this proprietary RTTI implementation is that there's
+a little more to do when implementing new classes (C++'s RTTI works more
+or less automatic) but don't worry, most of the work is simplified by
+macros.
+
+@subsection A minimalistic example
+
+Now we will start implementing a new class @code{mystring} that allows
+placing character strings in algebraic expressions (this is not very useful,
+but it's just an example). This class will be a direct subclass of
+@code{basic}. You can use this sample implementation as a starting point
+for your own classes.
+
+The code snippets given here assume that you have included some header files
+as follows:
+
+@example
+#include <iostream>
+#include <string>
+#include <stdexcept>
+using namespace std;
+
+#include <ginac/ginac.h>
+using namespace GiNaC;
+@end example
+
+The first thing we have to do is to define a @code{tinfo_key} for our new
+class. This can be any arbitrary unsigned number that is not already taken
+by one of the existing classes but it's better to come up with something
+that is unlikely to clash with keys that might be added in the future. The
+numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
+which is not a requirement but we are going to stick with this scheme:
+
+@example
+const unsigned TINFO_mystring = 0x42420001U;
+@end example
+
+Now we can write down the class declaration. The class stores a C++
+@code{string} and the user shall be able to construct a @code{mystring}
+object from a C or C++ string:
+
+@example
+class mystring : public basic
+@{
+ GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
+
+public:
+ mystring(const string &s);
+ mystring(const char *s);
+
+private:
+ string str;
+@};
+
+GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
+@end example
+
+The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
+macros are defined in @file{registrar.h}. They take the name of the class
+and its direct superclass as arguments and insert all required declarations
+for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
+the first line after the opening brace of the class definition. The
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
+source (at global scope, of course, not inside a function).
+
+@code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
+declarations of the default and copy constructor, the destructor, the
+assignment operator and a couple of other functions that are required. It
+also defines a type @code{inherited} which refers to the superclass so you
+don't have to modify your code every time you shuffle around the class
+hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
+constructor, the destructor and the assignment operator.
+
+Now there are nine member functions we have to implement to get a working
+class:
+
+@itemize
+
+@item
+@code{mystring()}, the default constructor.
+
+@item
+@code{void destroy(bool call_parent)}, which is used in the destructor and the
+assignment operator to free dynamically allocated members. The @code{call_parent}
+specifies whether the @code{destroy()} function of the superclass is to be
+called also.
+
+@item
+@code{void copy(const mystring &other)}, which is used in the copy constructor
+and assignment operator to copy the member variables over from another
+object of the same class.
+
+@item
+@code{void archive(archive_node &n)}, the archiving function. This stores all
+information needed to reconstruct an object of this class inside an
+@code{archive_node}.
+
+@item
+@code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
+constructor. This constructs an instance of the class from the information
+found in an @code{archive_node}.
+
+@item
+@code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
+unarchiving function. It constructs a new instance by calling the unarchiving
+constructor.
+
+@item
+@code{int compare_same_type(const basic &other)}, which is used internally
+by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
+-1, depending on the relative order of this object and the @code{other}
+object. If it returns 0, the objects are considered equal.
+@strong{Note:} This has nothing to do with the (numeric) ordering
+relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
+for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
+may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
+must provide a @code{compare_same_type()} function, even those representing
+objects for which no reasonable algebraic ordering relationship can be
+defined.
+
+@item
+And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
+which are the two constructors we declared.
+
+@end itemize
+
+Let's proceed step-by-step. The default constructor looks like this:
+
+@example
+mystring::mystring() : inherited(TINFO_mystring)
+@{
+ // dynamically allocate resources here if required
+@}
+@end example
+
+The golden rule is that in all constructors you have to set the
+@code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
+it will be set by the constructor of the superclass and all hell will break
+loose in the RTTI. For your convenience, the @code{basic} class provides
+a constructor that takes a @code{tinfo_key} value, which we are using here
+(remember that in our case @code{inherited = basic}). If the superclass
+didn't have such a constructor, we would have to set the @code{tinfo_key}
+to the right value manually.
+
+In the default constructor you should set all other member variables to
+reasonable default values (we don't need that here since our @code{str}
+member gets set to an empty string automatically). The constructor(s) are of
+course also the right place to allocate any dynamic resources you require.
+
+Next, the @code{destroy()} function:
+
+@example
+void mystring::destroy(bool call_parent)
+@{
+ // free dynamically allocated resources here if required
+ if (call_parent)
+ inherited::destroy(call_parent);
+@}
+@end example
+
+This function is where we free all dynamically allocated resources. We don't
+have any so we're not doing anything here, but if we had, for example, used
+a C-style @code{char *} to store our string, this would be the place to
+@code{delete[]} the string storage. If @code{call_parent} is true, we have
+to call the @code{destroy()} function of the superclass after we're done
+(to mimic C++'s automatic invocation of superclass destructors where
+@code{destroy()} is called from outside a destructor).
+
+The @code{copy()} function just copies over the member variables from
+another object:
+
+@example
+void mystring::copy(const mystring &other)
+@{
+ inherited::copy(other);
+ str = other.str;
+@}
+@end example
+
+We can simply overwrite the member variables here. There's no need to worry
+about dynamically allocated storage. The assignment operator (which is
+automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
+recall) calls @code{destroy()} before it calls @code{copy()}. You have to
+explicitly call the @code{copy()} function of the superclass here so
+all the member variables will get copied.
+
+Next are the three functions for archiving. You have to implement them even
+if you don't plan to use archives, but the minimum required implementation
+is really simple. First, the archiving function:
+
+@example
+void mystring::archive(archive_node &n) const
+@{
+ inherited::archive(n);
+ n.add_string("string", str);
+@}
+@end example
+
+The only thing that is really required is calling the @code{archive()}
+function of the superclass. Optionally, you can store all information you
+deem necessary for representing the object into the passed
+@code{archive_node}. We are just storing our string here. For more
+information on how the archiving works, consult the @file{archive.h} header
+file.
+
+The unarchiving constructor is basically the inverse of the archiving
+function:
+
+@example
+mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+@{
+ n.find_string("string", str);
+@}
+@end example
+
+If you don't need archiving, just leave this function empty (but you must
+invoke the unarchiving constructor of the superclass). Note that we don't
+have to set the @code{tinfo_key} here because it is done automatically
+by the unarchiving constructor of the @code{basic} class.
+
+Finally, the unarchiving function:
+
+@example
+ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
+@{
+ return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
+@}
+@end example
+
+You don't have to understand how exactly this works. Just copy these four
+lines into your code literally (replacing the class name, of course). It
+calls the unarchiving constructor of the class and unless you are doing
+something very special (like matching @code{archive_node}s to global
+objects) you don't need a different implementation. For those who are
+interested: setting the @code{dynallocated} flag puts the object under
+the control of GiNaC's garbage collection. It will get deleted automatically
+once it is no longer referenced.
+
+Our @code{compare_same_type()} function uses a provided function to compare
+the string members:
+
+@example
+int mystring::compare_same_type(const basic &other) const
+@{
+ const mystring &o = static_cast<const mystring &>(other);
+ int cmpval = str.compare(o.str);
+ if (cmpval == 0)
+ return 0;
+ else if (cmpval < 0)
+ return -1;
+ else
+ return 1;
+@}
+@end example
+
+Although this function takes a @code{basic &}, it will always be a reference
+to an object of exactly the same class (objects of different classes are not
+comparable), so the cast is safe. If this function returns 0, the two objects
+are considered equal (in the sense that @math{A-B=0}), so you should compare
+all relevant member variables.
+
+Now the only thing missing is our two new constructors:
+
+@example
+mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
+@{
+ // dynamically allocate resources here if required
+@}
+
+mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
+@{
+ // dynamically allocate resources here if required
+@}
+@end example
+
+No surprises here. We set the @code{str} member from the argument and
+remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
+
+That's it! We now have a minimal working GiNaC class that can store
+strings in algebraic expressions. Let's confirm that the RTTI works:
+
+@example
+ex e = mystring("Hello, world!");
+cout << is_ex_of_type(e, mystring) << endl;
+ // -> 1 (true)
+
+cout << e.bp->class_name() << endl;
+ // -> mystring
+@end example
+
+Obviously it does. Let's see what the expression @code{e} looks like:
+
+@example
+cout << e << endl;
+ // -> [mystring object]
+@end example
+
+Hm, not exactly what we expect, but of course the @code{mystring} class
+doesn't yet know how to print itself. This is done in the @code{print()}
+member function. Let's say that we wanted to print the string surrounded
+by double quotes:
+
+@example
+class mystring : public basic
+@{
+ ...
+public:
+ void print(const print_context &c, unsigned level = 0) const;
+ ...
+@};
+
+void mystring::print(const print_context &c, unsigned level) const
+@{
+ // print_context::s is a reference to an ostream
+ c.s << '\"' << str << '\"';
+@}
+@end example
+
+The @code{level} argument is only required for container classes to
+correctly parenthesize the output. Let's try again to print the expression:
+
+@example
+cout << e << endl;
+ // -> "Hello, world!"
+@end example
+
+Much better. The @code{mystring} class can be used in arbitrary expressions:
+
+@example
+e += mystring("GiNaC rulez");
+cout << e << endl;
+ // -> "GiNaC rulez"+"Hello, world!"
+@end example
+
+(note that GiNaC's automatic term reordering is in effect here), or even
+
+@example
+e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
+cout << e << endl;
+ // -> "One string"^(2*sin(-"Another string"+Pi))
+@end example
+
+Whether this makes sense is debatable but remember that this is only an
+example. At least it allows you to implement your own symbolic algorithms
+for your objects.
+
+Note that GiNaC's algebraic rules remain unchanged:
+
+@example
+e = mystring("Wow") * mystring("Wow");
+cout << e << endl;
+ // -> "Wow"^2
+
+e = pow(mystring("First")-mystring("Second"), 2);
+cout << e.expand() << endl;
+ // -> -2*"First"*"Second"+"First"^2+"Second"^2
+@end example
+
+There's no way to, for example, make GiNaC's @code{add} class perform string
+concatenation. You would have to implement this yourself.
+
+@subsection Automatic evaluation
+
+@cindex @code{hold()}
+@cindex evaluation
+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.
+This is done in the evaluation member function @code{eval()}. Let's say that
+we wanted all strings automatically converted to lowercase with
+non-alphabetic characters stripped, and empty strings removed:
+
+@example
+class mystring : public basic
+@{
+ ...
+public:
+ ex eval(int level = 0) const;
+ ...
+@};
+
+ex mystring::eval(int level) const
+@{
+ string new_str;
+ for (int i=0; i<str.length(); i++) @{
+ char c = str[i];
+ if (c >= 'A' && c <= 'Z')
+ new_str += tolower(c);
+ else if (c >= 'a' && c <= 'z')
+ new_str += c;
+ @}
+
+ if (new_str.length() == 0)
+ return _ex0();
+ else
+ return mystring(new_str).hold();
+@}
+@end example
+
+The @code{level} argument is used to limit the recursion depth of the
+evaluation. We don't have any subexpressions in the @code{mystring} class
+so we are not concerned with this. If we had, we would call the @code{eval()}
+functions of the subexpressions with @code{level - 1} as the argument if
+@code{level != 1}. The @code{hold()} member function sets a flag in the
+object that prevents further evaluation. Otherwise we might end up in an
+endless loop. When you want to return the object unmodified, use
+@code{return this->hold();}.
+
+Let's confirm that it works:
+
+@example
+ex e = mystring("Hello, world!") + mystring("!?#");
+cout << e << endl;
+ // -> "helloworld"
+
+e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
+cout << e << endl;
+ // -> 3*"wow"
+@end example
+
+@subsection Other member functions
+
+We have implemented only a small set of member functions to make the class
+work in the GiNaC framework. For a real algebraic class, there are probably
+some more functions that you will want to re-implement, such as
+@code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
+or the header file of the class you want to make a subclass of to see
+what's there. You can, of course, also add your own new member functions.
+In this case you will probably want to define a little helper function like
+
+@example
+inline const mystring &ex_to_mystring(const ex &e)
+@{
+ return static_cast<const mystring &>(*e.bp);
+@}
+@end example
+
+that let's you get at the object inside an expression (after you have verified
+that the type is correct) so you can call member functions that are specific
+to the class.
+
That's it. May the source be with you!
-@node A Comparison With Other CAS, Advantages, Symbolic functions, Top
+@node A Comparison With Other CAS, Advantages, Adding classes, Top
@c node-name, next, previous, up
@chapter A Comparison With Other CAS
@cindex advocacy
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@example
#include <ginac/ginac.h>
-using namespace GiNaC;
int main(void)
@{
- symbol x("x");
- ex a = sin(x);
- cout << "Derivative of " << a << " is " << a.diff(x) << endl;
+ GiNaC::symbol x("x");
+ GiNaC::ex a = GiNaC::sin(x);
+ std::cout << "Derivative of " << a
+ << " is " << a.diff(x) << std::endl;
return 0;
@}
@end example
AC_PROG_INSTALL
AC_LANG_CPLUSPLUS
-AM_PATH_GINAC(0.4.0, [
+AM_PATH_GINAC(0.7.0, [
LIBS="$LIBS $GINACLIB_LIBS"
- CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
+ CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
], AC_MSG_ERROR([need to have GiNaC installed]))
AC_OUTPUT(Makefile)
The only command in this which is not standard for automake
is the @samp{AM_PATH_GINAC} macro.
-That command does the following:
-
-@display
-If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
-@env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
-with the error message `need to have GiNaC installed'
-@end display
+That command does the following: If a GiNaC version greater or equal
+than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
+and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
+the error message `need to have GiNaC installed'
And the @file{Makefile.am}, which will be used to build the Makefile.
@printindex cp
@bye
-
git://www.ginac.de / ginac.git / blobdiff