SymbolicC++ is a general purpose computer algebra system written in the programming language C++. It is free software released under the terms of the GNU General Public License. SymbolicC++ is used by including a C++ header file or by linking against a library.
Examples
#include <iostream>
#include "symbolicc++.h" using namespace std;
int main(void) { Symbolic x("x"); cout << integrate(x+1, x); // => 1/2*x^(2)+x Symbolic y("y"); cout << df(y, x); // => 0 cout << df(y[x], x); // => df(y[x],x) cout << df(exp(cos(y[x])), x); // => -sin(y[x])*df(y[x],x)*e^cos(y[x]) return 0; }
The following program fragment inverts the matrix
symbolically.
Symbolic theta("theta"); Symbolic R = ( ( cos(theta), sin(theta) ), ( -sin(theta), cos(theta) ) ); cout << R(0,1); // sin(theta) Symbolic RI = R.inverse(); cout << RI[ (cos(theta)^2) == 1 - (sin(theta)^2) ];
The output is
[ cos(theta) −sin(theta) ]
[ sin(theta) cos(theta) ]
The next program illustrates non-commutative symbols in SymbolicC++. Here b is a Bose annihilation operator and bd is a Bose creation operator. The variable vs denotes the vacuum state . The ~ operator toggles the commutativity of a variable, i.e. if b is commutative that ~b is non-commutative and if b is non-commutative ~b is commutative.
#include <iostream>
#include "symbolicc++.h" using namespace std;
int main(void) { // The operator b is the annihilation operator and bd is the creation operator Symbolic b("b"), bd("bd"), vs("vs");
b = ~b; bd = ~bd; vs = ~vs;
Equations rules = (b*bd == bd*b + 1, b*vs == 0);
// Example 1
Symbolic result1 = b*bd*b*bd; cout << "result1 = " << result1.subst_all(rules) << endl; cout << "result1*vs = " << (result1*vs).subst_all(rules) << endl;
// Example 2
Symbolic result2 = (b+bd)^4; cout << "result2 = " << result2.subst_all(rules) << endl; cout << "result2*vs = " << (result2*vs).subst_all(rules) << endl;
return 0; }
Further examples can be found in the books listed below.
History
SymbolicC++ is described in a series of books on computer algebra. The first book described the first version of SymbolicC++. In this version the main data type for symbolic computation was the Sum class. The list of available classes included
Verylong : An unbounded integer implementation
Rational : A template class for rational numbers
Quaternion : A template class for quaternions
Derive : A template class for automatic differentiation
Vector : A template class for vectors (see vector space) Matrix : A template class for matrices (see matrix (mathematics)) Sum : A template class for symbolic expressions
Example: #include <iostream> #include "rational.h" #include "msymbol.h" using namespace std;
int main(void) { Sum<int> x("x",1); Sum<Rational<int> > y("y",1); cout << Int(y, y); // => 1/2 yˆ2 y.depend(x); cout << df(y, x); // => df(y,x) return 0; }
The second version of SymbolicC++ featured new classes such as the Polynomial class and initial support for simple integration. Support for the algebraic computation of Clifford algebras was described in using SymbolicC++ in 2002. Subsequently, support for Gröbner bases was added. The third version features a complete rewrite of SymbolicC++ and was released in 2008. This version encapsulates all symbolic expressions in the Symbolic class.
Newer versions are available from the SymbolicC++ website.