SymbolicC++

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.

#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 ${\displaystyle {\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}}$ 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 ${\displaystyle |0\rangle }$. 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.^[1][2][3][4]

SymbolicC++ is described in a series of books on computer algebra. The first book^[5] 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^[6] 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.^[7] Subsequently, support for Gröbner bases was added.^[8] The third version^[4] 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.

• Comparison of computer algebra systems
• GiNaC

• Official website
• Programming exercises in SymbolicC++