From Wikipedia, the free encyclopedia
Jump to: navigation, search
Developer(s) Yorick Hardy, Willi-Hans Steeb and Tan Kiat Shi
Stable release 3.35 / September 15, 2010; 4 years ago (2010-09-15)
Written in C++
Operating system Cross-platform
Type Mathematical software
License GPL

SymbolicC++ is a general purpose computer algebra system embedded 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 
 \cos\theta & \sin\theta\\
-\sin\theta & \cos\theta

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 |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


#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
 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.

See also[edit]


  1. ^ Steeb, W.-H. (2010). Quantum Mechanics Using Computer Algebra, second edition, World Scientific Publishing, Singapore.
  2. ^ Steeb, W.-H. (2008). The Nonlinear Workbook: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithm, Gene Expression Programming, Wavelets, Fuzzy Logic with C++, Java and SymbolicC++ Programs, fourth edition, World Scientific Publishing, Singapore.
  3. ^ Steeb, W.-H. (2007). Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, second edition, World Scientific Publishing, Singapore.
  4. ^ a b Hardy, Y, Tan Kiat Shi and Steeb, W.-H. (2008). Computer Algebra with SymbolicC++, World Scientific Publishing, Singapore.
  5. ^ Tan Kiat Shi and Steeb, W.-H. (1997). SymbolicC++: An introduction to Computer Algebra Using Object-Oriented Programming Springer-Verlag, Singapore.
  6. ^ Tan Kiat Shi, Steeb, W.-H. and Hardy, Y (2000). SymbolicC++: An Introduction to Computer Algebra using Object-Oriented Programming, 2nd extended and revised edition, Springer-Verlag, London.
  7. ^ Fletcher, J.P. (2002). Symbolic Processing of Clifford Numbers in C++
    in Doran C., Dorst L. and Lasenby J. (eds.) Applied Geometrical Algebras in computer Science and Engineering AGACSE 2001, Birkhauser, Basel.
  8. ^ Kruger, P.J.M (2003). Gröbner bases with Symbolic C++, M. Sc. Dissertation, Rand Afrikaans University.

External links[edit]