# SymbolicC++

Developer(s) Yorick Hardy, Willi-Hans Steeb and Tan Kiat Shi 3.35 / September 15, 2010; 6 years ago C++ Cross-platform Mathematical software GPL http://issc.uj.ac.za/symbolic/symbolic.html

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.

## 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 ${\displaystyle {\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}}$ 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]

## History

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.

## References

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.
http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper25/index.php
8. ^ Kruger, P.J.M (2003). Gröbner bases with Symbolic C++, M. Sc. Dissertation, Rand Afrikaans University.