Asymptotic theory (statistics)

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 ${\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}$ 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 $|0\rangle$. 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.

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.

See also

• Comparison of computer algebra systems
• GiNaC

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Template:Official website
• Programming exercises in SymbolicC++