GiNaC is Not a CAS

GiNaC is a C++ library. It has been developed to become a replacement engine for xloops which in the past was powered by the Maple CAS. Its design is revolutionary in a sense that contrary to other CAS it does not try to provide extensive algebraic capabilities and a simple programming language but instead accepts a given language (C++) and extends it by a set of algebraic capabilities. The name GiNaC is an iterated and recursive abbreviation for GiNaC is Not a CAS, where CAS stands for Computer Algebra System.

As a little example that shows some syntactical candy possible with GiNaC, here is a complete C++-program that uses the defining equation for (normalized) Laguerre Polynomials L_n(z)==e^z(d/dz)ⁿ(zⁿe^-z)/n! and the associated Laguerre Polynomials L_n,m(z)==(d/dz)^mL_n(z) to compute the first few Laguerre polynomials in the symbolic variable x. Of course it is not optimized for efficiency. A faster version would not use Rodrigues' definition with the derivative. Instead, this version is optimized for elegance:

#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

/* Laguerre Polynomial L_n(z) == exp(z)*(d/dz)^n (z^n*exp(-z)) / n! */
ex LaguerrePoly(int n, const symbol & z)
{
    const ex LKer = exp(-z);
    return normal(diff(pow(z, n)*LKer, z, n) / LKer) / factorial(n);
}

/* Associated Laguerre Polynomial L_{n,m}(x) == (d/dx)^m L_n(x). */
ex LaguerrePoly(int n, int m, const symbol & z)
{
    return LaguerrePoly(n, z).diff(z, m);
}

int main()
{
    symbol x("x");
    
    for (int i=0; i<7; ++i) {
        cout << "    L_" << i << "(" << x << ") == "
             << LaguerrePoly(i, x) << endl;
        for (int j=1; j<=i; ++j) {
            cout << "L_{" << i << ',' << j << "}(" << x << ") == "
                 << LaguerrePoly(i, j, x) << endl;
        }
        cout << endl;
    }
    
    return 0;
}

When you compile this program and run it, it will type out:

    L_0(x) == 1

    L_1(x) == 1-x
L_{1,1}(x) == -1

    L_2(x) == 1-2*x+1/2*x^2
L_{2,1}(x) == -2+x
L_{2,2}(x) == 1

    L_3(x) == 1-3*x+3/2*x^2-1/6*x^3
L_{3,1}(x) == -3+3*x-1/2*x^2
L_{3,2}(x) == 3-x
L_{3,3}(x) == -1

    L_4(x) == 1-4*x+3*x^2-2/3*x^3+1/24*x^4
L_{4,1}(x) == -4+6*x-2*x^2+1/6*x^3
L_{4,2}(x) == 6-4*x+1/2*x^2
L_{4,3}(x) == -4+x
L_{4,4}(x) == 1

    L_5(x) == 1-5*x+5*x^2-5/3*x^3+5/24*x^4-1/120*x^5
L_{5,1}(x) == -5+10*x-5*x^2+5/6*x^3-1/24*x^4
L_{5,2}(x) == 10-10*x+5/2*x^2-1/6*x^3
L_{5,3}(x) == -10+5*x-1/2*x^2
L_{5,4}(x) == 5-x
L_{5,5}(x) == -1

    L_6(x) == 1-6*x+15/2*x^2-10/3*x^3+5/8*x^4-1/20*x^5+1/720*x^6
L_{6,1}(x) == -6+15*x-10*x^2+5/2*x^3-1/4*x^4+1/120*x^5
L_{6,2}(x) == 15-20*x+15/2*x^2-x^3+1/24*x^4
L_{6,3}(x) == -20+15*x-3*x^2+1/6*x^3
L_{6,4}(x) == 15-6*x+1/2*x^2
L_{6,5}(x) == -6+x
L_{6,6}(x) == 1

Currently, GiNaC's capabilities include:

Very fast manipulation of large integers and rationals owing to the CLN library (for instance, it uses Karatsuba multiplication and Schönhage-Strassen multiplication for very large integers).
Efficient handling of multivariate polynomials and rational functions.
Support for linear algebra includes symbolic matrices, vectors and solving equations.
Very fast heuristic polynomial GCD.
Many built in functions (sin, cos, atan, sinh, factorial, etc.)
Symbolic differentiation and series expansion of all built-in functions.
Several forms of output (also as optimized C++, for numerical postprocessing).
Memory-efficiency and -safety through the internal use of reference counting for all expressions.

The capabilites still missing (but would be nice to see implemented) are listed in the ToDo List.

For a more in-depth description of GiNaC we refer to the tutorial which contains many more examples.

About GiNaC