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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 ( cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ) {\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 | 0 ⟩ {\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.¹²³⁴

History

SymbolicC++ is described in a series of books on computer algebra. The first book⁵ 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 version⁶ 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 version⁹ 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

External links

• Official website
• Programming exercises in SymbolicC++

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. 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 /wiki/Joan_Lasenby ↩

8. Kruger, P.J.M (2003). Gröbner bases with Symbolic C++, M. Sc. Dissertation, Rand Afrikaans University. ↩

9. Hardy, Y, Tan Kiat Shi and Steeb, W.-H. (2008). Computer Algebra with SymbolicC++, World Scientific Publishing, Singapore. ↩