SymbolicC++

<h2 id="examples">Examples</h2>
#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 <a href="/facts/Matrix_inversion/fPqXk3V8">inverts</a> 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 <a href="/facts/Annihilation_operator/CwxD5TnL">annihilation operator</a> and bd is a Bose <a href="/facts/Creation_operator/CwxD5TnL">creation operator</a>. The variable vs denotes the <a href="/facts/Vacuum_state/dTvXYPad">vacuum state</a> 
 
 
 
 
 |
 
 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.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="history">History</h2>
SymbolicC++ is described in a series of books on <a href="/facts/Computer_algebra/hjun1KAX">computer algebra</a>. The first book<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> 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

<ul><li>Verylong  : An unbounded <a href="/facts/Integer/PUwwrawx">integer</a> implementation</li>
<li>Rational  : A template class for <a href="/facts/Rational_number/VJQmyY05">rational numbers</a></li>
<li>Quaternion : A template class for <a href="/facts/Quaternion/T3tnu7mX">quaternions</a></li>
<li>Derive  : A template class for <a href="/facts/Automatic_differentiation/dQcdhBAM">automatic differentiation</a></li>
<li>Vector  : A template class for vectors (see <a href="/facts/Vector_space/M1pxMLx2">vector space</a>)</li>
<li>Matrix  : A template class for matrices (see <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix (mathematics)</a>)</li>
<li>Sum  : A template class for symbolic expressions</li></ul>
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<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> of SymbolicC++ featured new classes such as the Polynomial class and initial support for simple integration. Support for the algebraic computation of <a href="/facts/Clifford_algebras/qnqSCz5h">Clifford algebras</a> was described in using SymbolicC++ in 2002.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Subsequently, support for Gröbner bases was added.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
The third version<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> 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++ <a href="http://issc.uj.ac.za/symbolic/symbolic.html">website</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Comparison_of_computer_algebra_systems/e5m8nyK2">Comparison of computer algebra systems</a></li>
<li><a href="/facts/GiNaC/P81a6eNs">GiNaC</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://issc.uj.ac.za/symbolic/symbolic.html">Official website</a></li>
<li><a href="http://issc.uj.ac.za/downloads/problems/advancedP.pdf">Programming exercises in SymbolicC++</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Steeb, W.-H. (2010).
Quantum Mechanics Using Computer Algebra, second edition,
World Scientific Publishing, Singapore. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Steeb, W.-H. (2007).
Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, second edition,
World Scientific Publishing, Singapore. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hardy, Y, Tan Kiat Shi and Steeb, W.-H. (2008). Computer Algebra with SymbolicC++, World Scientific Publishing, Singapore. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Tan Kiat Shi and Steeb, W.-H. (1997). SymbolicC++: An introduction to Computer Algebra Using Object-Oriented Programming Springer-Verlag, Singapore. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="/wiki/Joan_Lasenby" target="_blank">/wiki/Joan_Lasenby</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kruger, P.J.M (2003). Gröbner bases with Symbolic C++, M. Sc. Dissertation, Rand Afrikaans University. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hardy, Y, Tan Kiat Shi and Steeb, W.-H. (2008). Computer Algebra with SymbolicC++, World Scientific Publishing, Singapore. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

SymbolicC++ open-in-new

SymbolicC++