Asymptotic theory (statistics)

2013-11-12T16:59:25Z

50.169.3.181: /* Asymptotic theorems */

{{Infobox Software
| name = SymbolicC++
| logo =
| screenshot =
| caption =
| developer = Yorick Hardy, Willi-Hans Steeb and Tan Kiat Shi
| latest_release_version = 3.35
| latest_release_date = {{release date and age|2010|09|15}}
| programming language = [[C++]]
| operating_system = [[Cross-platform]]
| genre = [[Mathematical software]]
| license = [[GNU General Public License|GPL]]
| website = 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 ==
<source lang="cpp">
#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;
}
</source>

The following program fragment inverts the matrix
<math>
\begin{pmatrix}
\cos\theta & \sin\theta\\
-\sin\theta & \cos\theta
\end{pmatrix}
</math>
symbolically.

<source lang="cpp">
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) ];
</source>

The output is

<pre>
[ cos(theta) −sin(theta) ]
[ sin(theta) cos(theta) ]
</pre>

The next program illustrates non-commutative symbols in SymbolicC++. Here <code>b</code> is a Bose [[annihilation operator]] and <code>bd</code> is a Bose [[creation operator]]. The variable <code>vs</code> denotes the [[vacuum state]] <math>|0\rangle</math>. The <code>~</code> operator toggles the commutativity of a variable, i.e. if <code>b</code> is commutative that <code>~b</code> is non-commutative and if <code>b</code> is non-commutative <code>~b</code> is commutative.

<source lang="cpp">
#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;
}
</source>

Further examples can be found in the books listed below.<ref>
Steeb, W.-H. (2010).
''Quantum Mechanics Using Computer Algebra, second edition,''
World Scientific Publishing, Singapore.
</ref><ref>
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.
</ref><ref>
Steeb, W.-H. (2007).
''Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, second edition,''
World Scientific Publishing, Singapore.
</ref><ref name="symcpp3"/>

== History ==
SymbolicC++ is described in a series of books on [[computer algebra]]. The first book<ref>Tan Kiat Shi and Steeb, W.-H. (1997). ''SymbolicC++: An introduction to Computer Algebra Using Object-Oriented Programming'' Springer-Verlag, Singapore.</ref> described the first version of SymbolicC++. In this version the main data type for symbolic computation was the <code>Sum</code> class. The list of available classes included

* <code>Verylong</code> : An unbounded [[integer]] implementation
* <code>Rational</code> : A template class for [[rational number]]s
* <code>Quaternion</code> : A template class for [[quaternion]]s
* <code>Derive</code> : A template class for [[automatic differentiation]]
* <code>Vector</code> : A template class for vectors (see [[vector space]])
* <code>Matrix</code> : A template class for matrices (see [[matrix (mathematics)]])
* <code>Sum</code> : A template class for symbolic expressions

Example:
<source lang="cpp">
#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;
}
</source>

The second version<ref>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.</ref> of SymbolicC++ featured new classes such as the <code>Polynomial</code> class and initial support for simple integration. Support for the algebraic computation of [[Clifford algebras]] was described in using SymbolicC++ in 2002.<ref>Fletcher, J.P. (2002). Symbolic Processing of Clifford Numbers in C++ <br>in Doran C., Dorst L. and Lasenby J. (eds.) ''Applied Geometrical Algebras in computer Science and Engineering AGACSE 2001'', Birkhauser, Basel.
<br>http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper25/index.php</ref> Subsequently support for Gröbner bases was added.<ref>Kruger, P.J.M (2003). ''Gröbner bases with Symbolic C++'', M. Sc. Dissertation, Rand Afrikaans University.</ref>
The third version<ref name="symcpp3">Hardy, Y, Tan Kiat Shi and Steeb, W.-H. (2008). ''Computer Algebra with SymbolicC++'', World Scientific Publishing, Singapore.</ref> features a complete rewrite of SymbolicC++ and was released in 2008. This version encapsulates all symbolic expressions in the <code>Symbolic</code> class.

Newer versions are available from the SymbolicC++ [http://issc.uj.ac.za/symbolic/symbolic.html website].

==See also==
*[[Comparison of computer algebra systems]]
*[[GiNaC]]

== References ==

{{Reflist}}

== External links ==
* {{Official website|http://issc.uj.ac.za/symbolic/symbolic.html}}
* [http://issc.uj.ac.za/downloads/problems/advancedP.pdf Programming exercises in SymbolicC++]

{{Computer algebra systems}}

{{DEFAULTSORT:Symbolicc}}
[[Category:Free computer algebra systems]]
[[Category:Free software programmed in C++]]
[[Category:C++ libraries]]

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Asymptotic theory (statistics)