Voir la source de User Manual 4.1 Numerical differentiation and integration

__NOTOC__
== Introduction ==
=== Scope ===
This section détails numerical differentialtion and integration (not to be misunderstood with numerical integration of ODE). A focus is realised on:
* differentiation methods of real univariate functions: Ridders and finite difference.
* integration methods of real univariate functions : Trapezoidal and Simpson.

=== Javadoc ===
The numerical differentiator objects are available in the package <code>fr.cnes.sirius.patrius.math.analysis.differentiation</code>.
The numerical integrator objects are available in the package <code>fr.cnes.sirius.patrius.math.analysis.integration</code>.

{| class="wikitable"
|-
! scope="col"| Library
! scope="col"| Javadoc
|-
|Patrius
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/differentiation/package-summary.html Package fr.cnes.sirius.patrius.math.analysis.differentiation]
|-
|Patrius
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/integration/package-summary.html Package fr.cnes.sirius.patrius.math.analysis.integration]
|}

=== Links ===
Links to the implemented integration methods :

http://mathworld.wolfram.com/TrapezoidalRule.html

http://mathworld.wolfram.com/SimpsonsRule.html

=== Useful Documents ===
None as of now.

=== Package Overview ===
The numerical differentiation conception is described hereafter : 

[[File:differentiators.png|center]]

Accuracy of integration method (all examples are based on sinus function):

{| class="wikitable"
|-
! scope="col"| Default setting
! scope="col"| Value
|-
|Maximum absolute error
|1.0e-15
|-
|Maximum relative error
|1.0e-6
|-
|Maximum number of iterations
|64
|-
|Minimum number of iterations
|3
|}

Note : the accuracy of the results and the computing time are directly dependent on the value of maximum relative error.

== Features Description ==

=== Numerical differentiation ===

==== Finite difference ====
Finite difference is the discrete analog of the derivative. The user can choose the number of points to use and the step size (the gap between each point).

==== Ridders algorithm ====
The derivative of a function at a point x is computed using the Ridders method of polynomial extrapolation.

=== Numerical Integration ===
=== Trapezoidal method ===
The trapezoidal rule works by approximating the region under the graph of the function as a trapezoid and calculating its area.

<syntaxhighlight lang="java">
UnivariateFunction f = new SinFunction();
UnivariateIntegrator integrator = new TrapezoidIntegrator();
double r = integrator.integrate(10000, f, 0, FastMath.PI);
</syntaxhighlight>

Result : r = 1.9999996078171345 (exact result = 2)

And :

<syntaxhighlight lang="java">
integrator = new TrapezoidIntegrator(1.e-12, 1.0e-15, 3, 64);
r = integrator.integrate(10000000, f, 0, FastMath.PI);
</syntaxhighlight>

Result : r = 1.9999999999904077

But :
<syntaxhighlight lang="java">integrator = new TrapezoidIntegrator(1.e-13, 1.0e-15, 3, 64);
r = integrator.integrate(10000000, f, 0, FastMath.PI);</syntaxhighlight>

Result : r = no result (infinite time !)

=== Simpson method ===
Simpson's rule is a Newton-Cotes formula for approximating the integral of a function using quadratic polynomials (i.e., parabolic arcs instead of the straight line segments used in the trapezoidal rule).
In general, this method has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases.

<syntaxhighlight lang="java">
UnivariateFunction f = new SinFunction();
UnivariateIntegrator integrator = new SimpsonIntegrator();
double r = integrator.integrate(10000, f, 0, FastMath.PI);
</syntaxhighlight>

Result : r = 2.000000064530001 (exact result = 2)

== Getting Started ==
{{specialInclusion prefix=$theme_sub section="GettingStarted"/}}

== Contents ==
=== Interfaces ===
{| class="wikitable"
|-
! scope="col"| Interface
! scope="col"| Summary
! scope="col"| Javadoc
|-
|'''UnivariateFunctionDifferentiator'''
|This interface represents a generic numerical differentiator.
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/differentiation/UnivariateFunctionDifferentiator.html ...]
|-
|UnivariateIntegrator
|Interface for univariate integration algorithms.
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/integration/UnivariateIntegrator.html ...]
|}

=== Classes ===
{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Summary
! scope="col"| Javadoc
|-
|'''FiniteDifferencesDifferentiator'''
|Apply the finite difference method to differentiate a function.
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/differentiation/FiniteDifferencesDifferentiator.html ...]
|-
|'''RiddersDifferentiator'''
|Apply the Ridders algorithm to differentiate a function.
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/differentiation/RiddersDifferentiator.html ...]
|-
|TrapezoidIntegrator
|The class implements the Trapezoidal rule
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/integration/TrapezoidIntegrator.html ...]
|-
|SimpsonIntegrator
|The class implements the Simpson rule
|[{{JavaDoc4.1}}/fr/cnes/sirius/patrius/math/analysis/integration/SimpsonIntegrator.html ...]
|}

[[Category:User_Manual_4.1_Mathematics]]