User Manual 3.3 Numerical differentiation and integration - Historique des versions

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2018-02-27T11:03:45Z

Contents

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 == Contents ==
 === Interfaces ===
-|=(% colspan="3" %)Interface|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
+{| class="wikitable"
-|(% colspan="3" %)'''UnivariateFunctionDifferentiator'''|(% colspan="6" %)This interface represents a generic numerical differentiator.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/UnivariateFunctionDifferentiator.html ...]
+|-
-|(% colspan="3" %)UnivariateIntegrator|(% colspan="6" %)Interface for univariate integration algorithms.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/UnivariateIntegrator.html ...]
+! scope="col"| Interface
 === Classes ===
-|=(% colspan="3" %)Class|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
+{| class="wikitable"
-|(% colspan="3" %)'''FiniteDifferencesDifferentiator'''|(% colspan="6" %)Apply the finite difference method to differentiate a function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html ...]
+|-
-|(% colspan="3" %)'''RiddersDifferentiator'''|(% colspan="6" %)Apply the Ridders algorithm to differentiate a function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/RiddersDifferentiator.html ...]
+! scope="col"| Class
-|(% colspan="3" %)TrapezoidIntegrator|(% colspan="6" %)The class implements the Trapezoidal rule|(% colspan="1" %)[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/TrapezoidIntegrator.html ...]
+! scope="col"| Summary
-|(% colspan="3" %)SimpsonIntegrator|(% colspan="6" %)The class implements the Simpson rule|(% colspan="1" %)[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/SimpsonIntegrator.html ...]
+! scope="col"| Javadoc

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Ridders algorithm

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	==== Ridders algorithm ====		==== Ridders algorithm ====
	~~{{specialInclusion prefix=$theme_sub section="~~Ridders ~~algorithm"/}}~~		The derivative of a function at a point x is computed using the Ridders method of polynomial extrapolation.

	=== Numerical Integration ===		=== Numerical Integration ===

Admin : /* Finite difference */

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Finite difference

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	==== Finite difference ====		==== Finite difference ====
	~~{{specialInclusion prefix=$theme_sub section="~~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 ====		==== Ridders algorithm ====

Admin : /* Package Overview */

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Package Overview

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	The numerical differentiation conception is described hereafter :		The numerical differentiation conception is described hereafter :

	[[File:differentiators.~~bmp~~\|center]]		[[File:differentiators.png\|center]]

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

	\|=~~(% colspan~~="3" %)Default setting~~\|=(% colspan~~="3" %)Value		{\| class="wikitable"
	\|~~(% colspan="3" %)~~Maximum absolute error\|~~(% colspan="6" %)~~1.0e-15		\|-
	\|~~(% colspan="3" %)~~Maximum relative error\|~~(% colspan="6" %)~~1.0e-6		! scope="col"\| Default setting
	\|~~(% colspan="3" %)~~Maximum number of iterations\|~~(% colspan="6" %)~~64		! scope="col"\| Value
	\|~~(% colspan="3" %)~~Minimum number of iterations\|~~(% colspan="6" %)~~3		\|-
			\|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.		Note : the accuracy of the results and the computing time are directly dependent on the value of maximum relative error.

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Javadoc

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	The numerical integrator objects are available in the package <code>org.apache.commons.math3.analysis.integration</code> in the Commons Math Library.		The numerical integrator objects are available in the package <code>org.apache.commons.math3.analysis.integration</code> in the Commons Math Library.

	\|=~~(% colspan~~="3" %)Library~~\|=(% colspan~~="6" %)Javadoc		{\| class="wikitable"
	\|~~(% colspan="3" %)~~Commons Math\|~~(% colspan="6" %)~~[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/package-summary.html Package org.apache.commons.math3.analysis.differentiation]		\|-
	\|~~(% colspan="3" %)~~Commons Math\|~~(% colspan="6" %)~~[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/package-summary.html Package org.apache.commons.math3.analysis.integration]		! scope="col"\| Library
			! scope="col"\| Javadoc
			\|-
			\|Commons Math
			\|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/package-summary.html Package org.apache.commons.math3.analysis.differentiation]
			\|-
			\|Commons Math
			\|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/package-summary.html Package org.apache.commons.math3.analysis.integration]
			\|}

	=== Links ===		=== Links ===

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Nouvelle page

== 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>org.apache.commons.math3.analysis.differentiation</code> in the Commons Math Library.
The numerical integrator objects are available in the package <code>org.apache.commons.math3.analysis.integration</code> in the Commons Math Library.

|=(% colspan="3" %)Library|=(% colspan="6" %)Javadoc
|(% colspan="3" %)Commons Math|(% colspan="6" %)[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/package-summary.html Package org.apache.commons.math3.analysis.differentiation]
|(% colspan="3" %)Commons Math|(% colspan="6" %)[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/package-summary.html Package org.apache.commons.math3.analysis.integration]

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

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

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

=== Useful Documents ===
{{specialInclusion prefix=$theme_sub section="UsefulDocs"/}}

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

[[File:differentiators.bmp|center]]

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

|=(% colspan="3" %)Default setting|=(% colspan="3" %)Value
|(% colspan="3" %)Maximum absolute error|(% colspan="6" %)1.0e-15
|(% colspan="3" %)Maximum relative error|(% colspan="6" %)1.0e-6
|(% colspan="3" %)Maximum number of iterations|(% colspan="6" %)64
|(% colspan="3" %)Minimum number of iterations|(% colspan="6" %)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 ====
{{specialInclusion prefix=$theme_sub section="Finite difference"/}}

==== Ridders algorithm ====
{{specialInclusion prefix=$theme_sub section="Ridders algorithm"/}}

=== 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 ===
|=(% colspan="3" %)Interface|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
|(% colspan="3" %)'''UnivariateFunctionDifferentiator'''|(% colspan="6" %)This interface represents a generic numerical differentiator.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/UnivariateFunctionDifferentiator.html ...]
|(% colspan="3" %)UnivariateIntegrator|(% colspan="6" %)Interface for univariate integration algorithms.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/UnivariateIntegrator.html ...]

=== Classes ===
|=(% colspan="3" %)Class|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
|(% colspan="3" %)'''FiniteDifferencesDifferentiator'''|(% colspan="6" %)Apply the finite difference method to differentiate a function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html ...]
|(% colspan="3" %)'''RiddersDifferentiator'''|(% colspan="6" %)Apply the Ridders algorithm to differentiate a function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/differentiation/RiddersDifferentiator.html ...]
|(% colspan="3" %)TrapezoidIntegrator|(% colspan="6" %)The class implements the Trapezoidal rule|(% colspan="1" %)[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/TrapezoidIntegrator.html ...]
|(% colspan="3" %)SimpsonIntegrator|(% colspan="6" %)The class implements the Simpson rule|(% colspan="1" %)[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/integration/SimpsonIntegrator.html ...]

User Manual 3.3 Numerical differentiation and integration - Historique des versions

Admin : /* Contents */

Admin : /* Ridders algorithm */

Admin : /* Finite difference */

Admin : /* Package Overview */

Admin : /* Javadoc */

Admin : Admin a déplacé la page Numerical differentiation and integration vers User Manual 3.3 Numerical differentiation and integration sans laisser de redirection

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