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__NOTOC__
== Introduction ==
=== Scope ===
In this section, a focus is realised on the following interpolation methods: spline, bicubic, tricubic, Lagrange and Newton, covariance matrix and linear in 1D, 2D or 3D interpolation.

=== Javadoc ===
The interpolation objects are available in the package <code>fr.cnes.sirius.patrius.math.analysis.interpolation</code> and in the package <code>fr.cnes.sirius.patrius.propagation.analytical.covariance</code>.

{| class="wikitable"
|-
! scope="col"| Library
! scope="col"| Javadoc
|-
| Patrius
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/package-summary.html Package fr.cnes.sirius.patrius.math.analysis.interpolation]
|-
| Patrius
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/package-summary.html Package fr.cnes.sirius.patrius.math.analysis.interpolation]
|-
| Patrius
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/propagation/analytical/covariance/package-summary.html Package fr.cnes.sirius.patrius.propagation.analytical.covariance]
|}

=== Links ===
None as of now.

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

=== Package Overview ===
The package <code>fr.cnes.sirius.patrius.math.analysis.interpolation</code> contains all the interpolation classes described in this section.

[[File:PATRIMOINESIRIUSSUMDiagInterpolation.png]]

== Features Description ==
=== Spline interpolation ===
The '''''spline interpolator''''' generates an interpolating function <math>f(x): \mathbb{R} \rightarrow \mathbb{R}</math>. The user gives as entries 2 sets of values, the values of x, y. The interpolator gives the function f such as <math>y=f(x)</math>.

For the linear equation <math>y=2x+1</math>

<syntaxhighlight lang="java">
double x[] = { 0.0, 1.0, 2.0 };
double y[] = { 1.0, 3.0, 5.0 };

UnivariateInterpolator interpolator = new SplineInterpolator();
UnivariateFunction function = interpolator.interpolate(x, y);
double value = function .value(0.5);
</syntaxhighlight>

=== Bicubic interpolation ===
The '''''bicubic interpolator''''' generates an interpolating function <math>f(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}</math>. The interpolator computes internally the coefficients of the bicubic function that is the interpolating function. The user gives as entries 3 sets of values, the values of x, y and z. The interpolator gives the function f such as <math>z=f(x,y)</math>.

For the equation of the plane <math>z=2x-3y + 5</math>

<syntaxhighlight lang="java">
double x[] = { 3, 4, 5, 6.5 };
double y[] = {-4, -3, -1, 2, 2.5 };
double z[][] = {{ 23, 20, 14, 5, 3.5 },
{ 25, 22, 16, 7, 5.5 },
{ 27, 24, 18, 9, 7.5 },
{ 30, 27, 21, 12, 10.5 }};
BivariateGridInterpolator interpolator = new BicubicSplineInterpolator();
BivariateFunction function = interpolator.interpolate(x, y, z);
</syntaxhighlight>

=== Tricubic interpolation ===
The '''''tricubic interpolator''''' generates an interpolating function <math>f(x,y,z): \mathbb{R}^3 \rightarrow \mathbb{R}</math>. The interpolator computes internally the coefficients of the tricubic function that is the interpolating function. The user gives as entries 4 sets of values, the values of x, y, z and w. The interpolator gives the function f such as <math>w=f(x,y,z)</math>.

For the equation of the plane <math>w=2x- 3y - z + 5</math>

<syntaxhighlight lang="java">
double x[] = { 3.0, 4.0, 5.0, 6.5 };
double y[] = {-4.0, -3.0, -1.0, 2.0, 2.5 };
double z[] = {-12.0, -8.0, -5.5, -3.0, 0.0, 2.5 };
double w[][][] = {{{ 35, 31, 28.5, 26, 23, 20.5 },
{ 32, 28, 25.5, 23, 20, 17.5 },
{ 26, 22, 19.5, 17, 14, 11.5 },
{ 17, 13, 10.5, 8, 5, 2.5 },
{ 15.5, 11.5, 9, 6.5, 3.5, 1 }},
{{ 37, 33, 30.5, 28, 25, 22.5 },
{ 34, 30, 27.5, 25, 22, 19.5 },
{ 28, 24, 21.5, 19, 16, 13.5 },
{ 19, 15, 12.5, 10, 7, 4.5 },
{ 17.5, 13.5, 11, 8.5, 5.5, 3 }},
{{ 39, 35, 32.5, 30, 27, 24.5 },
{ 36, 32, 39.5, 27, 24, 21.5 },
{ 30, 26, 23.5, 21, 18, 15.5 },
{ 21, 17, 14.5, 12, 9, 6.5 },
{ 19.5, 15.5, 13, 10.5, 7.5, 5 }},
{{ 42, 38, 35.5, 33, 30, 27.5 },
{ 39, 35, 32.5, 30, 27, 24.5 },
{ 33, 29, 26.5, 24, 21, 18.5 },
{ 24, 20, 17.5, 15, 12, 9.5 },
{ 22.5, 18.5, 16, 13.5, 10.5, 8 }}};

TrivariateGridInterpolator interpolator = new TricubicSplineInterpolator();
TrivariateFunction function = interpolator.interpolate(x, y, z, w);
</syntaxhighlight>

=== Lagrange interpolation ===
The '''''Lagrange interpolator''''' generates an interpolating function <math>f(x): \mathbb{R} \rightarrow \mathbb{R}</math>. The user gives as entries 2 sets of values, the values of x, y. The interpolator gives the function f such as <math>y=f(x)</math>.

For the linear equation <math>y=2x+1</math>

<syntaxhighlight lang="java">
double x[] = { 0.0, 1.0, 2.0 };
double y[] = { 1.0, 3.0, 5.0 };

UnivariateFunction interpolator = new PolynomialFunctionLagrangeForm(x,y);
double value = interpolator.value(0.5);
</syntaxhighlight>

=== Newton interpolation ===
The '''''Newton interpolator''''' generates an interpolating function <math>f(x): \mathbb{R} \rightarrow \mathbb{R}</math>. The user gives as entries 2 sets of values, the coefficients <math>c_i</math>and the centers <math>x_i</math>such as the polynomial function <math>P(x)=c_0 + c_1 (x - x_0) + ... + c_n (x - x_n)</math>. The interpolator gives the function f such as <math>y=P(x)</math>.

For the linear equation <math>y=2x+1</math>

<syntaxhighlight lang="java">
double c_i[] = { 3.0, 2.0 };
double x_i[] = { 1.0 };

UnivariateFunction interpolator = new PolynomialFunctionNewtonForm(c_i,x_i);
double value = interpolator.value(0.5);
</syntaxhighlight>

=== Covariance matrix interpolation ===
The purpose of this interpolation algorithm is to compute the covariance matrix at a given date through a simplified model of the transition matrix. When a covariance in PV coordinates is searched for an object orbiting around an celestial body, a simple dynamical model can be used, meaning limited to the newtonian attraction, plus a constant acceleration. The value of this constant acceleration will not change the transition matrix.

The transition matrix between a date <math>t_1</math> and a date <math>t</math> can be approximated :

* at order 0 : by <math>\phi_1(t_1, t) = I_{3 \times 3}</math>

* at order 1 : by <math>\phi_1(t_1, t) = I_{3 \times 3} + J_{PV} ( t- t_1)</math>

* at order 2 : by<math>\phi_1(t_1, t) =I_{3 \times 3} + J_{PV} ( t- t_1)+ 0.5 * J_{PV}^2 ( t - t_1)^2</math>

where <math>J_{PV} = \left(\begin{array}{cc} 0_{3 \times 3} & I_{3 \times 3} \\ A & 0_{3 \times 3} \end{array} \right)</math>, <math>J_{PV}^2 = \left(\begin{array}{cc} A & 0_{3 \times 3} \\ 0_{3 \times 3} & A \end{array} \right)</math> and <math>A =- \frac{ GM}{r^3}\left(I_{3 \times 3} - 3 \frac{ PP^T}{r^2}\right)</math>, where <math>A</math> is considered as a constant on the interval <math>[t_1,t]</math> and <math>P</math> is the satellite position vector.

We denote by <math>M(t)</math> the covariance matrix at instant t. Let <math>t \in [t_1,t]</math> . The transition matrices <math>\phi_1(t_1, t)</math> and <math>\phi_2(t_2, t)</math> are given by the above formula, and since matrix <math>A</math> is constant on <math>[t_1,t_2]</math>, we have that the covariance matrix at instant <math>t</math> is given by
<math>M(t) = (1- \alpha) \phi_1(t_1, t) M(t_1)\phi_1^T(t_1, t) + \alpha \phi_2(t_2, t) M(t_2)\phi_2^T(t_2, t),</math>
with <math>\alpha = \frac{t-t_1}{t_2-t_1}</math>.

=== Linear interpolation ===
These classes allow linear piecewise interpolations in dimensions 1, 2 or 3.

==== 1D interpolation ====

Let <math>f</math> be a real function <math>\mathbb{R} \rightarrow \mathbb{R}</math> and <math>[x_1,x_2]</math> the interpolation interval, where <math>f(x_1),f(x_2)</math> are known. For all <math>x \in [x_1,x_2]</math>, the interpolated value <math>f(x)</math> is given by
<math>f(x) = f(x_1) + (x-x_1) \frac{f(x_2)- f(x_1)}{x_2-x_1}.</math>

==== 2D interpolation ====
The two dimensional interpolation will be two successive 1D interpolations.
Let <math>f</math> be a real function <math>\mathbb{R}^2 \rightarrow \mathbb{R}</math> and <math>[x_1,x_2] \times [y_1,y_2]</math> the interpolation interval.
First, a 1D interpolation in the <math>y</math> direction is made, leading to
<math>f(x,y_1) = f(x_1,y_1) + (y-y_1) \frac{f(x_2,y_1)- f(x_1,y_1)}{y_2-y_1},</math>

<math>f(x,y_2) = f(x_1,y_2) + (y-y_1) \frac{f(x_2,y_2)- f(x_1,y_2)}{y_2-y_1}.</math>

Then a second 1D interpolation is made in the <math>x</math> direction with the previous two interpolated values :
<math>f(x,y) = f(x,y_1) + (x-x_1) \frac{f(x,y_2)- f(x, y_1)}{x_2-x_1}.</math>

==== 3D interpolation ====

Let <math>f</math> be a real function <math>\mathbb{R}^3 \rightarrow \mathbb{R}</math> and <math>[x_1,x_2] \times [y_1,y_2] \times [z_1,z_2]</math> the interpolation interval. There will be <math>2^3- 1</math> successives 1D interpolations.

<math>f(x,y,z)</math> is interpolated from <math>f(x,y,z_1)</math> and <math>f(x,y,z_2)</math>.

<math>f(x,y,z_1)</math> is interpolated from <math>f(x,y_1,z_1)</math> and <math>f(x,y_2,z_1)</math>.

<math>f(x,y,z_2)</math> is interpolated from <math>f(x,y_1,z_2)</math> and <math>f(x,y_2,z_2)</math>.

<math>f(x,y_1,z_1)</math> is interpolated from <math>f(x_1,y_1,z_1)</math> and <math>f(x_2,y_1,z_1)</math>.

<math>f(x,y_2,z_1)</math> is interpolated from <math>f(x_1,y_2,z_1)</math> and <math>f(x_2,y_2,z_1)</math>.

<math>f(x,y_1,z_2)</math> is interpolated from <math>f(x_1,y_1,z_2)</math> and <math>f(x_2,y_1,z_1)</math>.

<math>f(x,y_2,z_2)</math> is interpolated from <math>f(x_1,y_2,z_2)</math> and <math>f(x_2,y_2,z_2)</math>.

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

== Contents ==
=== Interfaces ===
The library defines the following interfaces related to interpolation :

{| class="wikitable"
|-
! scope="col"| Interface
! scope="col"| Summary
! scope="col"| Javadoc
|-
|'''UnivariateInterpolator'''
|Interface for a univariate interpolating function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/UnivariateInterpolator.html ...]
|-
|'''BivariateGridInterpolator'''
|Interface for a bivariate interpolating function where the sample points must be specified on a regular grid.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/BivariateGridInterpolator.html ...]
|-
|'''TrivariateGridInterpolator'''
|Interface for a trivariate interpolating function where the sample points must be specified on a regular grid.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/TrivariateGridInterpolator.html ...]
|-
|'''UnivariateFunction'''
|Interface for a univariate function
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/UnivariateFunction.html ...]
|}

=== Classes ===
This section is about the following classes related to interpolation :

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Summary
! scope="col"| Javadoc
|-
|'''SplineInterpolator'''
|Spline interpolator for a univariate real function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/SplineInterpolator.html ...]
|-
|'''BicubicSplineInterpolator'''
|Bicubic spline interpolator for a bivariate real function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/BicubicSplineInterpolator.html ...]
|-
|'''TricubicSplineInterpolator'''
|Tricubic spline interpolator for a trivariate real function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/TricubicSplineInterpolator.html ...]
|-
|'''PolynomialFunctionLagrangeForm'''
|Lagrange interpolator, directly usable as a univariate real function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/polynomials/PolynomialFunctionLagrangeForm.html ...]
|-
|'''PolynomialFunctionNewtonForm'''
|Newton interpolator, directly usable as a univariate real function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/polynomials/PolynomialFunctionNewtonForm.html ...]
|}

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Summary
! scope="col"| Javadoc
|-
|'''CovarianceInterpolation'''
|Interpolator of a covariance matrix based on its two surrounding covariance matrices.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/propagation/analytical/covariance/CovarianceInterpolation.html ...]
|-
|'''OrbitCovariance'''
|Class containing a covariance matrix and its associated AbsoluteDate. New class replacing older class CovarianceMatrix
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/propagation/analytical/covariance/OrbitCovariance.html ...]
|}

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Summary
! scope="col"| Javadoc
|-
|'''CovarianceInterpolation'''
|Interpolator of a covariance matrix based on its two surrounding covariance matrices.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/propagation/analytical/covariance/CovarianceInterpolation.html ...]
|-
|'''OrbitCovariance'''
|Class containing a covariance matrix and its associated AbsoluteDate. New class replacing older class CovarianceMatrix
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/propagation/analytical/covariance/OrbitCovariance.html ...]
|}

{| class="wikitable"
|-
! scope="col"| Class
! scope="col"| Summary
! scope="col"| Javadoc
|-
|'''AbstractLinearIntervalsFunction'''
|Abstract class for linear interpolations.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/AbstractLinearIntervalsFunction.html ...]
|-
|'''UniLinearIntervalsFunction'''
|Linear one-dimensional function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/UniLinearIntervalsFunction.html ...]
|-
|'''BiLinearIntervalsFunction'''
|Linear two-dimensional function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/BiLinearIntervalsFunction.html ...]
|-
|'''TriLinearIntervalsFunction'''
|Linear three-dimensional function.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/TriLinearIntervalsFunction.html ...]
|-
|'''UniLinearIntervalsInterpolator'''
|Interpolator of linear one-dimensional functions.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/UniLinearIntervalsInterpolator.html ...]
|-
|'''BiLinearIntervalsInterpolator'''
|Interpolator of linear two-dimensional functions.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/BiLinearIntervalsInterpolator.html ...]
|-
|'''TriLinearIntervalsInterpolator'''
|Interpolator of linear three-dimensional functions.
|[{{JavaDoc4.4}}/fr/cnes/sirius/patrius/math/analysis/interpolation/TriLinearIntervalsInterpolator.html ...]
|}

[[Category:User_Manual_4.4_Mathematics]]

User Manual 4.4 Interpolation Methods - Historique des versions

Admin : Page créée avec « __NOTOC__ == Introduction == === Scope === In this section, a focus is realised on the following interpolation methods: spline, bicubic, tricubic, Lagrange and Newton, co... »

Admin : Page créée avec « NOTOC == Introduction == === Scope === In this section, a focus is realised on the following interpolation methods: spline, bicubic, tricubic, Lagrange and Newton, co... »