User Manual 4.0 Interpolation Methods

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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 fr.cnes.sirius.patrius.math.analysis.interpolation and in the package fr.cnes.sirius.patrius.propagation.analytical.covariance. |=Library|=Javadoc | Patrius|Package fr.cnes.sirius.patrius.math.analysis.interpolation | Patrius|Package fr.cnes.sirius.patrius.math.analysis.interpolation | Patrius|Package fr.cnes.sirius.patrius.propagation.analytical.covariance

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Package Overview

The package

fr.cnes.sirius.patrius.math
.analysis.interpolation

contains all the interpolation classes described in this section.

Features Description

Spline interpolation

Thespline 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]

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);

Bicubic interpolation

Thebicubic 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]

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);

Tricubic interpolation

Thetricubic 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]

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);

Lagrange interpolation

TheLagrange 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]

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);

Newton interpolation

TheNewton 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]

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);

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].

Contents

Interfaces

The library defines the following interfaces related to interpolation :

|=Interface|=Summary|=Javadoc |UnivariateInterpolator|Interface for a univariate interpolating function.|... |BivariateGridInterpolator|Interface for a bivariate interpolating function where the sample points must be specified on a regular grid.|... |TrivariateGridInterpolator|Interface for a trivariate interpolating function where the sample points must be specified on a regular grid.|... |UnivariateFunction|Interface for a univariate function|...

Classes

This section is about the following classes related to interpolation :

|=Class|=Summary|=Javadoc |SplineInterpolator|Spline interpolator for a univariate real function.|... |BicubicSplineInterpolator|Bicubic spline interpolator for a bivariate real function.|... |TricubicSplineInterpolator|Tricubic spline interpolator for a trivariate real function.|... |PolynomialFunctionLagrangeForm|Lagrange interpolator, directly usable as a univariate real function.|... |PolynomialFunctionNewtonForm|Newton interpolator, directly usable as a univariate real function.|...

|=Class|=Summary|=Javadoc |CovarianceInterpolation|Interpolator of a covariance matrix based on its two surrounding covariance matrices.|... |OrbitCovariance|Class containing a covariance matrix and its associated AbsoluteDate. New class replacing older class CovarianceMatrix|...

|=Class|=Summary|=Javadoc |AbstractLinearIntervalsFunction|Abstract class for linear interpolations.|... |UniLinearIntervalsFunction|Linear one-dimensional function.|... |BiLinearIntervalsFunction|Linear two-dimensional function.|... |TriLinearIntervalsFunction|Linear three-dimensional function.|... |UniLinearIntervalsInterpolator|Interpolator of linear one-dimensional functions.|... |BiLinearIntervalsInterpolator|Interpolator of linear two-dimensional functions.|... |TriLinearIntervalsInterpolator|Interpolator of linear three-dimensional functions.|...