User Manual 3.4.1 Interpolation Methods : Différence entre versions

Version actuelle en date du 2 mars 2018 à 16:35

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 org.apache.commons.math3.analysis.interpolation both in the Commons Math library and the Commons Math Addons library and in the package org.orekit.propagation.analytical.covariance in the Orekit library.

Links

None as of now.

Useful Documents

None as of now.

Package Overview

The package org.apache.commons.math3.analysis.interpolation contains all the interpolation classes described in this section.

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]

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

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]

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

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]

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

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]

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

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]

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].

Getting Started

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Contents

Interfaces

The library defines the following interfaces related to interpolation :

Classes

This section is about the following classes related to interpolation :

In Commons Math

In Orekit

In Commons Math Addons