CodeCogs

Approximates an arbitrary function using orthogonal polynomials.

You're viewing an older version of this page (#1007). View the current version.

Interface

#include <codecogs/maths/approximation/regression/orthogonal.h>

using namespace Maths::Approximation::Regression;

Orthogonal(int n, double *x, double *y, int degree)
~Orthogonal()
double getValue(double x)
static double prod(int n, double *x, double *y, double *z)
double Orthogonal_once(int n, double *x, double *y, int degree, double a )

Overview

This class approximates an arbitrary discrete function by least squares fitting orthogonal polynomials. In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative <em> weight function </em> w precisely if

$\int_{x_1}^{x_2} f(x) g(x) w(x) \mathrm{d}x = 0$

(1)

In other words, if polynomials are treated as vectors and the inner product of two polynomials $f$ and $g$ is defined as

$<f, g> = \int_{x_1}^{x_2} f(x) g(x) w(x) \mathrm{d}x$

(2)

then the orthogonal polynomials are simply orthogonal vectors in this inner product space. The algorithm uses this class of polynomials to achieve the best curve fitting by minimization in terms of least squares.

Below you will find the regression graph for a set of points obtained by evaluating the function $f(x) = \sin(x) + x$ , displayed in light blue, at particular abscissas. The regression polynomial, displayed in red, has been calculated using this class. The root mean squared error is also displayed.

As a general rule, the degree parameter should always be strictly less than the the total number of sample points.

References

Jean-Pierre Moreau's Home Page, http://perso.wanadoo.fr/jean-pierre.moreau/
Tuan Dang Trong, "Numath Library" in Fortran 77

Example 1

The following example displays 10 approximated values (you may change this amount through the N_out variable) for the given function $f(x)$ with abscissas equally spaced in the $[ \pi, 3\pi]$ interval. The absolute approximation errors are also displayed. The X and Y coordinate arrays are initialized by evaluating this function for N = 13 points equally spaced in the domain from $\pi$ to $5 \pi$ . The degree parameter (the number of orthogonal polynomials to use) is equal to 12 in this example.

#include <codecogs/maths/regression/orthogonal.h>

#include "orthogonal.h"

#include <cmath>
#include <iostream>
#include <iomanip>
using namespace std;

#define PI 3.1415926535897932384626433832795
#define N  13

int main()
{
    // Declare and initialize two arrays to hold the coordinates of the initial data points
    double x[N], y[N];

    double xx = PI, step = 4 * PI / (N - 1);
    for (int i = 0; i < N; i++, xx += step) {
      x[i] = xx;
      y[i] = sin(xx) + xx;
    }

    // Initialize the regression approximation routine with known data points
    Maths::Regression::Orthogonal A(N, x, y, N - 1);

    // Interrogate the regression function to find approximated values
    int N_out = 10;
    xx = PI, step = (3 * PI) / (N_out - 1);
    for (int i = 0; i < N_out; ++i, xx += step) {
        cout << "x = " << setw(7) << xx << "  y = ";
        cout << setw(7) << A.getValue(xx);
        cout << setw(10) << "error = " << fabs(sin(xx) + xx - A.getValue(xx)) << endl;
    }

    return 0;
}

Output:

x = 3.14159  y = 3.14159  error = 1.33227e-15
x = 4.18879  y = 3.32276  error = 1.77636e-15
x = 5.23599  y = 4.36996  error = 1.86517e-14
x = 6.28319  y = 6.28319  error = 7.19425e-14
x = 7.33038  y = 8.19641  error = 1.1191e-13
x = 8.37758  y = 9.24361  error = 2.41585e-13
x = 9.42478  y = 9.42478  error = 2.91323e-13
x =  10.472  y = 9.60595  error = 2.62901e-13
x = 11.5192  y = 10.6531  error = 2.2915e-13
x = 12.5664  y = 12.5664  error = 1.04805e-13

Members of Orthogonal

CLASS METHOD

Orthogonal

Initializes the necessary data for following evaluations of the polynomial.

Parameters

The number of initial points in the arrays x and y

The x-coordinates for the initial points

The y-coordinates for the initial points

degree

The number of orthogonal polynomials to use in the approximation (must be strictly less than the number of points n)

CLASS METHOD

Orthogonal

Class destructor

CLASS METHOD

getValue

Returns the approximated ordinate at the given abscissa.

Parameters

The abscissa of the approximation point

STATIC CLASS METHOD

prod

CLASS METHOD

Orthogonal_once

This function implements the Orthogonal class for one off calculations, thereby avoid the need to instantiate the Orthogonal class yourself.

Example 1

The following graphs are constructed from the interpolation of the following values, using a 3rd order polynomial and then 8th order polynomial, respectively:

x = 1  y = 0.22
x = 2  y = 0.04
x = 3  y = -0.13
x = 4  y = -0.17
x = 5  y = -0.04
x = 6  y = 0.09
x = 7  y = 0.11

\graph n=7 x="1 2 3 4 5 6 7" y="0.22 0.04 -0.13 -0.17 -0.04 0.09 0.11" degree=3 a=1:7 .input \graph n=7 x="1 2 3 4 5 6 7" y="0.22 0.04 -0.13 -0.17 -0.04 0.09 0.11" degree=6 a=1:7 .input

Parameters

The number of initial points in the arrays x and y

The x-coordinates for the initial points

The y-coordinates for the initial points

degree

The number of polynomials to use in the approximation (must be strictly less than the number of points n)

The x-coordinate for the output point

Returns

the interpolated y-coordinate that corresponds to a.