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# Forsythe

Approximates an arbitrary function using Forsythe orthogonal polynomials.
Controller: CodeCogs

C++
Excel

## Class Forsythe

This class approximates an arbitrary function by least squares fitting using Forsythe orthogonal polynomials. It uses a generalization of the three-term relation by G. Forsythe (1957), generating recursively a system of orthonormal polynomials over arbitrary sets of data points.

Below you will find the regression graph for a set of points obtained by evaluating the function , 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.

## References:

• F.R. Ruckdeschel, "BASIC Scientific Subroutines", Vol. II, BYTE/McGRAWW-HILL, 1981

### Example 1

The following example displays 10 approximated values (you may change this amount through the N_out variable) for the given function  with abscissas equally spaced in the  interval. The X and Y coordinate arrays are initialized by evaluating this function for N = 12 points equally spaced in the domain from π to 5 π .
#include <codecogs/maths/regression/forsythe.h>

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

#define PI  3.1415926535897932384626433832795
#define N   12

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

// Generate the points
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::Forsythe A(N, x, y, 11);

// 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(11) << A.getValue(xx) << endl;
}
return 0;
}
Output:
x = 3.14159  y =     3.14159
x = 4.18879  y =     3.32358
x = 5.23599  y =     4.36968
x = 6.28319  y =     6.28329
x = 7.33038  y =     8.19637
x = 8.37758  y =     9.24362
x = 9.42478  y =     9.42478
x =  10.472  y =     9.60593
x = 11.5192  y =     10.6532
x = 12.5664  y =     12.5663

### Authors

Lucian Bentea (August 2005)
##### Source Code

Source code is available when you buy a Commercial licence.

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## Members of Forsythe

#### Forsythe

 Forsythe( int n double* x double* y int degree )[constructor]
Initializes the necessary data for following evaluations of the polynomial.
 n The number of initial points in the arrays x and y x The x-coordinates for the initial points y The y-coordinates for the initial points degree The number of polynomials to use in the approximation

#### GetValue

 doublegetValue( double x )
Returns the approximated ordinate at the given abscissa.
 x The abscissa of the approximation point

## Forsythe Once

 doubleForsythe_once( int n double* x double* y int degree double a )
This function implements the Forsythe class for one off calculations, thereby avoid the need to instantiate the Forsythe class yourself.

### Example 2

The following graph are constructed by forming a regression of the following values, using a 3rd order orthogonal polynomials
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
There is an error with your graph parameters for Forsythe_once with options 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

Error Message:Function Forsythe_once failed. Ensure that: Invalid C++

### Parameters

 n The number of initial points in the arrays x and y x The x-coordinates for the initial points y The y-coordinates for the initial points degree The number of polynomials to use in the approximation a The x-coordinate for the output point

### Returns

the interpolated y-coordinate that corresponds to a.
##### Source Code

Source code is available when you buy a Commercial licence.

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