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Discrete

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Approximates a discrete function using least squares polynomial fitting.
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C++
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Class Discrete

This class approximates an arbitrary discrete function using polynomial least squares fitting.

The algorithm finds the coefficients a_i, with 0 \leq i \leq n such that the following polynomial fits the given set of points with minimum error, using leasts squares minimization For this function the residual (or error between y and that calculationed using the coefficients) is given by

From which the rate of change of this error with respect to each constant are, which ideally we want to make zero:

Equating to zero and rearranging to seperate the constants a from y, gives: which in matrix form, yields Solving this solutions using a matrix transpose, yields the coefficients a in terms of x and y.

Below you will find the regression graph for a set of points obtained by evaluating the function f(x) = \sin(x) / x. The regression polynomial using a variety of orders are displayed (same results are shown in example below)

1/discrete3.png
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Fitting 3rd and 5th order polynomials

References:

Example 1

The following example displays 10 approximated values (you may change this amount through the N_out variable) for the function g(x) = \sin(x) / x with abscissas equally spaced in the [ \pi/2, 4\pi] interval. The X and Y coordinate arrays are initialized by evaluating this function for N = 20 points equally spaced in the domain from \pi/2 to 4\pi.
#include <codecogs/maths/approximation/regression/discrete.h>
 
#include <cmath>
#include <stdio.h>
using namespace std;
 
#define PI  3.1415926535897932384626433832795
#define N   30
 
int main() 
{
  // Delvare two arrays to hold the coordinates of initial data points
  double x[N], y[N];

// Generate the points double xx = PI/2; double step = 2 * PI / (N - 1);

for (int i = 0; i < N; ++i, xx += step) { double x2=xx+sin(xx); // vary x spacing x[i] = x2; y[i] = sin(x2)/x2; }

// Initialize the regression approximation routine with known data points Maths::Regression::Discrete A(N, x, y, 3); Maths::Regression::Discrete B(N, x, y, 5); Maths::Regression::Discrete C(N, x, y, 10);

// Interrogate the regression function to find approximated values int N_out =50; xx = PI/2 ; step = 2 * PI / (N_out - 1);

printf("\nx, exact, discrete_3, discrete_5, discrete_10");

for (int i = 0; i < N_out; ++i, xx += step) { double x2=xx+sin(xx); printf("\n%.4lf, %.6lf, %.6lf, %.6lf, %.6lf", x2, sin(x2)/x2, A.getValue(x2), B.getValue(x2), C.getValue(x2)); } return 0; }
Output (first 10 numbers):
x, exact, discrete_3,  discrete_5,  discrete_10
2.5708, 0.210169, 0.235747, 0.210570, 0.210190
2.6908, 0.161909, 0.175569, 0.162336, 0.161899
2.7945, 0.121709, 0.127760, 0.122034, 0.121692
2.8824, 0.088920, 0.090229, 0.089115, 0.088905
2.9550, 0.062769, 0.061196, 0.062848, 0.062759
3.0134, 0.042441, 0.039160, 0.042431, 0.042435
3.0585, 0.027131, 0.022864, 0.027058, 0.027128
3.0919, 0.016071, 0.011248, 0.015955, 0.016070
3.1150, 0.008531, 0.003406, 0.008388, 0.008532
3.1296, 0.003821, -0.001463, 0.003663, 0.003822

Authors

Lucian Bentea (August 2005)
Will Bateman (Mar 2006)
Source Code

Source code is available when you buy a Commercial licence.

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Members of Discrete

Discrete

 
Discreteintn
double*x
double*y
intdegree )[constructor]
Initializes the necessary data for following evaluations of the polynomial.
nTotal number of data points to analyse.
xAn array [0 to n-1] with x-coordinates of points.
yAn array [0 to n-1] with y-coordinates of points.
degreeThe number of coefficient to be used in the polynomial fitting.

~Discrete

 
~Discrete )
Detailed Description...

GetValue

 
doublegetValuedoublex )
Returns the approximated ordinate at the given abscissa.
xThe abscissa of the approximation point

GetCoefficent

 
doublegetCoefficentinti )
Returns individual coefficient from the computed polynomial, i.e. a_i in the following equation:

Example 2

...
Maths::Regression::Discrete A(N, x, y, 7); 
for(int i=0;i<7;i++) printf("\n coefficient %d is %lf", A.getCoefficient(i)); 
...
iThe ith coefficient, starting at i=0 to degree.


Discrete Once

 
doubleDiscrete_onceintN
double*x
double*y
intdegree
doublea )
This function implements the Discrete class for one off calculations, thereby avoid the need to instantiate the Discrete class yourself.

Example 3

The following graph is constructed from interpolating the following values:
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=5, a=1:7

Parameters

NThe number of initial points
xThe x-coordinates for the initial points
yThe y-coordinates for the initial points
degreeThe number of coefficient to be used in the polynomial fitting (the order)
athe x-coordinate for the point to be computed

Returns

the y-coordinate that corresponds to a.
Source Code

Source code is available when you buy a Commercial licence.

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