# Discrete

Approximates a discrete function using least squares polynomial fitting.

Controller: **CodeCogs**

## Dependents

## Interface

C++

HTML

## Class Discrete

This class approximates an arbitrary discrete function using polynomial least squares fitting. The algorithm finds the coefficients , with 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 . The regression polynomial using a variety of orders are displayed (same results are shown in example below)

## References:

### Example 1

- The following example displays 10 approximated values (you may change this amount through
the N_out variable) for the function with abscissas equally spaced in the
interval. The X and Y coordinate arrays are initialized by evaluating
this function for N = 20 points equally spaced in the domain from to .
#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

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## Members of Discrete

#### Discrete

Initializes the necessary data for following evaluations of the polynomial.Discrete( int `n`double* `x`double* `y`int `degree`) *[constructor]*n Total number of data points to analyse. x An array [0 to n-1] with x-coordinates of points. y An array [0 to n-1] with y-coordinates of points. degree The number of coefficient to be used in the polynomial fitting.

#### ~Discrete

Detailed Description...~Discrete( ) #### GetValue

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

#### GetCoefficent

Returns individual coefficient from the computed polynomial, i.e. in the following equation:doublegetCoefficent( int `i`) ### 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)); ...

i The ith coefficient, starting at i=0 to degree.

## Discrete Once

doubleDiscrete_once( | int | N | |

double* | x | ||

double* | y | ||

int | degree | ||

double | a | ) |

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

### Parameters

N The number of initial points x The x-coordinates for the initial points y The y-coordinates for the initial points degree The number of coefficient to be used in the polynomial fitting (the order) a the x-coordinate for the point to be computed

### Returns

- the y-coordinate that corresponds to
*a*.

##### Source Code

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