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Parabolic

viewed 3983 times and licensed 139 times
Approximates an arbitrary function using parabolic least squares fitting.
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Interface

C++
Excel
HTML

Class Parabolic

This class approximates an arbitrary function using a polynomial of degree 2, which makes it more suitable for approximating parabola-shaped graphs.

The algorithm finds the coefficients a, b and c such that the following quadratic function fits the given set of points with a minimum error, in terms of leasts squares minimization

Below you will find the regression graph for a set of points obtained by evaluating the function f(x) = \sin(2x) \cdot x, displayed in light blue, at particular abscissas. The regression quadratic, displayed in red, has been calculated using this class. The root mean squared error is also displayed.

1/parabolic-378.png
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References:

  • Jean-Pierre Moreau's Home Page, http://perso.wanadoo.fr/jean-pierre.moreau/
  • 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 f(x) with abscissas equally spaced in the [ \pi, 3\pi] interval. The X and Y coordinate arrays are initialized by evaluating this function for N = 12 points equally spaced in the domain from \pi to 5 \pi.
#include <codecogs/maths/regression/parabolic.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(2 * xx) * xx;
    }
 
    // Initialize the regression approximation routine with known data points
    Maths::Regression::Parabolic A(N, x, y);
 
    // 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 =   0.0202138
x = 4.18879  y =    0.196758
x = 5.23599  y =    0.313613
x = 6.28319  y =    0.370778
x = 7.33038  y =    0.368253
x = 8.37758  y =    0.306039
x = 9.42478  y =    0.184135
x =  10.472  y =  0.00254086
x = 11.5192  y =   -0.238743
x = 12.5664  y =   -0.539716

Authors

Lucian Bentea (August 2005)
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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Members of Parabolic

Parabolic

 
Parabolicintn
double*x
double*y )[constructor]
Initializes the necessary data for following evaluations of the quadratic.
nThe number of initial points in the arrays x and y
xThe x-coordinates for the initial points
yThe y-coordinates for the initial points

GetValue

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


Parabolic Once

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

Example 2

The following graph fits a parabola to these 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", a=1:7

Parameters

nThe number of initial points in the arrays x and y
xThe x-coordinates for the initial points
yThe y-coordinates for the initial points
aThe x-coordinate for the output point

Returns

the interpolated y-coordinate that corresponds to a
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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