Cubic
Interpolates a given set of points using cubic spline fitting.
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Interface
#include <codecogs/maths/approximation/interpolation/cubic.h>
using namespace Maths::Approximation::Interpolation;
Overview
A cubic spline is a piecewise cubic polynomial such that the function, its derivative and its second derivative are continuous at the interpolation nodes. The natural cubic spline has zero second derivatives at the endpoints. It is the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative. For more information on this algorithm, please see references.
An important detail when using this class is that the abscissas array (the x-axis) given as argument to the constructor needs to be sorted in ascending order and its elements need to be equally spaced, meaning that:
where N is the size of the array. In other words, the associated function needs to be discrete.
Below you will find the interpolation graphs for a set of points obtained by evaluating the function , displayed in light blue, at particular abscissas. The spline fitting curve, displayed in red, has been calculated using this class. In the first graph there had been chosen a number of 12 points, while in the second 36 points were considered. You may notice the root mean squared error in each of the cases.
References
- PlanetMath, http://planetmath.org/encyclopedia/CubicSplinInterpolation.html
- Jean-Pierre Moreau's Home Page, http://perso.wanadoo.fr/jean-pierre.moreau/
- Tuan Dang Trong, "Numath Library"
- Forsythe, G.E. "Computer methods for mathematical computations", PRENTICE-HALL, INC (1977)
Example 1
The following example displays 20 interpolated 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
to
.
#include <codecogs/maths/interpolation/cubic.h>
#include <math.h>
#include <iostream>
#include <iomanip>
using namespace std;
#define PI 3.1415
#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 Cubic spline interpolation routine with known data points
Maths::Interpolation::Cubic A(N, x, y);
// Interrogate spline curve to find interpolated values
int N_out = 20;
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(13) << A.getValue(xx) << endl;
}
return 0;
}Output:
x = 3.1415 y = -5.89868e-005
x = 3.63753 y = 0.233061
x = 4.13355 y = 0.216427
x = 4.62958 y = 0.0486148
x = 5.12561 y = -0.133157
x = 5.62163 y = -0.172031
x = 6.11766 y = -0.0456079
x = 6.61368 y = 0.0906686
x = 7.10971 y = 0.116462
x = 7.60574 y = 0.0557287
x = 8.10176 y = -0.03875
x = 8.59779 y = -0.10346
x = 9.09382 y = -0.0734111
x = 9.58984 y = 0.0298435
x = 10.0859 y = 0.094886
x = 10.5819 y = 0.0588743
x = 11.0779 y = -0.0171021
x = 11.5739 y = -0.0630512
x = 12.07 y = -0.0601684
x = 12.566 y = -0.00994154Members of Cubic
CLASS METHOD
Cubic
Initializes the necessary data for following evaluations of the spline.
Parameters
CLASS METHOD
Cubic
m_class destructor
CLASS METHOD
getValue
Returns the approximated ordinate at the given abscissa.
Parameters
The value of the x parameter needs to be in the X[1]...X[N - 1] interval (including endpoints).
CLASS METHOD
Cubic_once
This function implements the Cubic class for one off calculations, thereby avoid the need to instantiate the Cubic class yourself.
Example 1
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" a=1:7:100