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Maths › Approximation › Interpolation ›

Linear

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Linearly interpolates a given set of points.

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Contents

Interface
Class Linear
Linear Once
Page Comments

Interface

C++

Class Linear

Linear interpolation is a process employed in mathematics, and numerous applications thereof including computer graphics. It is a very simple form of interpolation. In numerical analysis a linear interpolation of certain points that are in reality values of some function f is typically used to approximate the function f. Linear interpolation can be regarded as a trivial example of polynomial interpolation. The error of this approximation is defined as

$R_T = f(x) - p(x)$

(2)

where p denotes the linear interpolation polynomial defined as follows

$p(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0} (x - x_0)$

(3)

It can be proven using Rolle's theorem that if f has two continuous derivatives, the error is bounded by

$|R_T| \leq \frac{(x_1 - x_0) ^ 2}{8} \mathrm{max}_{x_0 \leq x \leq x_1} |f''(x)|$

(4)

As you see, the approximation between two points on a given function gets worse with the second derivative of the function that is approximated. This is intuitively correct as well: the "curvier" the function is, the worse is the approximations made with simple linear interpolation.

Below you will find the interpolation graphs for a set of points obtained by evaluating the function $\inline f(x) = \sin(2x) / x$ , displayed in light blue, at particular abscissas. The linear interpolating function, 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.

MISSING IMAGE!

1/linear-378.png cannot be found in /users/1/linear-378.png. Please contact the submission author.

References:

Wikipedia, http://en.wikipedia.org/wiki/Linear_interpolation

Example 1

The following example displays 20 interpolated values (you may change this amount through the N_out variable) for the given function $\inline f(x)$ with abscissas equally spaced in the $\inline [ \pi, 3\pi]$ interval. The X and Y coordinate arrays are initialized by evaluating this function for N = 12 points equally spaced in the domain from $\inline \pi$ to $\inline 5 \pi$ .

#include <codecogs/maths/approximation/interpolation/linear.h>
 
#include <cmath>
#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 linear interpolation routine with known data points
    Maths::Interpolation::Linear A(N, x, y);
 
    // Interrogate linear fitting 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 =     0.0765858
x = 4.13355  y =      0.153231
x = 4.62958  y =     0.0678533
x = 5.12561  y =    -0.0879685
x = 5.62163  y =     -0.137135
x = 6.11766  y =     -0.022215
x = 6.61368  y =     0.0804548
x = 7.10971  y =      0.060627
x = 7.60574  y =     0.0407992
x = 8.10176  y =    -0.0110834
x = 8.59779  y =    -0.0715961
x = 9.09382  y =    -0.0619804
x = 9.58984  y =     0.0221467
x = 10.0859  y =      0.081803
x = 10.5819  y =     0.0313408
x = 11.0779  y =    -0.0191214
x = 11.5739  y =    -0.0324255
x =   12.07  y =    -0.0406044
x =  12.566  y =    -0.0146181

Members of Linear

Linear

Linear(	int	`n`
	double*	`x`
	double*	`y`	)[constructor]

Initializes the necessary data for following evaluations of the fitting lines.

n	The number of initial points
x	The x-coordinates for the initial points
y	The y-coordinates for the initial points

GetValue

doublegetValue( double x )

Returns the approximated ordinate at the given abscissa.

Note

This function is not designed to provide extrapolation points, thus you need to keep the value of x in the interval from X[0] to X[N - 1].

x	The abscissa of the interpolation point

Linear Once

doubleLinear_once(	int	`N`
	double*	`x`
	double*	`y`
	double	`a`	)

This function implements the Linear class for one off calculations, thereby avoid the need to instantiate the Linear class yourself.

Example 2

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

There is an error with your graph parameters for Linear_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" a=1:7 .input

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

Parameters

N	The number of initial points
x	The x-coordinates for the initial points (evenly spaced!)
y	The y-coordinates for the initial points
a	The x-coordinate for the output point

Returns

the interpolated y-coordinate that corresponds to a.

Source Code

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Linear

Interface

Class Linear

MISSING IMAGE!

References:

Example 1

See Also

Authors

Source Code

Members of Linear

Linear

GetValue

Note

Linear Once

Example 2

Parameters

Returns

Source Code