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Maths › Approximation › Interpolation ›

Lagrange

Interpolates a given set of points using the Lagrange polynomial.

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Contents

Interface
Class Lagrange
Lagrange Once
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Interface

C++

Class Lagrange

The Lagrange interpolating polynomial is the polynomial of degree n - 1 that passes through the n points

$y_1 = f(x_1), y_2 = f(x_2), \ldots, y_n = f(x_n)$

(2)

It is given by

$P(x) = \sum_{j = 1} ^ n P_j(x)$

(3)

where

$P_j(x) = y_j \prod_{k=1 \\ k \neq j} ^ n \frac{x - x_k} {x_j - x_k}$

(4)

The formula was first published by Waring (1779), rediscovered by Euler in 1783, and published by Lagrange in 1795 (Jeffreys and Jeffreys 1988). An important detail when using this class is that the abscissas array given as argument to the constructor needs to be sorted in ascending order.

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 Lagrange polynomial, 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. The level of interpolation in both graphs is 3. The root mean squared error is also displayd in each of the cases.

MISSING IMAGE!

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

References:

MathWorld, http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html
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 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/interpolation/lagrange.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 Lagrange interpolation routine with known data points
    Maths::Interpolation::Lagrange A(N, x, y);
 
    // Interrogate Lagrange polynomial 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, 3) << endl;
  }
    return 0;
}

Output:

x =  3.1415  y = -5.89868e-005
x = 3.63753  y =      0.216649
x = 4.13355  y =      0.208793
x = 4.62958  y =    -0.0536974
x = 5.12561  y =     -0.186543
x = 5.62163  y =      -0.10577
x = 6.11766  y =     0.0268879
x = 6.61368  y =     0.0875189
x = 7.10971  y =     0.0993752
x = 7.60574  y =     0.0512131
x = 8.10176  y =    -0.0885626
x = 8.59779  y =     -0.123293
x = 9.09382  y =    -0.0160297
x = 9.58984  y =     0.0787203
x = 10.0859  y =     0.0791771
x = 10.5819  y =     0.0216086
x = 11.0779  y =    -0.0212055
x = 11.5739  y =    -0.0727429
x =   12.07  y =    -0.0621462
x =  12.566  y =     0.0312161

Members of Lagrange

Lagrange

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

Initializes the necessary data for following evaluations of the polynomial.

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`
	int	`l`	)

Returns the approximated ordinate at the given abscissa.

Note

The value of the x parameter needs to be in the X[0]...X[N - L + 1] interval (including endpoints), where l > 1 is the level of interpolation. For example a level 3 interpolation would have the maximum working interval between X[0] and X[N - 2].

x	The abscissa of the interpolation point
l	The level of interpolation (2 means quadratic)

Lagrange Once

doubleLagrange_once(	int	`N`
	double*	`x`
	double*	`y`
	double	`a`
	int	`l`	)

This function implements the Lagrange class for one off calculations, thereby avoid the need to instantiate the Lagrange 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 Lagrange_once with options N=7 l=2 x="1 2 3 4 5 6 7" y="0.22 0.04 -0.13 -0.17 -0.04 0.09 0.11" a=1:6 .input

Error Message:Function Lagrange_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
l	The level of interpolation (2 means quadratic)

Returns

the interpolated y-coordinate that corresponds to a.

Source Code

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Lagrange

Interface

Class Lagrange

MISSING IMAGE!

References:

Example 1

See Also

Authors

Source Code

Members of Lagrange

Lagrange

GetValue

Note

Lagrange Once

Example 2

Parameters

Returns

Source Code