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# Lagrange

Interpolates a given set of points using the Lagrange polynomial.
Controller: CodeCogs

C++

## Class Lagrange

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

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

It is given by

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

where

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

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 $\inline&space;&space;f(x)&space;=&space;\sin(2x)&space;/&space;x$, 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.

## 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 $\inline&space;&space;f(x)$ with abscissas equally spaced in the $\inline&space;&space;[&space;\pi,&space;3\pi]$ interval. The X and Y coordinate arrays are initialized by evaluating this function for N = 12 points equally spaced in the domain from $\inline&space;&space;\pi$ to $\inline&space;&space;5&space;\pi$.
#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

## See Also

Also consider the regression methods: Regression/Discrete, Regression/Forsythe, Regression/Orthogonal, Regression/Stiefel

### Authors

Lucian Bentea (August 2005)
##### Source Code

Source code is available when you buy a Commercial licence.

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## 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

Source code is available when you buy a Commercial licence.

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Last Modified: 5 Apr 10 @ 17:17     Page Rendered: 2022-03-14 17:42:23