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5. Lagrange Interpolation

# Lagrange Interpolation

## Introduction

We can write the second order polynomial in the form Let us assume are the three interpolating points then,   Using above we have   Putting the values together we have, This form can be written as where Generalizing we have ## Algorithm

1. Start
2. Read number of points, let us assume (n)
3. Read the value at which the value is needed, let us assume (x)
4. Read given data points
5. Calculate the value of Li
for I = 1 to n
for j = 1 to n
if (j!=i)
L[i] = L[i]*((x-x[i])/(x[i]-x[j]))
Endif
endfor
endfor
6. Calculate the interpolated point at x
For i= 1 to n
t=t+fx[i]*lx[i]
Endfor
7. Print the interpolated value v at x
8. End


## Codes

C ProgramJS
#include <stdio.h>
#include <conio.h>

void main()
{
float x, y, xp, yp = 0, p;
int i, j, n;
clrscr();
/* Input Section */
printf("Enter number of data: ");
scanf("%d", &n);
printf("Enter data:\n");
for (i = 1; i <= n; i++)
{
printf("x[%d] = ", i);
scanf("%f", &x[i]);
printf("y[%d] = ", i);
scanf("%f", &y[i]);
}
printf("Enter interpolation point: ");
scanf("%f", &xp);
/* Implementing Lagrange Interpolation */
for (i = 1; i <= n; i++)
{
p = 1;
for (j = 1; j <= n; j++)
{
if (i != j)
{
p = p * (xp - x[j]) / (x[i] - x[j]);
}
}
yp = yp + p * y[i];
}
printf("Interpolated value at %.3f is %.3f.", xp, yp);
getch();
}