In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points Xi and numbers Yi, the Lagrange polynomial is the polynomial of the least degree that at each point Xj assumes the corresponding value Yj (i.e. the functions coincide at each point). The interpolating polynomial of the least degree is unique, however, and it is therefore more appropriate to speak of “the Lagrange form” of that unique polynomial rather than “the Lagrange interpolation polynomial”, since the same polynomial can be arrived at through multiple methods.
The Lagrange interpolating polynomial is the polynomial P(X) of degree <=(n – 1) that passes through the n points (x1, y1 = f(x1)),(x2, y2 = f(x2)) ,.., (xn, yn = f(xn)), , and is given by
(1)

where
(2)

Written explicitly,
(3)

The formula was first published by Waring (1779), rediscovered by Euler in 1783, and published by Lagrange in 1795 (Jeffreys and Jeffreys 1988).
Objective: To write a program in C to find the functional value of any value entered by user( using Lagrange’s Interpolation)
Algorithm for Lagrange’s Interpolation
 Scan for the number of data available. (data)
 Scan value for which f(x) – (datay[]) is to be calculated
 loop for i=0 to number_of_data
scan datax[i], scan datay[i], next i  loop for i to number_of_data
factor[i] = 1.0
loop for j to number_of_data
if i != j
factor[i] = factor[i] * (value – datax[j])/(datax[i]datax[j])
end if, next j, next i  loop for i to number_of_data
sum = sum + factor[i] * datay[i]
next i  print the results
 Stop
C Program implementing Lagrange Interpolation Formula
#include 'studio.h' #include 'math.h' #define SIZE 40 int main(){ int i, j, data; float datax[SIZE], datay[SIZE], coeff[SIZE], value, sum=0.0, factor[SIZE]; printf("How many data are available? \n"); scanf("%d",&data); printf("\nEnter the value of x whose value is to be calculated?"); scanf("%f",&value); for(i=0; i<data ; i++){ printf("\nEnter X[%d]",i); scanf("%f",&datax[i]); printf("Enter Y[%d]",i); scanf("%f",&datay[i]); } for(i=0; i<data; i++){ factor[i] = 1.0; for(j=0; j<data; j++){ if(i!=j){ factor[i] = factor[i] * (value  datax[j])/(datax[i]datax[j]); } } } printf("Coeff are: \n"); for(i=0;i<data;i++){ printf("\nfactor[%d] = %f",i, factor[i]); sum = sum + factor[i] * datay[i]; } printf("\n%f\n",sum); }
Formula source: Lagrange Interpolating Polynomial
Nice article which helps me to understand the concept of Lagrange interpolation formula.