Implement a MATLAB function LagrangeInter of the form function y=LagrangeInter(x

Question

Implement a MATLAB function LagrangeInter of the form function y=LagrangeInter(x,xi,yi) to get the value of the Lagrange interpolating polynomial at x:

1. compute the vector L with Lk = Ln;k(x); k = 1; : : : ; n, where n is the length of xi, by implementing a function

function L=Lagrange(x,xi).

2. obtain the value of the Lagrange interpolating polynomial at x from y = Pn k=1 ykLk.

3. in both functions, allow x to be a vector (by vectorizing or by using ìforîloop).

Run your function LagrangeInter for xi = i; yi = sin(xi); i = 0, 1, ... 6, and x = [0:5; 1:5; 2:5; 3:5; 4:5]:

xi=[0:1:6]; yi=sin(xi); x=[0.5,1.5, 2.5, 3.5, 4.5];

yL=LagrangeInter(x,xi,yi);

Present your results in a table showing the values of sin(x) and yL, along with you MATLAB functions.

Explanation / Answer

% Orginal function: yi = LagrangeInter(x,y,xi) located at % http://dmpeli.math.mcmaster.ca/Matlab/Math4Q3/Lecture2-1/LagrangeInter.m function yi = LagrangeInter(x,y,xi) % Lagrange interpolation algorithm % x,y - row-vectors of (n+1) data values (x,y) % xi - a row-vector of x-values, where interpolation is to be found (could be a single value) % yi - a row-vector of interpolated y-values n = length(x) - 1; % the degree of interpolation polynomial ni = length(xi); % the number of x-values, where interpolation is to be found L = ones(n+1,ni); % the matrix for Lagrange interpolating polynomials L_(n,k)(x) % has (n+1) rows for each polynomial at k = 0,1,...,n % has ni column for each x-value of xi % Note: the algorithm uses the MATLAB capacities for matrix handling! % The two nested loops below are designed to compute in parallel % the values of Lagrange interpolating polynomials at each x-value of xi ! for k = 0 : n % start the outer loop through the data values for x for kk = 0 : (k-1) % start the inner loop through the data values for x (if k = 0 this loop is not executed) L(kk+1,:) = L(kk+1,:).*(xi - x(k+1))/(x(kk+1)-x(k+1)); % see the Lagrange interpolating polynomials end % end of the inner loop for kk = k+1 : n % start the inner loop through the data values (if k = n this loop is not executed) L(kk+1,:) = L(kk+1,:).*(xi - x(k+1))/(x(kk+1)-x(k+1)); end % end of the inner loop end % the end of the outer loop % Now multiply the values for Lagrange interpolating polynomials by the data values for y yi = y * L; % use matrix multiplication of row-vector y and the matrix L, the output is the row-vector

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