function [yi, ypi, P, D] = neville(x, y, xi)

% NEVILLE Interpolation using Neville's Method% NEVILLE(X,Y,XI) interpolates to find YI, the value of% the underlying function Y at the point XI, using Neville's% Method. X and Y must be vectors of length N.%% [YI,YPI] = NEVILLE() also returns the interpolated derivative% YPI.%% [YI,YPI,P,D] = NEVILLE() also returns the polynomial table% P and derivative polynomial table D calculated for the last XI.

% Joe Henning - Fall 2011

% An Iterative Method of Numerical Differentiation% D. B. Hunter

n = length(x);

for k = 1:length(xi) xd = []; for i = 1:n xd(i) = abs(x(i) - xi(k)); end

[xds,i] = sort(xd);

x = x(i); y = y(i);

P = zeros(n,n); P(:,1) = y(:);

for i = 1:n-1 for j = 1:(n-i) P(j,i+1) = ((xi(k)-x(j))*P(j+1,i) + (x(j+i)-xi(k))*P(j,i))/(x(j+i)-x(j)); end end

yi(k) = P(1,n);

D = zeros(n,n); D(:,1) = y(:);

for i = 1:n-1 D(i,2) = (D(i+1,1)-D(i,1))/(x(i+1)-x(i)); end

for i = 2:n-1 for j = 1:(n-i) D(j,i+1) = (P(j+1,i) + (xi(k)-x(j))*D(j+1,i) - P(j,i) + (x(j+i)-xi(k))*D(j,i))/(x(j+i)-x(j)); end end

ypi(k) = D(1,n);end

A Neville's algorithm example in Matlab

Neville's algorithm as given in the book is easy to implement in Matlab, if one realizes that there is a quick way to deal with the "0 vs. 1" indexing issue. That is, arrays in other languages are frequently indexed from i=0 to i=n. This is what the book assumes. But since the Matlab language has the concept of a matrix so firmly imbedded, all indices start at i=1. So, to write Algorithm 3.1 in Matlab, one trick is to add one to indices, but keep the "for" loops the same. That is, the only line that needs to change is the main line: Q(i+1,j+1) = ((xx-x(i-j+1))*Q(i+1,j) - (xx-x(i+1))*Q(i,j))/... (x(i+1)-x(i-j+1));Again, note that the only change is that the indices in this line have all been increased by one. The loops are still: for i = 1:n for j = 1:iThere is a better way to write Algorithm 3.1. Namely, to compute each new column of the Neville's algorithm table, so that the program doesn't have to store away the whole table as a matrix, but rather just a vector which is a column. This is what the code below does. There is a disadvantage--the program can not be modified to add nodes as needed, as suggested on page 119. It should be saved as "nev.m": function y = nev(xx,n,x,Q)% Neville's algorithm as a function (save as "nev.m")%% inputs:% n = order of interpolation (n+1 = # of points)% x(1),...,x(n+1) x coords% Q(1),...,Q(n+1) y coords% xx=evaluation point for interpolating polynomial p%% output: p(xx)

for i = n:-1:1 for j = 1:i Q(j) = (xx-x(j))*Q(j+1) - (xx-x(j+n+1-i))*Q(j); Q(j) = Q(j)/(x(j+n+1-i)-x(j)); endend

y = Q(1);

Now, to run this on problem I. in assignment #2 would look like: x=1940:10:1990

x =

1940 1950 1960 1970 1980 1990

Q=[132165 151326 179323 203302 226542 249633]

Q =

132165 151326 179323 203302 226542 249633

nev(1940,5,x,Q) % just a check

ans =

132165

nev(1930,5,x,Q)

ans =

169649

nev(2000,5,x,Q)

ans =

251654

But to actually plot out some polynomials, as in problem II. on assignment #2, you probably want to actually write a new program, for instance: % save in something like "probII.m"% --the name doesn't matter since it's not a function% plot out the n=2, 4, 8, 16 polynomials which interpolate the% function f(x) = 1/(1+25x^2) on the interval [-1,1] using% equally spaced points

xx=[-1:.01:1]; % plot a couple hundred points in each case

% first plot the original functionyy=1./(1+25*xx.^2);plot(xx,yy);hold on; % this makes the plots that follow pile up

for n=[2 4 8 16] x=-1:2/n:1; % (check this works for n=4, for instance) Q=1./(1+25*x.^2); % get the interpolation points for i=1:201 yy(i)=nev(xx(i),n,x,Q); %interpolate end; plot(xx,yy);end;

% inessential embellishments:axis([-1 1 -2 2]); % experimentation showed this was niceplot(x,Q,'o'); % show the 16 interpolation points--last values of x and Qtitle('Polynomial interpolation of y=1/(1+25*x^2)--the Runge example');

hold off; % so future plots start over