(Matlab) Write a Mat lab program that inputs: number of nodes m; vector x of nod

Question

(Matlab) Write a Mat lab program that inputs: number of nodes m; vector x of nodes, ordered from smallest to largest; vector y of values at nodes; vector yprime of first derivative values at nodes: location z; and outputs the value of the piecewise cubic Hermite interpolant at z.

Explanation / Answer

function pp = pchipd(x,y,d,xx) %PCHIPD Piecewise Cubic Hermite Interpolating Polynomial with Derivatives. % PP = PCHIPD(X,Y,D) provides the piecewise cubic polynomial which % interpolates values Y and derivatives D at the sites X. This is meant % to augment the built-in Matlab function PCHIP, which does not allow the % user to specify derivatives. % % X must be a vector. % % If Y and D are vectors, then Y(i) and D(i) are the value and derivative % to be matched at X(i). % % If Y and D are matrices, then size(Y,2) == size(D,2) == length(X). % Also, size(Y,1) == size(D,1). Use this for interpolating vector valued % functions. % % YY = PCHIPD(X,Y,D,XX) is the same as YY = PPVAL(PCHIPD(X,Y,D),XX), thus % providing, in YY, the values of the interpolant at XX. % % Example comparing SPLINE, PCHIP, and PCHIPD % a = -10; % b = 10; % x = linspace(a,b,7); % f = @(x) 1./(1+exp(-x)); % logistic function % df = @(x) f(x).*(1-f(x)); % derivative of the logistic function % t = linspace(a,b,50); % r = f(t); % p = pchip(x,f(x),t); % s = spline(x,f(x),t); % q = pchipd(x,f(x),df(x),t); % plot(t,r,'k',x,f(x),'o',t,p,'-',t,s,'-.',t,q,'--') % legend('true','data','pchip','spline','pchipd',4) % % check inputs % x must be a vector if ~isvector(x) error('pchipd:input_error','x must be a vector of length > 2.') end % get size and orient n = length(x); x = x(:); % make sure x is long enough, we can't construct an interpolating % polynomial with just one point if n < 2 error('pchipd:input_error','x must be a vector of length > 2.') end % check y and d if isvector(y) && isvector(d) && length(y) == n && length(d) == n % orient y = y(:); d = d(:); m = 1; elseif size(y,2) == n && size(d,2) == n && size(y,1) == size(d,1) m = size(y,1); y = y'; d = d'; else error('pchipd:input_error','y and d must be vectors or matrices of same size with length(x) columns.') end % sort breaks & data if needed if ~issorted(x) [x x_ix] = sort(x); if m == 1 y = y(x_ix); d = d(x_ix); else y = y(x_ix,:); d = d(x_ix,:); end end % compute coefficients coef = zeros(m,n-1,4); dx = diff(x); for i = 1:m dy = diff(y(:,i)); coef(i,:,4) = y(1:end-1,i)'; coef(i,:,3) = d(1:end-1,i)'; coef(i,:,2) = 3*dy./(dx.^2) - (2*d(1:end-1,i)+d(2:end,i))./dx; coef(i,:,1) = -2*dy./(dx.^3) + (d(1:end-1,i)+d(2:end,i))./(dx.^2); end % create the piecewise polynomial structure pp = mkpp(x,coef,m); % if user requests evaluations if nargin > 3 pp = ppval(pp,xx); end

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