cc.rar_matlab 三次样条插值_matlab 样条插值_三次样条插值_三次样条插值 MATLAB_样条插值

function [yi,yxi,yxxi] = csinterp( x, y, xi ) % CSINTERP 1-D piecewise cubic spline interpolation of tabulated % data and calculation of first and second derivatives. % [YI,YXI,YXXI] = CSINTERP(X,Y,XI) interpolates function values, % first and second derivatives using the rows/columns (X,Y) at % the nodes specified in XI. X and Y should be row or column % vectors, each containing at least three values, otherwise the % function will return an empty output. % The vector X specifies the points at which the data Y is given. % Output data YI, YXI and YXXI are same size as XI. % X and XI can be non-uniformly spaced, but repeated values in X % will be removed and the resulting vector will be sorted. % This function also extrapolates the input data (no NaNs are % returned); use extrapolated values at your own risk. % % Requires bindex.m (available at Matlab FEX). % % Examples: % x = [0.1:0.1:0.3 1:3]; y = sin( x ); % xi = linspace(0,3,200); % [yi,yxi,yxxi] = csinterp(x,y,xi); % figure(1) % plot(x,y,'o',xi,yi,'r-'), box on, grid on % xlabel('x') % ylabel('y(x)') % title('Continuity test') % legend('exact','csinterp') % % n = 101; m = n; % x = linspace(0,2*pi,n); % y = sin( x ); yx = cos( x ); yxx = -y; % xi = 2*pi*rand(1,m); % [yi,yxi,yxxi] = csinterp(x,y,xi); % figure(2) % subplot(311) % plot(x,y,xi,yi,'o'), box on, grid on % title('y(x)') % legend('exact','csinterp') % subplot(312) % plot(x,yx,xi,yxi,'o'), box on, grid on % title('dy/dx') % legend('exact','csinterp') % subplot(313) % plot(x,yxx,xi,yxxi,'o'), box on, grid on % title('d^2y/dx^2') % legend('exact','csinterp') % % n = 2001; m = 5001; % x = linspace(0,2*pi,n); % y = sin( x ); yx = cos( x ); yxx = -y; % xi = sort( 2*pi*rand(1,m) ); % [yi,yxi,yxxi] = csinterp(x,y,xi); % figure(3) % subplot(311) % plot(x,y,'o',xi,yi,'r-'), box on, grid on % ylabel('y(x)') % title('Large dataset test') % legend('exact','csinterp') % subplot(312) % plot(x,yx,'o',xi,yxi,'r-'), box on, grid on % ylabel('dy/dx') % legend('exact','csinterp') % subplot(313) % plot(x,yxx,'o',xi,yxxi,'r-'), box on, grid on % ylabel('d^2y/dx^2') % legend('exact','csinterp') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % First version: 20/11/2007 % % Contact: [email protected] % % Any suggestions to improve the performance of this % code will be greatly appreciated. % % Reference: % C. Moler, Numerical Computing with Matlab (NCM) % http://www.mathworks.com/moler % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Let's go: % Declare an empty output just in case calculations won't be % performed: yi = []; yxi = []; yxxi = []; % Error checking: sx = size(x); sy = size(y); sxi = size(xi); nx = length(x); ny = length(y); if ( nargin ~= 3 ) error('This function requires three input arguments...'); end if nx ~= ny error('The length of x must match the length of y...') end if min(sx) ~= 1 error('Input x should be a row or column vector...') end if min(sy) ~= 1 error('Input y should be a row or column vector...') end % Convert x and y to column vectors to avoid problems % with mixed row/column input data: x = x(:); y = y(:); % Remove repeated elements: [x,ui] = unique(x); y = y(ui); nx = length(x); % Proceed with calculations only when the # of function points % is greater than 2: if nx > 2 yi = zeros( sxi ); yxi = zeros( sxi ); yxxi = zeros( sxi ); dy = diff( y ); h = diff( x ); hsq = h.*h; h1 = h(1); h2 = h(2); h3 = h(nx-2); h4 = h(nx-1); hb = h(1:nx-2); hf = h(2:nx-1); % Bracketting: ix = bindex(xi,x,1); % Calculate the local variable: if sxi(1) == 1 s = xi(:) - x(ix); else s = xi - x( ix ); end % Calculate the slopes: slopes = dy./h; s1 = slopes(1); s2 = slopes(2); s3 = slopes(nx-2); s4 = slopes(nx-1); sb = slopes(1:nx-2); sf = slopes(2:nx-1); % Calculate the vectors filling the matrix of the system: lodiag(1:nx-2) = hf; lodiag( nx-1 ) = h3 + h4; l4 = lodiag( nx-1 ); cendiag( 1 ) = h2; cendiag(2:nx-1) = 2*( hb + hf ); cendiag( nx ) = h3; updiag( 1 ) = h1 + h2; u1 = updiag(1); updiag(2:nx-1) = hb; % Right side of the system (not-a-knot end conditions): r1 = ( ( h1 + 2*u1 )*h2*s1 + h1^2*s2 )/u1; rn = ( ( h4 + 2*l4 )*h3*s4 + h4^2*s3 )/l4; r = 3*( hf.*sb + hb.*sf ); r = [r1;r;rn]; % Fill the matrix of the system (it is tridiagonal, so let us use a sparse matrix): M = sparse( diag( cendiag ) + diag( lodiag, -1 ) + diag( updiag, 1 ) ); % Solve the linear system: d = M\r; db = d(1:nx-1); df = d(2: nx ); % Chapter 3, page 14: % Calculate the remaining spline coefficients c = ( 3*slopes - 2*db - df )./h ; b = ( db - 2*slopes + df )./hsq; % Interpolate the function and its derivatives s2 = s.*s; s3 = s2.*s; yi = y( ix ) + s.*d( ix ) + s2.*c( ix ) + s3.*b( ix ); yxi = d( ix ) + 2*s.*c( ix ) + 3*s2.*b( ix ); yxxi = 2*c( ix ) + 6*s.*b( ix ); if sxi(1) == 1 yi = yi(:)'; yxi = yxi(:)'; yxxi = yxxi(:)'; end end