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author | pkienzle |
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date | Wed, 10 Oct 2001 19:54:49 +0000 |
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## Copyright (C) 2000,2001 Kai Habel ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 2 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, write to the Free Software ## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA ## -*- texinfo -*- ## @deftypefn {Function File} {@var{pp} = } csape (@var{x}, @var{y}, @var{cond}, @var{valc}) ## cubic spline interpolation with various end conditions. ## creates the pp-form of the cubic spline. ## ## the following end conditions as given in @var{cond} are possible. ## @table @asis ## @item 'complete' ## match slopes at first and last point as given in @var{valc} ## @item 'not-a-knot' ## third derivatives are continuous at the second and second last point ## @item 'periodic' ## match first and second derivative of first and last point ## @item 'second' ## match second derivative at first and last point as given in @var{valc} ## @item 'variational' ## set second derivative at first and last point to zero (natural cubic spline) ## @end table ## ## @seealso{ppval, spline} ## @end deftypefn ## Author: Kai Habel <kai.habel@gmx.de> ## Date: 23. nov 2000 ## Algorithms taken from G. Engeln-Muellges, F. Uhlig: ## "Numerical Algorithms with C", Springer, 1996 ## Paul Kienzle, 19. feb 2001, csape supports now matrix y value function pp = csape (x, y, cond, valc) x = x(:); n = length(x); transpose = (columns(y) == n); if (transpose) y = y'; endif a = y; b = c = zeros (size (y)); h = diff (x); idx = ones (columns(y),1); if (nargin < 3 || strcmp(cond,"complete")) # specified first derivative at end point if (nargin < 4) valc = [0, 0]; endif dg = 2 * (h(1:n - 2) .+ h(2:n - 1)); dg(1) = dg(1) - 0.5 * h(1); dg(n - 2) = dg(n-2) - 0.5 * h(n - 1); e = h(2:n - 2); g = 3 * diff (a(2:n,:)) ./ h(2:n - 1,idx)\ - 3 * diff (a(1:n - 1,:)) ./ h(1:n - 2,idx); g(1,:) = 3 * (a(3,:) - a(2,:)) / h(2) \ - 3 / 2 * (3 * (a(2,:) - a(1,:)) / h(1) - valc(1)); g(n - 2,:) = 3 / 2 * (3 * (a(n,:) - a(n - 1,:)) / h(n - 1) - valc(2))\ - 3 * (a(n - 1,:) - a(n - 2,:)) / h(n - 2); c(2:n - 1,:) = trisolve(dg,e,g); c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * valc(1) - c(2,:) * h(1)) / (2 * h(1)); c(n,:) = - (3 / h(n - 1) * (a(n,:) - a(n - 1,:)) - 3 * valc(2) + c(n - 1,:) * h(n - 1)) / (2 * h(n - 1)); b(1:n - 1,:) = diff (a) ./ h(1:n - 1, idx)\ - h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); d = diff (c) ./ (3 * h(1:n - 1, idx)); elseif (strcmp(cond,"variational") || strcmp(cond,"second")) if ((nargin < 4) || strcmp(cond,"variational")) ## set second derivatives at end points to zero valc = [0, 0]; endif c(1,:) = valc(1) / 2; c(n,:) = valc(2) / 2; g = 3 * diff (a(2:n,:)) ./ h(2:n - 1, idx)\ - 3 * diff (a(1:n - 1,:)) ./ h(1:n - 2, idx); g(1,:) = g(1,:) - h(1) * c(1,:); g(n - 2,:) = g(n-2,:) - h(n - 1) * c(n,:); dg = 2 * (h(1:n - 2) .+ h(2:n - 1)); e = h(2:n - 2); c(2:n - 1,:) = trisolve (dg,e,g); b(1:n - 1,:) = diff (a) ./ h(1:n - 1,idx)\ - h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); d = diff (c) ./ (3 * h(1:n - 1, idx)); elseif (strcmp(cond,"periodic")) h = [h; h(1)]; ## XXX FIXME XXX --- the following gives a smoother periodic transition: ## a(n,:) = a(1,:) = ( a(n,:) + a(1,:) ) / 2; a(n,:) = a(1,:); tmp = diff (shift ([a; a(2,:)], -1)); g = 3 * tmp(1:n - 1,:) ./ h(2:n,idx)\ - 3 * diff (a) ./ h(1:n - 1,idx); if (n > 3) dg = 2 * (h(1:n - 1) .+ h(2:n)); e = h(2:n - 1); c(2:n,idx) = trisolve(dg,e,g,h(1),h(1)); elseif (n == 3) A = [2 * (h(1) + h(2)), (h(1) + h(2)); (h(1) + h(2)), 2 * (h(1) + h(2))]; c(2:n,idx) = A \ g; endif c(1,:) = c(n,:); b = diff (a) ./ h(1:n - 1,idx)\ - h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); b(n,:) = b(1,:); d = diff (c) ./ (3 * h(1:n - 1, idx)); d(n,:) = d(1,:); elseif (strcmp(cond,"not-a-knot")) if (n > 4) dg = 2 * (h(1:n - 2) .+ h(2:n - 1)); dg(1) = dg(1) - h(1); dg(n - 2) = dg(n-2) - h(n - 1); ldg = udg = h(2:n - 2); udg(1) = udg(1) - h(1); ldg(n - 3) = ldg(n-3) - h(n - 1); elseif (n == 4) dg = [h(1) + 2 * h(2), 2 * h(2) + h(3)]; ldg = h(2) - h(3); udg = h(2) - h(1); endif g = zeros(n - 2,columns(y)); g(1,:) = 3 / (h(1) + h(2)) * (a(3,:) - a(2,:)\ - h(2) / h(1) * (a(2,:) - a(1,:))); if (n > 4) g(2:n - 3,:) = 3 * diff (a(3:n - 1,:)) ./ h(2:n - 3,idx)\ - 3 * diff (a(2:n - 2,:)) ./ h(1:n - 4,idx); endif g(n - 2,:) = 3 / (h(n - 1) + h(n - 2)) *\ (h(n - 2) / h(n - 1) * (a(n,:) - a(n - 1,:)) -\ (a(n - 1,:) - a(n - 2,:))); c(2:n - 1,:) = trisolve(ldg,dg,udg,g); c(1,:) = c(2,:) + h(1) / h(2) * (c(2,:) - c(3,:)); c(n,:) = c(n - 1,:) + h(n - 1) / h(n - 2) * (c(n - 1,:) - c(n - 2,:)); b = diff (a) ./ h(1:n - 1, idx)\ - h(1:n - 1, idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); d = diff (c) ./ (3 * h(1:n - 1, idx)); else msg = sprintf("unknown end condition: %s",cond); error (msg); endif d = d(1:n-1,:); c=c(1:n-1,:); b=b(1:n-1,:); a=a(1:n-1,:); coeffs = [d(:), c(:), b(:), a(:)]; pp = mkpp (x, coeffs); endfunction %!shared x,y,cond %! x = linspace(0,2*pi,15)'; y = sin(x); %!assert (ppval(csape(x,y),x), y, 10*eps); %!assert (ppval(csape(x,y),x'), y', 10*eps); %!assert (ppval(csape(x',y'),x'), y', 10*eps); %!assert (ppval(csape(x',y'),x), y, 10*eps); %!assert (ppval(csape(x,[y,y]),x), \ %! [ppval(csape(x,y),x),ppval(csape(x,y),x)], 10*eps) %!test cond='complete'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y,y],cond),x), \ %! [ppval(csape(x,y,cond),x),ppval(csape(x,y,cond),x)], 10*eps) %!test cond='variational'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y,y],cond),x), \ %! [ppval(csape(x,y,cond),x),ppval(csape(x,y,cond),x)], 10*eps) %!test cond='second'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y,y],cond),x), \ %! [ppval(csape(x,y,cond),x),ppval(csape(x,y,cond),x)], 10*eps) %!test cond='periodic'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y,y],cond),x), \ %! [ppval(csape(x,y,cond),x),ppval(csape(x,y,cond),x)], 10*eps) %!test cond='not-a-knot'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y,y],cond),x), \ %! [ppval(csape(x,y,cond),x),ppval(csape(x,y,cond),x)], 10*eps)