--- /dev/null
+## 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, see <http://www.gnu.org/licenses/>.
+
+## -*- 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);
+ if (n < 3)
+ error("csape requires at least 3 points");
+ endif
+
+ ## Check the size and shape of y
+ ndy = ndims (y);
+ szy = size (y);
+ if (ndy == 2 && (szy(1) == n || szy(2) == n))
+ if (szy(2) == n)
+ a = y.';
+ else
+ a = y;
+ szy = fliplr (szy);
+ endif
+ else
+ a = shiftdim (reshape (y, [prod(szy(1:end-1)), szy(end)]), 1);
+ endif
+
+
+ b = c = zeros (size (a));
+ h = diff (x);
+ idx = ones (columns(a),1);
+
+ if (nargin < 3 || strcmp(cond,"complete"))
+ # specified first derivative at end point
+ if (nargin < 4)
+ valc = [0, 0];
+ endif
+
+ if (n == 3)
+ dg = 1.5 * h(1) - 0.5 * h(2);
+ c(2:n - 1,:) = 1/dg(1);
+ else
+ 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,:) = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-2,n-2) \ g;
+
+ end
+
+ 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,:);
+
+ if( n == 3)
+ dg = 2 * h(1);
+ c(2:n - 1,:) = g / dg;
+ else
+ dg = 2 * (h(1:n - 2) .+ h(2:n - 1));
+ e = h(2:n - 2);
+ c(2:n - 1,:) = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-2,n-2) \ g;
+ end
+
+ 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);
+
+ ## Use Sherman-Morrison formula to extend the solution
+ ## to the cyclic system. See Numerical Recipes in C, pp 73-75
+ gamma = - dg(1);
+ dg(1) -= gamma;
+ dg(end) -= h(1) * h(1) / gamma;
+ z = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-1,n-1) \ ...
+ [[gamma; zeros(n-3,1); h(1)],g];
+ fact = (z(1,2:end) + h(1) * z(end,2:end) / gamma) / ...
+ (1.0 + z(1,1) + h(1) * z(end,1) / gamma);
+
+ c(2:n,idx) = z(:,2:end) - z(:,1) * fact;
+ 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"))
+
+ g = zeros(n - 2,columns(a));
+ g(1,:) = 3 / (h(1) + h(2)) * (a(3,:) - a(2,:)\
+ - h(2) / h(1) * (a(2,:) - a(1,:)));
+ 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,:)));
+
+ if (n > 4)
+
+ g(2:n - 3,:) = 3 * diff (a(3:n - 1,:)) ./ h(3:n - 2,idx)\
+ - 3 * diff (a(2:n - 2,:)) ./ h(2:n - 3,idx);
+
+ 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);
+ c(2:n - 1,:) = spdiags([[ldg(:);0],dg,[0;udg(:)]],[-1,0,1],n-2,n-2) \ g;
+
+ elseif (n == 4)
+
+ dg = [h(1) + 2 * h(2), 2 * h(2) + h(3)];
+ ldg = h(2) - h(3);
+ udg = h(2) - h(1);
+ c(2:n - 1,:) = spdiags([[ldg(:);0],dg,[0;udg(:)]],[-1,0,1],n-2,n-2) \ g;
+
+ else # n == 3
+
+ dg= [h(1) + 2 * h(2)];
+ c(2:n - 1,:) = g/dg(1);
+
+ endif
+
+ 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,:);
+ pp = mkpp (x, cat (2, d'(:), c'(:), b'(:), a'(:)), szy(1:end-1));
+
+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)
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff