--- /dev/null
+## Copyright (C) 2000-2012 Kai Habel
+## Copyright (C) 2006 David Bateman
+##
+## This file is part of Octave.
+##
+## Octave 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 3 of the License, or (at
+## your option) any later version.
+##
+## Octave 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 Octave; see the file COPYING. If not, see
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{pp} =} spline (@var{x}, @var{y})
+## @deftypefnx {Function File} {@var{yi} =} spline (@var{x}, @var{y}, @var{xi})
+## Return the cubic spline interpolant of points @var{x} and @var{y}.
+##
+## When called with two arguments, return the piecewise polynomial @var{pp}
+## that may be used with @code{ppval} to evaluate the polynomial at specific
+## points. When called with a third input argument, @code{spline} evaluates
+## the spline at the points @var{xi}. The third calling form @code{spline
+## (@var{x}, @var{y}, @var{xi})} is equivalent to @code{ppval (spline
+## (@var{x}, @var{y}), @var{xi})}.
+##
+## The variable @var{x} must be a vector of length @var{n}. @var{y} can be
+## either a vector or array. If @var{y} is a vector it must have a length of
+## either @var{n} or @code{@var{n} + 2}. If the length of @var{y} is
+## @var{n}, then the "not-a-knot" end condition is used. If the length of
+## @var{y} is @code{@var{n} + 2}, then the first and last values of the
+## vector @var{y} are the values of the first derivative of the cubic spline
+## at the endpoints.
+##
+## If @var{y} is an array, then the size of @var{y} must have the form
+## @tex
+## $$[s_1, s_2, \cdots, s_k, n]$$
+## @end tex
+## @ifnottex
+## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n}]}
+## @end ifnottex
+## or
+## @tex
+## $$[s_1, s_2, \cdots, s_k, n + 2].$$
+## @end tex
+## @ifnottex
+## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n} + 2]}.
+## @end ifnottex
+## The array is reshaped internally to a matrix where the leading
+## dimension is given by
+## @tex
+## $$s_1 s_2 \cdots s_k$$
+## @end tex
+## @ifnottex
+## @code{@var{s1} * @var{s2} * @dots{} * @var{sk}}
+## @end ifnottex
+## and each row of this matrix is then treated separately. Note that this
+## is exactly opposite to @code{interp1} but is done for @sc{matlab}
+## compatibility.
+##
+## @seealso{pchip, ppval, mkpp, unmkpp}
+## @end deftypefn
+
+## This code is based on csape.m from octave-forge, but has been
+## modified to use the sparse solver code in octave that itself allows
+## special casing of tri-diagonal matrices, modified for NDArrays and
+## for the treatment of vectors y 2 elements longer than x as complete
+## splines.
+
+function ret = spline (x, y, xi)
+
+ x = x(:);
+ n = length (x);
+ if (n < 2)
+ error ("spline: requires at least 2 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
+
+ for k = (1:columns (a))(any (isnan (a)))
+ ok = ! isnan (a(:,k));
+ a(!ok,k) = spline (x(ok), a(ok,k), x(!ok));
+ endfor
+
+ complete = false;
+ if (size (a, 1) == n + 2)
+ complete = true;
+ dfs = a(1,:);
+ dfe = a(end,:);
+ a = a(2:end-1,:);
+ endif
+
+ if (~issorted (x))
+ [x, idx] = sort(x);
+ a = a(idx,:);
+ endif
+
+ b = c = zeros (size (a));
+ h = diff (x);
+ idx = ones (columns (a), 1);
+
+ if (complete)
+
+ if (n == 2)
+ d = (dfs + dfe) / (x(2) - x(1)) ^ 2 + ...
+ 2 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 3;
+ c = (-2 * dfs - dfe) / (x(2) - x(1)) - ...
+ 3 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 2;
+ b = dfs;
+ a = a(1,:);
+
+ d = d(1:n-1,:);
+ c = c(1:n-1,:);
+ b = b(1:n-1,:);
+ a = a(1:n-1,:);
+ else
+ 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) - dfs);
+ g(n-2,:) = 3 / 2 * (3 * (a(n,:) - a(n-1,:)) / h(n-1) - dfe) ...
+ - 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;
+ endif
+
+ c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * dfs
+ - c(2,:) * h(1)) / (2 * h(1));
+ c(n,:) = - (3 / h(n-1) * (a(n,:) - a(n-1,:)) - 3 * dfe
+ + 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));
+
+ d = d(1:n-1,:);
+ c = c(1:n-1,:);
+ b = b(1:n-1,:);
+ a = a(1:n-1,:);
+ endif
+ else
+
+ if (n == 2)
+ b = (a(2,:) - a(1,:)) / (x(2) - x(1));
+ a = a(1,:);
+ d = [];
+ c = [];
+ b = b(1:n-1,:);
+ a = a(1:n-1,:);
+ elseif (n == 3)
+
+ n = 2;
+ c = (a(1,:) - a(3,:)) / ((x(3) - x(1)) * (x(2) - x(3))) ...
+ + (a(2,:) - a(1,:)) / ((x(2) - x(1)) * (x(2) - x(3)));
+ b = (a(2,:) - a(1,:)) * (x(3) - x(1)) ...
+ / ((x(2) - x(1)) * (x(3) - x(2))) ...
+ + (a(1,:) - a(3,:)) * (x(2) - x(1)) ...
+ / ((x(3) - x(1)) * (x(3) - x(2)));
+ a = a(1,:);
+ d = [];
+ x = [min(x), max(x)];
+
+ c = c(1:n-1,:);
+ b = b(1:n-1,:);
+ a = a(1:n-1,:);
+ else
+
+ 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;
+
+ 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));
+
+ d = d(1:n-1,:);d = d.'(:);
+ c = c(1:n-1,:);c = c.'(:);
+ b = b(1:n-1,:);b = b.'(:);
+ a = a(1:n-1,:);a = a.'(:);
+ endif
+
+ endif
+ ret = mkpp (x, cat (2, d, c, b, a), szy(1:end-1));
+
+ if (nargin == 3)
+ ret = ppval (ret, xi);
+ endif
+
+endfunction
+
+%!demo
+%! x = 0:10; y = sin(x);
+%! xspline = 0:0.1:10; yspline = spline(x,y,xspline);
+%! title("spline fit to points from sin(x)");
+%! plot(xspline,sin(xspline),"r",xspline,yspline,"g-",x,y,"b+");
+%! legend("original","interpolation","interpolation points");
+%! %--------------------------------------------------------
+%! % confirm that interpolated function matches the original
+
+%!shared x,y,abserr
+%! x = [0:10]; y = sin(x); abserr = 1e-14;
+%!assert (spline(x,y,x), y, abserr);
+%!assert (spline(x,y,x'), y', abserr);
+%!assert (spline(x',y',x'), y', abserr);
+%!assert (spline(x',y',x), y, abserr);
+%!assert (isempty(spline(x',y',[])));
+%!assert (isempty(spline(x,y,[])));
+%!assert (spline(x,[y;y],x), [spline(x,y,x);spline(x,y,x)],abserr)
+%!assert (spline(x,[y;y],x'), [spline(x,y,x);spline(x,y,x)],abserr)
+%!assert (spline(x',[y;y],x), [spline(x,y,x);spline(x,y,x)],abserr)
+%!assert (spline(x',[y;y],x'), [spline(x,y,x);spline(x,y,x)],abserr)
+%! y = cos(x) + i*sin(x);
+%!assert (spline(x,y,x), y, abserr)
+%!assert (real(spline(x,y,x)), real(y), abserr);
+%!assert (real(spline(x,y,x.')), real(y).', abserr);
+%!assert (real(spline(x.',y.',x.')), real(y).', abserr);
+%!assert (real(spline(x.',y,x)), real(y), abserr);
+%!assert (imag(spline(x,y,x)), imag(y), abserr);
+%!assert (imag(spline(x,y,x.')), imag(y).', abserr);
+%!assert (imag(spline(x.',y.',x.')), imag(y).', abserr);
+%!assert (imag(spline(x.',y,x)), imag(y), abserr);
+%!test
+%! xnan = 5;
+%! y(x==xnan) = NaN;
+%! ok = ! isnan (y);
+%! assert (spline (x, y, x(ok)), y(ok), abserr);
+%!test
+%! ok = ! isnan (y);
+%! assert (! isnan (spline (x, y, x(!ok))));
+%!test
+%! x = [1,2];
+%! y = [1,4];
+%! assert (spline (x,y,x), [1,4], abserr);
+%!test
+%! x = [2,1];
+%! y = [1,4];
+%! assert (spline (x,y,x), [1,4], abserr);
+%!test
+%! x = [1,2];
+%! y = [1,2,3,4];
+%! pp = spline (x,y);
+%! [x,P] = unmkpp (pp);
+%! assert (norm (P-[3,-3,1,2]), 0, abserr);
+%!test
+%! x = [2,1];
+%! y = [1,2,3,4];
+%! pp = spline (x,y);
+%! [x,P] = unmkpp (pp);
+%! assert (norm (P-[7,-9,1,3]), 0, abserr);
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff