Mercurial > octave-nkf
view scripts/polynomial/spline.m @ 14868:5d3a684236b0
maint: Use Octave coding conventions for cuddling parentheses in scripts directory
* lin2mu.m, loadaudio.m, wavread.m, accumarray.m, bicubic.m, celldisp.m,
colon.m, cplxpair.m, dblquad.m, divergence.m, genvarname.m, gradient.m,
int2str.m, interp1.m, interp1q.m, interp2.m, interpn.m, loadobj.m, nthargout.m,
__isequal__.m, __splinen__.m, quadgk.m, quadl.m, quadv.m, rat.m, rot90.m,
rotdim.m, saveobj.m, subsindex.m, triplequad.m, delaunay3.m, griddata.m,
inpolygon.m, tsearchn.m, voronoi.m, get_first_help_sentence.m, which.m,
gray2ind.m, pink.m, dlmwrite.m, strread.m, textread.m, textscan.m, housh.m,
ishermitian.m, issymmetric.m, krylov.m, logm.m, null.m, rref.m,
compare_versions.m, copyfile.m, dump_prefs.m, edit.m, fileparts.m,
getappdata.m, isappdata.m, movefile.m, orderfields.m, parseparams.m,
__xzip__.m, rmappdata.m, setappdata.m, swapbytes.m, unpack.m, ver.m, fminbnd.m,
fminunc.m, fsolve.m, glpk.m, lsqnonneg.m, qp.m, sqp.m, configure_make.m,
copy_files.m, describe.m, get_description.m, get_forge_pkg.m, install.m,
installed_packages.m, is_architecture_dependent.m, load_package_dirs.m,
print_package_description.m, rebuild.m, repackage.m, save_order.m, shell.m,
allchild.m, ancestor.m, area.m, axes.m, axis.m, clabel.m, close.m, colorbar.m,
comet.m, comet3.m, contour.m, cylinder.m, ezmesh.m, ezsurf.m, findobj.m,
fplot.m, hist.m, isocolors.m, isonormals.m, isosurface.m, isprop.m, legend.m,
mesh.m, meshz.m, pareto.m, pcolor.m, peaks.m, plot3.m, plotmatrix.m, plotyy.m,
polar.m, print.m, __add_datasource__.m, __add_default_menu__.m,
__axes_limits__.m, __bar__.m, __clabel__.m, __contour__.m, __errcomm__.m,
__errplot__.m, __ezplot__.m, __file_filter__.m, __fltk_print__.m,
__ghostscript__.m, __gnuplot_print__.m, __go_draw_axes__.m,
__go_draw_figure__.m, __interp_cube__.m, __marching_cube__.m, __patch__.m,
__pie__.m, __plt__.m, __print_parse_opts__.m, __quiver__.m, __scatter__.m,
__stem__.m, __tight_eps_bbox__.m, __uigetdir_fltk__.m, __uigetfile_fltk__.m,
__uiputfile_fltk__.m, quiver.m, quiver3.m, rectangle.m, refreshdata.m,
ribbon.m, scatter.m, semilogy.m, shading.m, slice.m, subplot.m, surface.m,
surfl.m, surfnorm.m, text.m, uigetfile.m, uiputfile.m, whitebg.m, deconv.m,
mkpp.m, pchip.m, polyaffine.m, polyder.m, polygcd.m, polyout.m, polyval.m,
ppint.m, ppjumps.m, ppval.m, residue.m, roots.m, spline.m, splinefit.m,
addpref.m, getpref.m, setpref.m, ismember.m, setxor.m, arch_fit.m, arch_rnd.m,
arch_test.m, autoreg_matrix.m, diffpara.m, fftconv.m, filter2.m, hanning.m,
hurst.m, periodogram.m, triangle_sw.m, sinc.m, spectral_xdf.m, spencer.m,
stft.m, synthesis.m, unwrap.m, yulewalker.m, bicgstab.m, gmres.m, pcg.m, pcr.m,
__sprand_impl__.m, speye.m, spfun.m, sprandn.m, spstats.m, svds.m,
treelayout.m, treeplot.m, bessel.m, factor.m, legendre.m, perms.m, primes.m,
magic.m, toeplitz.m, corr.m, cov.m, mean.m, median.m, mode.m, qqplot.m,
quantile.m, ranks.m, zscore.m, logistic_regression_likelihood.m,
bartlett_test.m, chisquare_test_homogeneity.m, chisquare_test_independence.m,
kolmogorov_smirnov_test.m, run_test.m, u_test.m, wilcoxon_test.m, z_test.m,
z_test_2.m, bin2dec.m, dec2base.m, mat2str.m, strcat.m, strchr.m, strjust.m,
strtok.m, substr.m, untabify.m, assert.m, demo.m, example.m, fail.m, speed.m,
test.m, now.m: Use Octave coding conventions for cuddling parentheses in
scripts directory.
author | Rik <octave@nomad.inbox5.com> |
---|---|
date | Tue, 17 Jul 2012 07:08:39 -0700 |
parents | 2e23cd0a9e40 |
children | c2dbdeaa25df |
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## 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 g(1,:) = (a(2,:) - a(1,:)) / h(1) - dfs; g(2:n-1,:) = (a(3:n,:) - a(2:n-1,:)) ./ h(2:n-1) - ... (a(2:n-1,:) - a(1:n-2,:)) ./ h(1:n-2); g(n,:) = dfe - (a(n,:) - a(n-1,:)) / h(n-1); c = spdiags ([[h/6;0],[h(1)/3;(h(1:n-2)+h(2:n-1))/3;h(n-1)/3],[0;h/6]],... [-1,0,1],n,n) \ (g / 2); 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 (P, [3,-3,1,2], abserr); %!test %! x = [2,1]; %! y = [1,2,3,4]; %! pp = spline (x,y); %! [x,P] = unmkpp (pp); %! assert (P, [7,-9,1,3], abserr); %!test %! x = [0,1,2]; %! y = [0,0,1,0,0]; %! pp = spline (x,y); %! [x,P] = unmkpp (pp); %! assert (P, [-2,3,0,0;2,-3,0,1], abserr); %!test %! x = [0,1,2,3]; %! y = [0,0,1,1,0,0]; %! pp = spline (x,y); %! [x,P] = unmkpp (pp); %! assert (P, [-1,2,0,0;0,-1,1,1;1,-1,-1,1], abserr);