Mercurial > octave-antonio
annotate scripts/polynomial/spline.m @ 14363:f3d52523cde1
Use Octave coding conventions in all m-file %!test blocks
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csc.m, cscd.m, csch.m, sec.m, secd.m, sech.m, sind.m, tand.m, accumarray.m,
accumdim.m, bitcmp.m, bitget.m, bitset.m, blkdiag.m, cart2pol.m, cart2sph.m,
celldisp.m, chop.m, circshift.m, colon.m, common_size.m, cplxpair.m,
cumtrapz.m, curl.m, dblquad.m, deal.m, divergence.m, flipdim.m, fliplr.m,
flipud.m, genvarname.m, gradient.m, idivide.m, int2str.m, interp1.m,
interp1q.m, interp2.m, interp3.m, interpft.m, interpn.m, isa.m, isdir.m,
isequal.m, isequalwithequalnans.m, issquare.m, logspace.m, nargchk.m,
narginchk.m, nargoutchk.m, nextpow2.m, nthargout.m, num2str.m, pol2cart.m,
polyarea.m, postpad.m, prepad.m, profile.m, profshow.m, quadgk.m, quadv.m,
randi.m, rat.m, repmat.m, rot90.m, rotdim.m, shift.m, shiftdim.m, sph2cart.m,
structfun.m, trapz.m, triplequad.m, convhull.m, dsearch.m, dsearchn.m,
griddata3.m, griddatan.m, rectint.m, tsearchn.m, __makeinfo__.m, doc.m,
get_first_help_sentence.m, help.m, type.m, unimplemented.m, which.m, imread.m,
imwrite.m, dlmwrite.m, fileread.m, is_valid_file_id.m, strread.m, textread.m,
textscan.m, commutation_matrix.m, cond.m, condest.m, cross.m,
duplication_matrix.m, expm.m, housh.m, isdefinite.m, ishermitian.m,
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fullfile.m, getfield.m, gzip.m, info.m, inputname.m, isappdata.m, isdeployed.m,
ismac.m, ispc.m, isunix.m, list_primes.m, ls.m, mexext.m, namelengthmax.m,
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setappdata.m, setfield.m, substruct.m, symvar.m, ver.m, version.m,
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ginput.m, graphics_toolkit.m, gtext.m, hggroup.m, hist.m, hold.m, isfigure.m,
ishghandle.m, ishold.m, isocolors.m, isonormals.m, isosurface.m, isprop.m,
legend.m, line.m, loglog.m, loglogerr.m, meshgrid.m, ndgrid.m, newplot.m,
orient.m, patch.m, plot3.m, plotyy.m, __print_parse_opts__.m, quiver3.m,
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deconv.m, mkpp.m, mpoles.m, pchip.m, poly.m, polyaffine.m, polyder.m,
polyfit.m, polygcd.m, polyint.m, polyout.m, polyval.m, polyvalm.m, ppder.m,
ppint.m, ppjumps.m, ppval.m, residue.m, roots.m, spline.m, intersect.m,
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range.m, ranks.m, run_count.m, runlength.m, skewness.m, spearman.m,
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gaminv.m, gampdf.m, gamrnd.m, geocdf.m, geoinv.m, geopdf.m, geornd.m,
hygecdf.m, hygeinv.m, hygepdf.m, hygernd.m, kolmogorov_smirnov_cdf.m,
laplace_cdf.m, laplace_inv.m, laplace_pdf.m, laplace_rnd.m, logistic_cdf.m,
logistic_inv.m, logistic_pdf.m, logistic_rnd.m, logncdf.m, logninv.m,
lognpdf.m, lognrnd.m, nbincdf.m, nbininv.m, nbinpdf.m, nbinrnd.m, normcdf.m,
norminv.m, normpdf.m, normrnd.m, poisscdf.m, poissinv.m, poisspdf.m,
poissrnd.m, stdnormal_cdf.m, stdnormal_inv.m, stdnormal_pdf.m, stdnormal_rnd.m,
tcdf.m, tinv.m, tpdf.m, trnd.m, unidcdf.m, unidinv.m, unidpdf.m, unidrnd.m,
unifcdf.m, unifinv.m, unifpdf.m, unifrnd.m, wblcdf.m, wblinv.m, wblpdf.m,
wblrnd.m, kolmogorov_smirnov_test.m, kruskal_wallis_test.m, base2dec.m,
bin2dec.m, blanks.m, cstrcat.m, deblank.m, dec2base.m, dec2bin.m, dec2hex.m,
findstr.m, hex2dec.m, index.m, isletter.m, mat2str.m, rindex.m, str2num.m,
strcat.m, strjust.m, strmatch.m, strsplit.m, strtok.m, strtrim.m, strtrunc.m,
substr.m, validatestring.m, demo.m, example.m, fail.m, speed.m, addtodate.m,
asctime.m, clock.m, ctime.m, date.m, datenum.m, datetick.m, datevec.m,
eomday.m, etime.m, is_leap_year.m, now.m:
Use Octave coding conventions in all m-file %!test blocks
author | Rik <octave@nomad.inbox5.com> |
---|---|
date | Mon, 13 Feb 2012 07:29:44 -0800 |
parents | 11949c9795a0 |
children | 2e23cd0a9e40 |
rev | line source |
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1 ## Copyright (C) 2000-2012 Kai Habel |
5824 | 2 ## Copyright (C) 2006 David Bateman |
3 ## | |
4 ## This file is part of Octave. | |
5 ## | |
6 ## Octave is free software; you can redistribute it and/or modify it | |
7 ## under the terms of the GNU General Public License as published by | |
7016 | 8 ## the Free Software Foundation; either version 3 of the License, or (at |
9 ## your option) any later version. | |
5824 | 10 ## |
11 ## Octave is distributed in the hope that it will be useful, but | |
12 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
14 ## General Public License for more details. | |
15 ## | |
16 ## You should have received a copy of the GNU General Public License | |
7016 | 17 ## along with Octave; see the file COPYING. If not, see |
18 ## <http://www.gnu.org/licenses/>. | |
5824 | 19 |
20 ## -*- texinfo -*- | |
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21 ## @deftypefn {Function File} {@var{pp} =} spline (@var{x}, @var{y}) |
7650 | 22 ## @deftypefnx {Function File} {@var{yi} =} spline (@var{x}, @var{y}, @var{xi}) |
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23 ## Return the cubic spline interpolant of points @var{x} and @var{y}. |
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24 ## |
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25 ## When called with two arguments, return the piecewise polynomial @var{pp} |
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26 ## that may be used with @code{ppval} to evaluate the polynomial at specific |
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27 ## points. When called with a third input argument, @code{spline} evaluates |
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28 ## the spline at the points @var{xi}. The third calling form @code{spline |
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29 ## (@var{x}, @var{y}, @var{xi})} is equivalent to @code{ppval (spline |
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30 ## (@var{x}, @var{y}), @var{xi})}. |
5824 | 31 ## |
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32 ## The variable @var{x} must be a vector of length @var{n}. @var{y} can be |
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33 ## either a vector or array. If @var{y} is a vector it must have a length of |
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34 ## either @var{n} or @code{@var{n} + 2}. If the length of @var{y} is |
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35 ## @var{n}, then the "not-a-knot" end condition is used. If the length of |
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36 ## @var{y} is @code{@var{n} + 2}, then the first and last values of the |
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37 ## vector @var{y} are the values of the first derivative of the cubic spline |
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38 ## at the endpoints. |
5824 | 39 ## |
40 ## If @var{y} is an array, then the size of @var{y} must have the form | |
41 ## @tex | |
42 ## $$[s_1, s_2, \cdots, s_k, n]$$ | |
43 ## @end tex | |
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44 ## @ifnottex |
5824 | 45 ## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n}]} |
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46 ## @end ifnottex |
5824 | 47 ## or |
48 ## @tex | |
8828 | 49 ## $$[s_1, s_2, \cdots, s_k, n + 2].$$ |
5824 | 50 ## @end tex |
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51 ## @ifnottex |
5824 | 52 ## @code{[@var{s1}, @var{s2}, @dots{}, @var{sk}, @var{n} + 2]}. |
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53 ## @end ifnottex |
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54 ## The array is reshaped internally to a matrix where the leading |
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55 ## dimension is given by |
5824 | 56 ## @tex |
57 ## $$s_1 s_2 \cdots s_k$$ | |
58 ## @end tex | |
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59 ## @ifnottex |
5824 | 60 ## @code{@var{s1} * @var{s2} * @dots{} * @var{sk}} |
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61 ## @end ifnottex |
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62 ## and each row of this matrix is then treated separately. Note that this |
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63 ## is exactly opposite to @code{interp1} but is done for @sc{matlab} |
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64 ## compatibility. |
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65 ## |
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66 ## @seealso{pchip, ppval, mkpp, unmkpp} |
5824 | 67 ## @end deftypefn |
68 | |
69 ## This code is based on csape.m from octave-forge, but has been | |
70 ## modified to use the sparse solver code in octave that itself allows | |
71 ## special casing of tri-diagonal matrices, modified for NDArrays and | |
72 ## for the treatment of vectors y 2 elements longer than x as complete | |
73 ## splines. | |
74 | |
75 function ret = spline (x, y, xi) | |
76 | |
77 x = x(:); | |
78 n = length (x); | |
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79 if (n < 2) |
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80 error ("spline: requires at least 2 points"); |
5824 | 81 endif |
82 | |
83 ## Check the size and shape of y | |
84 ndy = ndims (y); | |
85 szy = size (y); | |
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86 if (ndy == 2 && (szy(1) == n || szy(2) == n)) |
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87 if (szy(2) == n) |
6014 | 88 a = y.'; |
5824 | 89 else |
90 a = y; | |
91 szy = fliplr (szy); | |
92 endif | |
93 else | |
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94 a = shiftdim (reshape (y, [prod(szy(1:end-1)), szy(end)]), 1); |
5824 | 95 endif |
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96 |
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97 for k = (1:columns (a))(any (isnan (a))) |
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98 ok = ! isnan (a(:,k)); |
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99 a(!ok,k) = spline (x(ok), a(ok,k), x(!ok)); |
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100 endfor |
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101 |
5824 | 102 complete = false; |
103 if (size (a, 1) == n + 2) | |
104 complete = true; | |
105 dfs = a(1,:); | |
106 dfe = a(end,:); | |
107 a = a(2:end-1,:); | |
108 endif | |
109 | |
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110 if (! issorted (x)) |
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111 [x, idx] = sort(x); |
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112 a = a(idx,:); |
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113 endif |
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114 |
5824 | 115 b = c = zeros (size (a)); |
116 h = diff (x); | |
5838 | 117 idx = ones (columns (a), 1); |
5824 | 118 |
119 if (complete) | |
120 | |
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121 if (n == 2) |
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122 d = (dfs + dfe) / (x(2) - x(1)) ^ 2 + ... |
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123 2 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 3; |
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124 c = (-2 * dfs - dfe) / (x(2) - x(1)) - ... |
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125 3 * (a(1,:) - a(2,:)) / (x(2) - x(1)) ^ 2; |
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126 b = dfs; |
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127 a = a(1,:); |
5824 | 128 |
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129 d = d(1:n-1,:); |
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130 c = c(1:n-1,:); |
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131 b = b(1:n-1,:); |
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132 a = a(1:n-1,:); |
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133 else |
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134 if (n == 3) |
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135 dg = 1.5 * h(1) - 0.5 * h(2); |
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136 c(2:n-1,:) = 1/dg(1); |
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137 else |
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138 dg = 2 * (h(1:n-2) .+ h(2:n-1)); |
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139 dg(1) = dg(1) - 0.5 * h(1); |
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140 dg(n-2) = dg(n-2) - 0.5 * h(n-1); |
5824 | 141 |
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142 e = h(2:n-2); |
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143 |
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144 g = 3 * diff (a(2:n,:)) ./ h(2:n-1,idx) ... |
5838 | 145 - 3 * diff (a(1:n-1,:)) ./ h(1:n-2,idx); |
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146 g(1,:) = 3 * (a(3,:) - a(2,:)) / h(2) ... |
5824 | 147 - 3 / 2 * (3 * (a(2,:) - a(1,:)) / h(1) - dfs); |
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148 g(n-2,:) = 3 / 2 * (3 * (a(n,:) - a(n-1,:)) / h(n-1) - dfe) ... |
5838 | 149 - 3 * (a(n-1,:) - a(n-2,:)) / h(n-2); |
5824 | 150 |
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151 c(2:n-1,:) = spdiags ([[e(:); 0], dg, [0; e(:)]], |
10549 | 152 [-1, 0, 1], n-2, n-2) \ g; |
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153 endif |
5824 | 154 |
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155 c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * dfs |
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156 - c(2,:) * h(1)) / (2 * h(1)); |
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157 c(n,:) = - (3 / h(n-1) * (a(n,:) - a(n-1,:)) - 3 * dfe |
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158 + c(n-1,:) * h(n-1)) / (2 * h(n-1)); |
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159 b(1:n-1,:) = diff (a) ./ h(1:n-1, idx) ... |
10549 | 160 - h(1:n-1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); |
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161 d = diff (c) ./ (3 * h(1:n-1, idx)); |
5824 | 162 |
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163 d = d(1:n-1,:); |
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164 c = c(1:n-1,:); |
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165 b = b(1:n-1,:); |
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166 a = a(1:n-1,:); |
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167 endif |
5824 | 168 else |
169 | |
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170 if (n == 2) |
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171 b = (a(2,:) - a(1,:)) / (x(2) - x(1)); |
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172 a = a(1,:); |
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173 d = []; |
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174 c = []; |
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175 b = b(1:n-1,:); |
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176 a = a(1:n-1,:); |
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177 elseif (n == 3) |
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179 n = 2; |
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180 c = (a(1,:) - a(3,:)) / ((x(3) - x(1)) * (x(2) - x(3))) ... |
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181 + (a(2,:) - a(1,:)) / ((x(2) - x(1)) * (x(2) - x(3))); |
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182 b = (a(2,:) - a(1,:)) * (x(3) - x(1)) ... |
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183 / ((x(2) - x(1)) * (x(3) - x(2))) ... |
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184 + (a(1,:) - a(3,:)) * (x(2) - x(1)) ... |
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185 / ((x(3) - x(1)) * (x(3) - x(2))); |
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186 a = a(1,:); |
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187 d = []; |
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188 x = [min(x), max(x)]; |
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189 |
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190 c = c(1:n-1,:); |
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191 b = b(1:n-1,:); |
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192 a = a(1:n-1,:); |
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193 else |
5824 | 194 |
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195 g = zeros (n-2, columns (a)); |
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196 g(1,:) = 3 / (h(1) + h(2)) ... |
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197 * (a(3,:) - a(2,:) - h(2) / h(1) * (a(2,:) - a(1,:))); |
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198 g(n-2,:) = 3 / (h(n-1) + h(n-2)) ... |
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199 * (h(n-2) / h(n-1) * (a(n,:) - a(n-1,:)) - (a(n-1,:) - a(n-2,:))); |
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200 |
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201 if (n > 4) |
5824 | 202 |
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203 g(2:n - 3,:) = 3 * diff (a(3:n-1,:)) ./ h(3:n-2,idx) ... |
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204 - 3 * diff (a(2:n-2,:)) ./ h(2:n - 3,idx); |
5824 | 205 |
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206 dg = 2 * (h(1:n-2) .+ h(2:n-1)); |
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207 dg(1) = dg(1) - h(1); |
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208 dg(n-2) = dg(n-2) - h(n-1); |
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209 |
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210 ldg = udg = h(2:n-2); |
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211 udg(1) = udg(1) - h(1); |
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212 ldg(n - 3) = ldg(n-3) - h(n-1); |
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213 c(2:n-1,:) = spdiags ([[ldg(:); 0], dg, [0; udg(:)]], |
10549 | 214 [-1, 0, 1], n-2, n-2) \ g; |
5824 | 215 |
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216 elseif (n == 4) |
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217 |
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218 dg = [h(1) + 2 * h(2); 2 * h(2) + h(3)]; |
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219 ldg = h(2) - h(3); |
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220 udg = h(2) - h(1); |
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221 c(2:n-1,:) = spdiags ([[ldg(:);0], dg, [0; udg(:)]], |
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222 [-1, 0, 1], n-2, n-2) \ g; |
5824 | 223 |
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224 endif |
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225 |
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226 c(1,:) = c(2,:) + h(1) / h(2) * (c(2,:) - c(3,:)); |
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227 c(n,:) = c(n-1,:) + h(n-1) / h(n-2) * (c(n-1,:) - c(n-2,:)); |
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228 b = diff (a) ./ h(1:n-1, idx) ... |
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229 - h(1:n-1, idx) / 3 .* (c(2:n,:) + 2 * c(1:n-1,:)); |
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230 d = diff (c) ./ (3 * h(1:n-1, idx)); |
5824 | 231 |
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232 d = d(1:n-1,:);d = d.'(:); |
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233 c = c(1:n-1,:);c = c.'(:); |
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234 b = b(1:n-1,:);b = b.'(:); |
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235 a = a(1:n-1,:);a = a.'(:); |
5824 | 236 endif |
237 | |
238 endif | |
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239 ret = mkpp (x, cat (2, d, c, b, a), szy(1:end-1)); |
5824 | 240 |
241 if (nargin == 3) | |
242 ret = ppval (ret, xi); | |
243 endif | |
244 | |
245 endfunction | |
246 | |
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247 |
5824 | 248 %!demo |
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249 %! x = 0:10; y = sin (x); |
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250 %! xspline = 0:0.1:10; yspline = spline (x,y,xspline); |
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251 %! title ("spline fit to points from sin (x)"); |
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252 %! plot (xspline,sin(xspline),"r", xspline,yspline,"g-", x,y,"b+"); |
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253 %! legend ("original", "interpolation", "interpolation points"); |
5824 | 254 %! %-------------------------------------------------------- |
255 %! % confirm that interpolated function matches the original | |
256 | |
6686 | 257 %!shared x,y,abserr |
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258 %! x = [0:10]; y = sin (x); abserr = 1e-14; |
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259 %!assert (spline (x,y,x), y, abserr) |
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260 %!assert (spline (x,y,x'), y', abserr) |
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261 %!assert (spline (x',y',x'), y', abserr) |
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262 %!assert (spline (x',y',x), y, abserr) |
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263 %!assert (isempty (spline (x',y',[]))) |
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264 %!assert (isempty (spline (x,y,[]))) |
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265 %!assert (spline (x,[y;y],x), [spline(x,y,x);spline(x,y,x)], abserr) |
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266 %!assert (spline (x,[y;y],x'), [spline(x,y,x);spline(x,y,x)], abserr) |
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267 %!assert (spline (x',[y;y],x), [spline(x,y,x);spline(x,y,x)], abserr) |
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268 %!assert (spline (x',[y;y],x'), [spline(x,y,x);spline(x,y,x)], abserr) |
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269 %! y = cos (x) + i*sin (x); |
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270 %!assert (spline (x,y,x), y, abserr) |
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271 %!assert (real (spline (x,y,x)), real (y), abserr) |
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272 %!assert (real (spline (x,y,x.')), real (y).', abserr) |
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273 %!assert (real (spline (x.',y.',x.')), real (y).', abserr) |
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274 %!assert (real (spline (x.',y,x)), real (y), abserr) |
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275 %!assert (imag (spline (x,y,x)), imag (y), abserr) |
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276 %!assert (imag (spline (x,y,x.')), imag (y).', abserr) |
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277 %!assert (imag (spline (x.',y.',x.')), imag (y).', abserr) |
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278 %!assert (imag (spline (x.',y,x)), imag (y), abserr) |
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279 %!test |
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280 %! xnan = 5; |
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281 %! y(x==xnan) = NaN; |
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282 %! ok = ! isnan (y); |
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283 %! assert (spline (x, y, x(ok)), y(ok), abserr); |
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284 %!test |
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285 %! ok = ! isnan (y); |
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286 %! assert (! isnan (spline (x, y, x(!ok)))); |
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287 %!test |
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288 %! x = [1,2]; |
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289 %! y = [1,4]; |
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290 %! assert (spline (x,y,x), [1,4], abserr); |
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291 %!test |
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292 %! x = [2,1]; |
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293 %! y = [1,4]; |
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294 %! assert (spline (x,y,x), [1,4], abserr); |
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295 %!test |
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296 %! x = [1,2]; |
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297 %! y = [1,2,3,4]; |
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298 %! pp = spline (x,y); |
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299 %! [x,P] = unmkpp (pp); |
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300 %! assert (norm (P-[3,-3,1,2]), 0, abserr); |
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301 %!test |
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302 %! x = [2,1]; |
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303 %! y = [1,2,3,4]; |
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304 %! pp = spline (x,y); |
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305 %! [x,P] = unmkpp (pp); |
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306 %! assert (norm (P-[7,-9,1,3]), 0, abserr); |
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307 |