Mercurial > octave-nkf
annotate doc/interpreter/interp.txi @ 17281:bc924baa2c4e
doc: Add new @qcode macro for code samples which are quoted.
Macro handles options ("on") or properties ("position") more elegantly
than @code{"text"}.
* doc/interpreter/macros.texi: Add new @qcode macro.
* doc/interpreter/tips.txi: Add documentation about @qcode macro.
* doc/interpreter/basics.txi, doc/interpreter/container.txi,
doc/interpreter/emacs.txi, doc/interpreter/errors.txi,
doc/interpreter/eval.txi, doc/interpreter/expr.txi,
doc/interpreter/external.txi, doc/interpreter/func.txi,
doc/interpreter/grammar.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/io.txi, doc/interpreter/matrix.txi,
doc/interpreter/numbers.txi, doc/interpreter/oop.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/quad.txi, doc/interpreter/sparse.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/vectorize.txi, libinterp/corefcn/balance.cc,
libinterp/corefcn/bitfcns.cc, libinterp/corefcn/cellfun.cc,
libinterp/corefcn/conv2.cc, libinterp/corefcn/data.cc,
libinterp/corefcn/debug.cc, libinterp/corefcn/defaults.cc,
libinterp/corefcn/dirfns.cc, libinterp/corefcn/dlmread.cc,
libinterp/corefcn/error.cc, libinterp/corefcn/file-io.cc,
libinterp/corefcn/find.cc, libinterp/corefcn/gammainc.cc,
libinterp/corefcn/graphics.cc, libinterp/corefcn/help.cc,
libinterp/corefcn/hex2num.cc, libinterp/corefcn/input.cc,
libinterp/corefcn/load-path.cc, libinterp/corefcn/load-save.cc,
libinterp/corefcn/ls-oct-ascii.cc, libinterp/corefcn/lu.cc,
libinterp/corefcn/luinc.cc, libinterp/corefcn/matrix_type.cc,
libinterp/corefcn/oct-hist.cc, libinterp/corefcn/pager.cc,
libinterp/corefcn/pr-output.cc, libinterp/corefcn/pt-jit.cc,
libinterp/corefcn/qz.cc, libinterp/corefcn/rand.cc,
libinterp/corefcn/regexp.cc, libinterp/corefcn/schur.cc,
libinterp/corefcn/sighandlers.cc, libinterp/corefcn/sparse.cc,
libinterp/corefcn/spparms.cc, libinterp/corefcn/str2double.cc,
libinterp/corefcn/svd.cc, libinterp/corefcn/symtab.cc,
libinterp/corefcn/syscalls.cc, libinterp/corefcn/toplev.cc,
libinterp/corefcn/tril.cc, libinterp/corefcn/typecast.cc,
libinterp/corefcn/utils.cc, libinterp/corefcn/variables.cc,
libinterp/dldfcn/__init_fltk__.cc, libinterp/dldfcn/chol.cc,
libinterp/dldfcn/colamd.cc, libinterp/dldfcn/fftw.cc, libinterp/dldfcn/qr.cc,
libinterp/dldfcn/symbfact.cc, libinterp/octave-value/ov-base.cc,
libinterp/octave-value/ov-fcn-handle.cc,
libinterp/octave-value/ov-fcn-inline.cc, libinterp/octave-value/ov-java.cc,
libinterp/octave-value/ov-range.cc, libinterp/octave-value/ov-struct.cc,
libinterp/octave-value/ov-usr-fcn.cc, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/pt-binop.cc, libinterp/parse-tree/pt-eval.cc,
libinterp/parse-tree/pt-mat.cc, scripts/@ftp/ftp.m,
scripts/deprecated/java_convert_matrix.m, scripts/deprecated/java_debug.m,
scripts/deprecated/java_unsigned_conversion.m, scripts/deprecated/shell_cmd.m,
scripts/general/dblquad.m, scripts/general/display.m,
scripts/general/genvarname.m, scripts/general/idivide.m,
scripts/general/interp1.m, scripts/general/interp2.m,
scripts/general/interp3.m, scripts/general/interpn.m, scripts/general/isa.m,
scripts/general/profexplore.m, scripts/general/profile.m,
scripts/general/quadgk.m, scripts/general/randi.m, scripts/general/structfun.m,
scripts/general/subsindex.m, scripts/general/triplequad.m,
scripts/geometry/griddata.m, scripts/geometry/griddata3.m,
scripts/geometry/griddatan.m, scripts/geometry/voronoi.m, scripts/help/help.m,
scripts/help/lookfor.m, scripts/image/cmpermute.m, scripts/image/colormap.m,
scripts/image/image.m, scripts/image/imagesc.m, scripts/image/imfinfo.m,
scripts/image/imformats.m, scripts/image/imread.m, scripts/image/imshow.m,
scripts/image/imwrite.m, scripts/image/ind2gray.m, scripts/image/lines.m,
scripts/image/rgb2ind.m, scripts/image/spinmap.m, scripts/io/dlmwrite.m,
scripts/io/strread.m, scripts/io/textread.m, scripts/io/textscan.m,
scripts/java/javaclasspath.m, scripts/java/usejava.m,
scripts/miscellaneous/bzip2.m, scripts/miscellaneous/computer.m,
scripts/miscellaneous/copyfile.m, scripts/miscellaneous/debug.m,
scripts/miscellaneous/dos.m, scripts/miscellaneous/edit.m,
scripts/miscellaneous/gzip.m, scripts/miscellaneous/license.m,
scripts/miscellaneous/mkoctfile.m, scripts/miscellaneous/movefile.m,
scripts/miscellaneous/parseparams.m, scripts/miscellaneous/unix.m,
scripts/optimization/fminbnd.m, scripts/optimization/fminsearch.m,
scripts/optimization/fminunc.m, scripts/optimization/fsolve.m,
scripts/optimization/fzero.m, scripts/optimization/glpk.m,
scripts/optimization/lsqnonneg.m, scripts/optimization/optimset.m,
scripts/optimization/pqpnonneg.m, scripts/pkg/pkg.m, scripts/plot/allchild.m,
scripts/plot/ancestor.m, scripts/plot/area.m, scripts/plot/axis.m,
scripts/plot/bar.m, scripts/plot/barh.m, scripts/plot/box.m,
scripts/plot/caxis.m, scripts/plot/cla.m, scripts/plot/clabel.m,
scripts/plot/clf.m, scripts/plot/close.m, scripts/plot/colorbar.m,
scripts/plot/daspect.m, scripts/plot/ezmesh.m, scripts/plot/ezmeshc.m,
scripts/plot/ezsurf.m, scripts/plot/ezsurfc.m, scripts/plot/findall.m,
scripts/plot/findobj.m, scripts/plot/gcbo.m, scripts/plot/gcf.m,
scripts/plot/gco.m, scripts/plot/grid.m, scripts/plot/guihandles.m,
scripts/plot/hdl2struct.m, scripts/plot/hidden.m, scripts/plot/hold.m,
scripts/plot/isonormals.m, scripts/plot/isosurface.m, scripts/plot/legend.m,
scripts/plot/mesh.m, scripts/plot/meshc.m, scripts/plot/meshz.m,
scripts/plot/newplot.m, scripts/plot/orient.m, scripts/plot/pareto.m,
scripts/plot/patch.m, scripts/plot/pbaspect.m, scripts/plot/pcolor.m,
scripts/plot/plot.m, scripts/plot/print.m,
scripts/plot/private/__add_default_menu__.m, scripts/plot/quiver.m,
scripts/plot/quiver3.m, scripts/plot/refreshdata.m, scripts/plot/saveas.m,
scripts/plot/scatter.m, scripts/plot/scatter3.m, scripts/plot/shading.m,
scripts/plot/shrinkfaces.m, scripts/plot/slice.m, scripts/plot/stem.m,
scripts/plot/stem3.m, scripts/plot/struct2hdl.m, scripts/plot/subplot.m,
scripts/plot/surf.m, scripts/plot/surfc.m, scripts/plot/surfl.m,
scripts/plot/tetramesh.m, scripts/plot/uigetfile.m, scripts/plot/uimenu.m,
scripts/plot/uiputfile.m, scripts/plot/waterfall.m, scripts/plot/whitebg.m,
scripts/plot/xlim.m, scripts/plot/ylim.m, scripts/plot/zlim.m,
scripts/polynomial/conv.m, scripts/polynomial/polyout.m,
scripts/polynomial/splinefit.m, scripts/set/ismember.m, scripts/set/powerset.m,
scripts/set/setdiff.m, scripts/set/union.m, scripts/set/unique.m,
scripts/signal/detrend.m, scripts/signal/filter2.m, scripts/signal/freqz.m,
scripts/signal/periodogram.m, scripts/signal/spectral_adf.m,
scripts/signal/spectral_xdf.m, scripts/sparse/eigs.m, scripts/sparse/svds.m,
scripts/specfun/legendre.m, scripts/special-matrix/gallery.m,
scripts/statistics/base/mean.m, scripts/statistics/base/moment.m,
scripts/statistics/tests/cor_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test.m,
scripts/statistics/tests/kolmogorov_smirnov_test_2.m,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/statistics/tests/prop_test_2.m, scripts/statistics/tests/sign_test.m,
scripts/statistics/tests/t_test.m, scripts/statistics/tests/t_test_2.m,
scripts/statistics/tests/t_test_regression.m,
scripts/statistics/tests/u_test.m, scripts/statistics/tests/var_test.m,
scripts/statistics/tests/welch_test.m,
scripts/statistics/tests/wilcoxon_test.m, scripts/statistics/tests/z_test.m,
scripts/statistics/tests/z_test_2.m, scripts/strings/base2dec.m,
scripts/strings/index.m, scripts/strings/isstrprop.m,
scripts/strings/mat2str.m, scripts/strings/regexptranslate.m,
scripts/strings/rindex.m, scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjust.m, scripts/strings/strmatch.m,
scripts/strings/validatestring.m, scripts/testfun/demo.m,
scripts/testfun/example.m, scripts/testfun/test.m, scripts/time/addtodate.m,
scripts/time/asctime.m, scripts/time/datestr.m, scripts/time/datetick.m,
scripts/time/weekday.m, scripts/ui/errordlg.m, scripts/ui/helpdlg.m,
scripts/ui/inputdlg.m, scripts/ui/listdlg.m, scripts/ui/msgbox.m,
scripts/ui/questdlg.m, scripts/ui/warndlg.m: Use new @qcode macro.
| author | Rik <rik@octave.org> |
|---|---|
| date | Mon, 19 Aug 2013 20:46:38 -0700 |
| parents | e7a059a9a644 |
| children | 1efe4c65c5cf |
| rev | line source |
|---|---|
|
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1 @c Copyright (C) 2007-2012 John W. Eaton |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 6549 | 18 |
| 19 @node Interpolation | |
| 20 @chapter Interpolation | |
| 21 | |
| 6702 | 22 @menu |
| 23 * One-dimensional Interpolation:: | |
| 24 * Multi-dimensional Interpolation:: | |
| 25 @end menu | |
| 26 | |
| 27 @node One-dimensional Interpolation | |
| 28 @section One-dimensional Interpolation | |
| 6549 | 29 |
| 6850 | 30 Octave supports several methods for one-dimensional interpolation, most |
| 31 of which are described in this section. @ref{Polynomial Interpolation} | |
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32 and @ref{Interpolation on Scattered Data} describe additional methods. |
| 6850 | 33 |
| 6549 | 34 @DOCSTRING(interp1) |
| 35 | |
| 6721 | 36 There are some important differences between the various interpolation |
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37 methods. The @qcode{"spline"} method enforces that both the first and second |
| 6721 | 38 derivatives of the interpolated values have a continuous derivative, |
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39 whereas the other methods do not. This means that the results of the |
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40 @qcode{"spline"} method are generally smoother. If the function to be |
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41 interpolated is in fact smooth, then @qcode{"spline"} will give excellent |
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42 results. However, if the function to be evaluated is in some manner |
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43 discontinuous, then @qcode{"pchip"} interpolation might give better results. |
| 6743 | 44 |
| 45 This can be demonstrated by the code | |
| 6721 | 46 |
| 47 @example | |
| 48 @group | |
| 6743 | 49 t = -2:2; |
| 50 dt = 1; | |
| 51 ti =-2:0.025:2; | |
| 52 dti = 0.025; | |
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53 y = sign (t); |
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54 ys = interp1 (t,y,ti,"spline"); |
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55 yp = interp1 (t,y,ti,"pchip"); |
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56 ddys = diff (diff (ys)./dti) ./ dti; |
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57 ddyp = diff (diff (yp)./dti) ./ dti; |
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58 figure (1); |
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59 plot (ti,ys,'r-', ti,yp,'g-'); |
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60 legend ("spline", "pchip", 4); |
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61 figure (2); |
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62 plot (ti,ddys,'r+', ti,ddyp,'g*'); |
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63 legend ("spline", "pchip"); |
| 6721 | 64 @end group |
| 65 @end example | |
| 66 | |
| 67 @ifnotinfo | |
| 68 @noindent | |
| 6743 | 69 The result of which can be seen in @ref{fig:interpderiv1} and |
| 70 @ref{fig:interpderiv2}. | |
| 6721 | 71 |
| 6743 | 72 @float Figure,fig:interpderiv1 |
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73 @center @image{interpderiv1,4in} |
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74 @caption{Comparison of @qcode{"pchip"} and @qcode{"spline"} interpolation methods for a |
| 6743 | 75 step function} |
| 76 @end float | |
| 77 | |
| 78 @float Figure,fig:interpderiv2 | |
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79 @center @image{interpderiv2,4in} |
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80 @caption{Comparison of the second derivative of the @qcode{"pchip"} and @qcode{"spline"} |
| 6743 | 81 interpolation methods for a step function} |
| 6721 | 82 @end float |
| 83 @end ifnotinfo | |
| 84 | |
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85 A simplified version of @code{interp1} that performs only linear |
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86 interpolation is available in @code{interp1q}. This argument is slightly |
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87 faster than @code{interp1} as to performs little error checking. |
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88 |
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89 @DOCSTRING(interp1q) |
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90 |
| 6702 | 91 Fourier interpolation, is a resampling technique where a signal is |
| 92 converted to the frequency domain, padded with zeros and then | |
| 93 reconverted to the time domain. | |
| 6549 | 94 |
| 95 @DOCSTRING(interpft) | |
| 96 | |
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97 There are two significant limitations on Fourier interpolation. Firstly, |
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98 the function signal is assumed to be periodic, and so non-periodic |
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99 signals will be poorly represented at the edges. Secondly, both the |
| 6702 | 100 signal and its interpolation are required to be sampled at equispaced |
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101 points. An example of the use of @code{interpft} is |
| 6549 | 102 |
| 6702 | 103 @example |
| 104 @group | |
| 105 t = 0 : 0.3 : pi; dt = t(2)-t(1); | |
| 106 n = length (t); k = 100; | |
| 107 ti = t(1) + [0 : k-1]*dt*n/k; | |
| 108 y = sin (4*t + 0.3) .* cos (3*t - 0.1); | |
| 109 yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1); | |
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110 plot (ti, yp, "g", ti, interp1 (t, y, ti, "spline"), "b", ... |
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111 ti, interpft (y, k), "c", t, y, 'r+'); |
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112 legend ('sin(4t+0.3)cos(3t-0.1', "spline", "interpft", "data"); |
| 6702 | 113 @end group |
| 114 @end example | |
| 115 | |
| 8828 | 116 @noindent |
| 6721 | 117 @ifinfo |
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118 which demonstrates the poor behavior of Fourier interpolation for non-periodic |
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119 functions. |
| 6721 | 120 @end ifinfo |
| 121 @ifnotinfo | |
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122 which demonstrates the poor behavior of Fourier interpolation for non-periodic |
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123 functions, as can be seen in @ref{fig:interpft}. |
| 6721 | 124 |
| 125 @float Figure,fig:interpft | |
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126 @center @image{interpft,4in} |
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127 @caption{Comparison of @code{interp1} and @code{interpft} for non-periodic data} |
| 6721 | 128 @end float |
| 129 @end ifnotinfo | |
| 6702 | 130 |
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131 In addition, the support functions @code{spline} and @code{lookup} that |
| 6702 | 132 underlie the @code{interp1} function can be called directly. |
| 6549 | 133 |
| 6558 | 134 @DOCSTRING(spline) |
| 135 | |
| 6702 | 136 @node Multi-dimensional Interpolation |
| 137 @section Multi-dimensional Interpolation | |
| 138 | |
| 6939 | 139 There are three multi-dimensional interpolation functions in Octave, with |
| 6850 | 140 similar capabilities. Methods using Delaunay tessellation are described |
| 141 in @ref{Interpolation on Scattered Data}. | |
| 6702 | 142 |
| 143 @DOCSTRING(interp2) | |
| 144 | |
| 145 @DOCSTRING(interp3) | |
| 146 | |
| 147 @DOCSTRING(interpn) | |
| 148 | |
| 149 A significant difference between @code{interpn} and the other two | |
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150 multi-dimensional interpolation functions is the fashion in which the |
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151 dimensions are treated. For @code{interp2} and @code{interp3}, the y-axis is |
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152 considered to be the columns of the matrix, whereas the x-axis corresponds to |
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153 the rows of the array. As Octave indexes arrays in column major order, the |
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154 first dimension of any array is the columns, and so @code{interpn} effectively |
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155 reverses the 'x' and 'y' dimensions. Consider the example, |
| 6702 | 156 |
| 157 @example | |
| 158 @group | |
| 159 x = y = z = -1:1; | |
| 160 f = @@(x,y,z) x.^2 - y - z.^2; | |
| 161 [xx, yy, zz] = meshgrid (x, y, z); | |
| 162 v = f (xx,yy,zz); | |
| 6721 | 163 xi = yi = zi = -1:0.1:1; |
| 6702 | 164 [xxi, yyi, zzi] = meshgrid (xi, yi, zi); |
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165 vi = interp3 (x, y, z, v, xxi, yyi, zzi, "spline"); |
| 6702 | 166 [xxi, yyi, zzi] = ndgrid (xi, yi, zi); |
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167 vi2 = interpn (x, y, z, v, xxi, yyi, zzi, "spline"); |
| 6723 | 168 mesh (zi, yi, squeeze (vi2(1,:,:))); |
| 6702 | 169 @end group |
| 170 @end example | |
| 171 | |
| 172 @noindent | |
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173 where @code{vi} and @code{vi2} are identical. The reversal of the |
| 6702 | 174 dimensions is treated in the @code{meshgrid} and @code{ndgrid} functions |
| 175 respectively. | |
| 6721 | 176 @ifnotinfo |
| 177 The result of this code can be seen in @ref{fig:interpn}. | |
| 178 | |
| 179 @float Figure,fig:interpn | |
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180 @center @image{interpn,4in} |
| 6721 | 181 @caption{Demonstration of the use of @code{interpn}} |
| 182 @end float | |
| 183 @end ifnotinfo | |
| 6702 | 184 |
| 185 In additional the support function @code{bicubic} that underlies the | |
| 186 cubic interpolation of @code{interp2} function can be called directly. | |
| 187 | |
| 188 @DOCSTRING(bicubic) |