Mercurial > octave-nkf
annotate doc/interpreter/interp.txi @ 18830:322eb69e30ad stable
doc: Fix some Latin wording in documentation.
* external.txi, interp.txi, sparse.txi: Use "a priori" rather than "a-priori".
Don't use "Firstly".
| author | Rik <rik@octave.org> |
|---|---|
| date | Mon, 02 Jun 2014 08:59:26 -0700 |
| parents | dae2230227a7 |
| children | 1c4994e66962 446c46af4b42 |
| rev | line source |
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John W. Eaton <jwe@octave.org>
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1 @c Copyright (C) 2007-2013 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 | |
| 6702 | 85 Fourier interpolation, is a resampling technique where a signal is |
| 86 converted to the frequency domain, padded with zeros and then | |
| 87 reconverted to the time domain. | |
| 6549 | 88 |
| 89 @DOCSTRING(interpft) | |
| 90 | |
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91 There are two significant limitations on Fourier interpolation. First, |
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92 the function signal is assumed to be periodic, and so non-periodic |
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93 signals will be poorly represented at the edges. Second, both the |
| 6702 | 94 signal and its interpolation are required to be sampled at equispaced |
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95 points. An example of the use of @code{interpft} is |
| 6549 | 96 |
| 6702 | 97 @example |
| 98 @group | |
| 99 t = 0 : 0.3 : pi; dt = t(2)-t(1); | |
| 100 n = length (t); k = 100; | |
| 101 ti = t(1) + [0 : k-1]*dt*n/k; | |
| 102 y = sin (4*t + 0.3) .* cos (3*t - 0.1); | |
| 103 yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1); | |
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104 plot (ti, yp, "g", ti, interp1 (t, y, ti, "spline"), "b", ... |
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105 ti, interpft (y, k), "c", t, y, "r+"); |
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106 legend ("sin(4t+0.3)cos(3t-0.1)", "spline", "interpft", "data"); |
| 6702 | 107 @end group |
| 108 @end example | |
| 109 | |
| 8828 | 110 @noindent |
| 6721 | 111 @ifinfo |
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112 which demonstrates the poor behavior of Fourier interpolation for non-periodic |
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113 functions. |
| 6721 | 114 @end ifinfo |
| 115 @ifnotinfo | |
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116 which demonstrates the poor behavior of Fourier interpolation for non-periodic |
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117 functions, as can be seen in @ref{fig:interpft}. |
| 6721 | 118 |
| 119 @float Figure,fig:interpft | |
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120 @center @image{interpft,4in} |
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121 @caption{Comparison of @code{interp1} and @code{interpft} for non-periodic data} |
| 6721 | 122 @end float |
| 123 @end ifnotinfo | |
| 6702 | 124 |
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125 In addition, the support functions @code{spline} and @code{lookup} that |
| 6702 | 126 underlie the @code{interp1} function can be called directly. |
| 6549 | 127 |
| 6558 | 128 @DOCSTRING(spline) |
| 129 | |
| 6702 | 130 @node Multi-dimensional Interpolation |
| 131 @section Multi-dimensional Interpolation | |
| 132 | |
| 6939 | 133 There are three multi-dimensional interpolation functions in Octave, with |
| 6850 | 134 similar capabilities. Methods using Delaunay tessellation are described |
| 135 in @ref{Interpolation on Scattered Data}. | |
| 6702 | 136 |
| 137 @DOCSTRING(interp2) | |
| 138 | |
| 139 @DOCSTRING(interp3) | |
| 140 | |
| 141 @DOCSTRING(interpn) | |
| 142 | |
| 143 A significant difference between @code{interpn} and the other two | |
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144 multi-dimensional interpolation functions is the fashion in which the |
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145 dimensions are treated. For @code{interp2} and @code{interp3}, the y-axis is |
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146 considered to be the columns of the matrix, whereas the x-axis corresponds to |
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147 the rows of the array. As Octave indexes arrays in column major order, the |
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148 first dimension of any array is the columns, and so @code{interpn} effectively |
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149 reverses the 'x' and 'y' dimensions. Consider the example, |
| 6702 | 150 |
| 151 @example | |
| 152 @group | |
| 153 x = y = z = -1:1; | |
| 154 f = @@(x,y,z) x.^2 - y - z.^2; | |
| 155 [xx, yy, zz] = meshgrid (x, y, z); | |
| 156 v = f (xx,yy,zz); | |
| 6721 | 157 xi = yi = zi = -1:0.1:1; |
| 6702 | 158 [xxi, yyi, zzi] = meshgrid (xi, yi, zi); |
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159 vi = interp3 (x, y, z, v, xxi, yyi, zzi, "spline"); |
| 6702 | 160 [xxi, yyi, zzi] = ndgrid (xi, yi, zi); |
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161 vi2 = interpn (x, y, z, v, xxi, yyi, zzi, "spline"); |
| 6723 | 162 mesh (zi, yi, squeeze (vi2(1,:,:))); |
| 6702 | 163 @end group |
| 164 @end example | |
| 165 | |
| 166 @noindent | |
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167 where @code{vi} and @code{vi2} are identical. The reversal of the |
| 6702 | 168 dimensions is treated in the @code{meshgrid} and @code{ndgrid} functions |
| 169 respectively. | |
| 6721 | 170 @ifnotinfo |
| 171 The result of this code can be seen in @ref{fig:interpn}. | |
| 172 | |
| 173 @float Figure,fig:interpn | |
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174 @center @image{interpn,4in} |
| 6721 | 175 @caption{Demonstration of the use of @code{interpn}} |
| 176 @end float | |
| 177 @end ifnotinfo | |
| 6702 | 178 |
| 179 In additional the support function @code{bicubic} that underlies the | |
| 180 cubic interpolation of @code{interp2} function can be called directly. | |
| 181 | |
| 182 @DOCSTRING(bicubic) |