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update Octave Project Developers copyright for the new year
In files that have the "Octave Project Developers" copyright notice,
update for 2021.
In all .txi and .texi files except gpl.txi and gpl.texi in the
doc/liboctave and doc/interpreter directories, change the copyright
to "Octave Project Developers", the same as used for other source
files. Update copyright notices for 2022 (not done since 2019). For
gpl.txi and gpl.texi, change the copyright notice to be "Free Software
Foundation, Inc." and leave the date at 2007 only because this file
only contains the text of the GPL, not anything created by the Octave
Project Developers.
Add Paul Thomas to contributors.in.
author | John W. Eaton <jwe@octave.org> |
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date | Tue, 28 Dec 2021 18:22:40 -0500 |
parents | 00f796120a6d |
children | 597f3ee61a48 |
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@c Copyright (C) 2007-2022 The Octave Project Developers @c @c This file is part of Octave. @c @c Octave is free software: you can redistribute it and/or modify it @c under the terms of the GNU General Public License as published by @c the Free Software Foundation, either version 3 of the License, or @c (at your option) any later version. @c @c Octave is distributed in the hope that it will be useful, but @c WITHOUT ANY WARRANTY; without even the implied warranty of @c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the @c GNU General Public License for more details. @c @c You should have received a copy of the GNU General Public License @c along with Octave; see the file COPYING. If not, see @c <https://www.gnu.org/licenses/>. @node Interpolation @chapter Interpolation @menu * One-dimensional Interpolation:: * Multi-dimensional Interpolation:: @end menu @node One-dimensional Interpolation @section One-dimensional Interpolation Octave supports several methods for one-dimensional interpolation, most of which are described in this section. @ref{Polynomial Interpolation} and @ref{Interpolation on Scattered Data} describe additional methods. @DOCSTRING(interp1) There are some important differences between the various interpolation methods. The @qcode{"spline"} method enforces that both the first and second derivatives of the interpolated values have a continuous derivative, whereas the other methods do not. This means that the results of the @qcode{"spline"} method are generally smoother. If the function to be interpolated is in fact smooth, then @qcode{"spline"} will give excellent results. However, if the function to be evaluated is in some manner discontinuous, then @qcode{"pchip"} interpolation might give better results. This can be demonstrated by the code @example @group t = -2:2; dt = 1; ti =-2:0.025:2; dti = 0.025; y = sign (t); ys = interp1 (t,y,ti,"spline"); yp = interp1 (t,y,ti,"pchip"); ddys = diff (diff (ys)./dti) ./ dti; ddyp = diff (diff (yp)./dti) ./ dti; figure (1); plot (ti,ys,"r-", ti,yp,"g-"); legend ("spline", "pchip", 4); figure (2); plot (ti,ddys,"r+", ti,ddyp,"g*"); legend ("spline", "pchip"); @end group @end example @ifnotinfo @noindent The result of which can be seen in @ref{fig:interpderiv1} and @ref{fig:interpderiv2}. @float Figure,fig:interpderiv1 @center @image{interpderiv1,4in} @caption{Comparison of @qcode{"pchip"} and @qcode{"spline"} interpolation methods for a step function} @end float @float Figure,fig:interpderiv2 @center @image{interpderiv2,4in} @caption{Comparison of the second derivative of the @qcode{"pchip"} and @qcode{"spline"} interpolation methods for a step function} @end float @end ifnotinfo Fourier interpolation, is a resampling technique where a signal is converted to the frequency domain, padded with zeros and then reconverted to the time domain. @DOCSTRING(interpft) There are two significant limitations on Fourier interpolation. First, the function signal is assumed to be periodic, and so non-periodic signals will be poorly represented at the edges. Second, both the signal and its interpolation are required to be sampled at equispaced points. An example of the use of @code{interpft} is @example @group t = 0 : 0.3 : pi; dt = t(2)-t(1); n = length (t); k = 100; ti = t(1) + [0 : k-1]*dt*n/k; y = sin (4*t + 0.3) .* cos (3*t - 0.1); yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1); plot (ti, yp, "g", ti, interp1 (t, y, ti, "spline"), "b", ... ti, interpft (y, k), "c", t, y, "r+"); legend ("sin(4t+0.3)cos(3t-0.1)", "spline", "interpft", "data"); @end group @end example @noindent @ifinfo which demonstrates the poor behavior of Fourier interpolation for non-periodic functions. @end ifinfo @ifnotinfo which demonstrates the poor behavior of Fourier interpolation for non-periodic functions, as can be seen in @ref{fig:interpft}. @float Figure,fig:interpft @center @image{interpft,4in} @caption{Comparison of @code{interp1} and @code{interpft} for non-periodic data} @end float @end ifnotinfo In addition, the support functions @code{spline} and @code{lookup} that underlie the @code{interp1} function can be called directly. @DOCSTRING(spline) @node Multi-dimensional Interpolation @section Multi-dimensional Interpolation There are three multi-dimensional interpolation functions in Octave, with similar capabilities. Methods using Delaunay tessellation are described in @ref{Interpolation on Scattered Data}. @DOCSTRING(interp2) @DOCSTRING(interp3) @DOCSTRING(interpn) A significant difference between @code{interpn} and the other two multi-dimensional interpolation functions is the fashion in which the dimensions are treated. For @code{interp2} and @code{interp3}, the y-axis is considered to be the columns of the matrix, whereas the x-axis corresponds to the rows of the array. As Octave indexes arrays in column major order, the first dimension of any array is the columns, and so @code{interpn} effectively reverses the 'x' and 'y' dimensions. Consider the example, @example @group x = y = z = -1:1; f = @@(x,y,z) x.^2 - y - z.^2; [xx, yy, zz] = meshgrid (x, y, z); v = f (xx,yy,zz); xi = yi = zi = -1:0.1:1; [xxi, yyi, zzi] = meshgrid (xi, yi, zi); vi = interp3 (x, y, z, v, xxi, yyi, zzi, "spline"); [xxi, yyi, zzi] = ndgrid (xi, yi, zi); vi2 = interpn (x, y, z, v, xxi, yyi, zzi, "spline"); mesh (zi, yi, squeeze (vi2(1,:,:))); @end group @end example @noindent where @code{vi} and @code{vi2} are identical. The reversal of the dimensions is treated in the @code{meshgrid} and @code{ndgrid} functions respectively. @ifnotinfo The result of this code can be seen in @ref{fig:interpn}. @float Figure,fig:interpn @center @image{interpn,4in} @caption{Demonstration of the use of @code{interpn}} @end float @end ifnotinfo