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Overhaul interp1, interp2, interp3 functions. * NEWS: Announce change in 'cubic' interpolation method for interp2 to match Matlab. * bicubic.m: Use interp2 (..., "spline") in %!tests. * interp1.m: Improve docstring. Use switch statement instead of if/elseif tree for simpler code. Use more informative error message than 'table too short'. Add titles to demo plots. Add new demo block showing difference between 'pchip' and 'spline' methods. * interp2.m: Rewrite docstring. Use variable 'extrap' instead of 'extrapval' to match documentation. Use clearer messages in error() calls. Make 'cubic' use the same algorithm as 'pchip' for Matlab compatibility. Use Octave coding conventions regarding spaces between variable and parenthesis. Added input validation tests. * interp3.m: Rewrite docstring. Use clearer messages in error() calls. Make 'cubic' use the same algorithm as 'pchip' for Matlab compatibility. Simplify input processing. Rewrite some %!tests for clarity. Added input validation tests.
author Rik <rik@octave.org>
date Sun, 30 Mar 2014 14:18:43 -0700
parents d63878346099
children 4197fc428c7d
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d63878346099 maint: Update copyright notices for release.
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1 ## Copyright (C) 2000-2013 Kai Habel
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {@var{dx} =} gradient (@var{m})
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21 ## @deftypefnx {Function File} {[@var{dx}, @var{dy}, @var{dz}, @dots{}] =} gradient (@var{m})
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22 ## @deftypefnx {Function File} {[@dots{}] =} gradient (@var{m}, @var{s})
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23 ## @deftypefnx {Function File} {[@dots{}] =} gradient (@var{m}, @var{x}, @var{y}, @var{z}, @dots{})
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24 ## @deftypefnx {Function File} {[@dots{}] =} gradient (@var{f}, @var{x0})
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25 ## @deftypefnx {Function File} {[@dots{}] =} gradient (@var{f}, @var{x0}, @var{s})
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26 ## @deftypefnx {Function File} {[@dots{}] =} gradient (@var{f}, @var{x0}, @var{x}, @var{y}, @dots{})
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27 ##
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28 ## Calculate the gradient of sampled data or a function. If @var{m}
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29 ## is a vector, calculate the one-dimensional gradient of @var{m}. If
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30 ## @var{m} is a matrix the gradient is calculated for each dimension.
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31 ##
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32 ## @code{[@var{dx}, @var{dy}] = gradient (@var{m})} calculates the one
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33 ## dimensional gradient for @var{x} and @var{y} direction if @var{m} is a
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34 ## matrix. Additional return arguments can be use for multi-dimensional
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35 ## matrices.
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36 ##
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37 ## A constant spacing between two points can be provided by the
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38 ## @var{s} parameter. If @var{s} is a scalar, it is assumed to be the spacing
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39 ## for all dimensions.
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40 ## Otherwise, separate values of the spacing can be supplied by
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41 ## the @var{x}, @dots{} arguments. Scalar values specify an equidistant
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42 ## spacing.
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43 ## Vector values for the @var{x}, @dots{} arguments specify the coordinate for
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44 ## that
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45 ## dimension. The length must match their respective dimension of @var{m}.
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46 ##
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47 ## At boundary points a linear extrapolation is applied. Interior points
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48 ## are calculated with the first approximation of the numerical gradient
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49 ##
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50 ## @example
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51 ## y'(i) = 1/(x(i+1)-x(i-1)) * (y(i-1)-y(i+1)).
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52 ## @end example
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53 ##
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54 ## If the first argument @var{f} is a function handle, the gradient of the
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55 ## function at the points in @var{x0} is approximated using central
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56 ## difference. For example, @code{gradient (@@cos, 0)} approximates the
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57 ## gradient of the cosine function in the point @math{x0 = 0}. As with
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58 ## sampled data, the spacing values between the points from which the
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59 ## gradient is estimated can be set via the @var{s} or @var{dx},
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60 ## @var{dy}, @dots{} arguments. By default a spacing of 1 is used.
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61 ## @seealso{diff, del2}
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62 ## @end deftypefn
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63
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64 ## Author: Kai Habel <kai.habel@gmx.de>
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65 ## Modified: David Bateman <dbateman@free.fr> Added NDArray support
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66
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67 function varargout = gradient (m, varargin)
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68
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69 if (nargin < 1)
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70 print_usage ();
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71 endif
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72
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73 nargout_with_ans = max (1,nargout);
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74 if (ismatrix (m))
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75 [varargout{1:nargout_with_ans}] = matrix_gradient (m, varargin{:});
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76 elseif (isa (m, "function_handle"))
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77 [varargout{1:nargout_with_ans}] = handle_gradient (m, varargin{:});
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78 elseif (ischar (m))
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79 [varargout{1:nargout_with_ans}] = handle_gradient (str2func (m), varargin{:});
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80 else
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81 error ("gradient: first input must be an array or a function");
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82 endif
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83
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84 endfunction
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85
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86 function varargout = matrix_gradient (m, varargin)
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87 transposed = false;
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88 if (isvector (m))
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89 ## make a row vector.
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90 transposed = (columns (m) == 1);
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91 m = m(:).';
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92 endif
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93
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94 nd = ndims (m);
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95 sz = size (m);
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96 if (length (sz) > 1)
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97 tmp = sz(1); sz(1) = sz(2); sz(2) = tmp;
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98 endif
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99
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100 if (nargin > 2 && nargin != nd + 1)
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101 print_usage ();
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102 endif
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103
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104 ## cell d stores a spacing vector for each dimension
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105 d = cell (1, nd);
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106 if (nargin == 1)
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107 ## no spacing given - assume 1.0 for all dimensions
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108 for i = 1:nd
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109 d{i} = ones (sz(i) - 1, 1);
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110 endfor
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111 elseif (nargin == 2)
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112 if (isscalar (varargin{1}))
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113 ## single scalar value for all dimensions
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114 for i = 1:nd
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115 d{i} = varargin{1} * ones (sz(i) - 1, 1);
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116 endfor
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117 else
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118 ## vector for one-dimensional derivative
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119 d{1} = diff (varargin{1}(:));
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120 endif
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121 else
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122 ## have spacing value for each dimension
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123 if (length(varargin) != nd)
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124 error ("gradient: dimensions and number of spacing values do not match");
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125 endif
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126 for i = 1:nd
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127 if (isscalar (varargin{i}))
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128 d{i} = varargin{i} * ones (sz(i) - 1, 1);
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129 else
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130 d{i} = diff (varargin{i}(:));
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131 endif
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132 endfor
55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
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133 endif
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134
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135 m = shiftdim (m, 1);
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136 for i = 1:min (nd, nargout)
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137 mr = rows (m);
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138 mc = numel (m) / mr;
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139 Y = zeros (size (m), class (m));
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140
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141 if (mr > 1)
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142 ## Top and bottom boundary.
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143 Y(1,:) = diff (m(1:2, :)) / d{i}(1);
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144 Y(mr,:) = diff (m(mr-1:mr, :) / d{i}(mr - 1));
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145 endif
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146
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147 if (mr > 2)
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148 ## Interior points.
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149 Y(2:mr-1,:) = ((m(3:mr,:) - m(1:mr-2,:))
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150 ./ kron (d{i}(1:mr-2) + d{i}(2:mr-1), ones (1, mc)));
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151 endif
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152
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153 ## turn multi-dimensional matrix in a way, that gradient
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154 ## along x-direction is calculated first then y, z, ...
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155
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156 if (i == 1)
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157 varargout{i} = shiftdim (Y, nd - 1);
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158 m = shiftdim (m, nd - 1);
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159 elseif (i == 2)
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160 varargout{i} = Y;
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161 m = shiftdim (m, 2);
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162 else
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163 varargout{i} = shiftdim (Y, nd - i + 1);
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164 m = shiftdim (m, 1);
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165 endif
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166 endfor
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167
55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
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168 if (transposed)
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169 varargout{1} = varargout{1}.';
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170 endif
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171 endfunction
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172
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173 function varargout = handle_gradient (f, p0, varargin)
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174 ## Input checking
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175 p0_size = size (p0);
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176
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177 if (numel (p0_size) != 2)
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178 error ("gradient: the second input argument should either be a vector or a matrix");
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179 endif
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180
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181 if (any (p0_size == 1))
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182 p0 = p0 (:);
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183 dim = 1;
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184 num_points = numel (p0);
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185 else
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186 num_points = p0_size (1);
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187 dim = p0_size (2);
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188 endif
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189
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190 if (length (varargin) == 0)
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191 delta = 1;
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192 elseif (length (varargin) == 1 || length (varargin) == dim)
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193 try
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194 delta = [varargin{:}];
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195 catch
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196 error ("gradient: spacing parameters must be scalars or a vector");
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197 end_try_catch
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198 else
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199 error ("gradient: incorrect number of spacing parameters");
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200 endif
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201
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202 if (isscalar (delta))
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203 delta = repmat (delta, 1, dim);
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204 elseif (!isvector (delta))
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205 error ("gradient: spacing values must be scalars or a vector");
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206 endif
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207
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208 ## Calculate the gradient
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209 p0 = mat2cell (p0, num_points, ones (1, dim));
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210 varargout = cell (1, dim);
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211 for d = 1:dim
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212 s = delta (d);
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213 df_dx = (f (p0{1:d-1}, p0{d}+s, p0{d+1:end})
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214 - f (p0{1:d-1}, p0{d}-s, p0{d+1:end})) ./ (2*s);
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215 if (dim == 1)
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216 varargout{d} = reshape (df_dx, p0_size);
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217 else
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218 varargout{d} = df_dx;
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219 endif
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220 endfor
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221 endfunction
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222
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223
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224 %!test
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225 %! data = [1, 2, 4, 2];
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226 %! dx = gradient (data);
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227 %! dx2 = gradient (data, 0.25);
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228 %! dx3 = gradient (data, [0.25, 0.5, 1, 3]);
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229 %! assert (dx, [1, 3/2, 0, -2]);
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230 %! assert (dx2, [4, 6, 0, -8]);
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231 %! assert (dx3, [4, 4, 0, -1]);
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232 %! assert (size_equal (data, dx));
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233
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234 %!test
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235 %! [Y,X,Z,U] = ndgrid (2:2:8,1:5,4:4:12,3:5:30);
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236 %! [dX,dY,dZ,dU] = gradient (X);
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237 %! assert (all (dX(:) == 1));
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238 %! assert (all (dY(:) == 0));
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239 %! assert (all (dZ(:) == 0));
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240 %! assert (all (dU(:) == 0));
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241 %! [dX,dY,dZ,dU] = gradient (Y);
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242 %! assert (all (dX(:) == 0));
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243 %! assert (all (dY(:) == 2));
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244 %! assert (all (dZ(:) == 0));
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245 %! assert (all (dU(:) == 0));
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246 %! [dX,dY,dZ,dU] = gradient (Z);
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247 %! assert (all (dX(:) == 0));
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248 %! assert (all (dY(:) == 0));
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249 %! assert (all (dZ(:) == 4));
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250 %! assert (all (dU(:) == 0));
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251 %! [dX,dY,dZ,dU] = gradient (U);
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252 %! assert (all (dX(:) == 0));
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253 %! assert (all (dY(:) == 0));
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254 %! assert (all (dZ(:) == 0));
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255 %! assert (all (dU(:) == 5));
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256 %! assert (size_equal (dX, dY, dZ, dU, X, Y, Z, U));
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257 %! [dX,dY,dZ,dU] = gradient (U, 5.0);
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258 %! assert (all (dU(:) == 1));
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259 %! [dX,dY,dZ,dU] = gradient (U, 1.0, 2.0, 3.0, 2.5);
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260 %! assert (all (dU(:) == 2));
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261
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262 %!test
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263 %! [Y,X,Z,U] = ndgrid (2:2:8,1:5,4:4:12,3:5:30);
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264 %! [dX,dY,dZ,dU] = gradient (X+j*X);
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265 %! assert (all (dX(:) == 1+1j));
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266 %! assert (all (dY(:) == 0));
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267 %! assert (all (dZ(:) == 0));
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268 %! assert (all (dU(:) == 0));
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269 %! [dX,dY,dZ,dU] = gradient (Y-j*Y);
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270 %! assert (all (dX(:) == 0));
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271 %! assert (all (dY(:) == 2-j*2));
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272 %! assert (all (dZ(:) == 0));
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273 %! assert (all (dU(:) == 0));
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274 %! [dX,dY,dZ,dU] = gradient (Z+j*1);
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275 %! assert (all (dX(:) == 0));
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276 %! assert (all (dY(:) == 0));
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277 %! assert (all (dZ(:) == 4));
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278 %! assert (all (dU(:) == 0));
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279 %! [dX,dY,dZ,dU] = gradient (U-j*1);
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280 %! assert (all (dX(:) == 0));
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281 %! assert (all (dY(:) == 0));
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282 %! assert (all (dZ(:) == 0));
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283 %! assert (all (dU(:) == 5));
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284 %! assert (size_equal (dX, dY, dZ, dU, X, Y, Z, U));
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285 %! [dX,dY,dZ,dU] = gradient (U, 5.0);
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286 %! assert (all (dU(:) == 1));
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287 %! [dX,dY,dZ,dU] = gradient (U, 1.0, 2.0, 3.0, 2.5);
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288 %! assert (all (dU(:) == 2));
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289
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290 %!test
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291 %! x = 0:10;
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292 %! f = @cos;
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293 %! df_dx = @(x) -sin (x);
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294 %! assert (gradient (f, x), df_dx (x), 0.2);
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295 %! assert (gradient (f, x, 0.5), df_dx (x), 0.1);
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296
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297 %!test
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298 %! xy = reshape (1:10, 5, 2);
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299 %! f = @(x,y) sin (x) .* cos (y);
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300 %! df_dx = @(x, y) cos (x) .* cos (y);
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301 %! df_dy = @(x, y) -sin (x) .* sin (y);
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302 %! [dx, dy] = gradient (f, xy);
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303 %! assert (dx, df_dx (xy (:, 1), xy (:, 2)), 0.1)
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304 %! assert (dy, df_dy (xy (:, 1), xy (:, 2)), 0.1)
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305