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gallery.m: clean and vectorize part of qmult. * scripts/special-matrix/gallery.m: make qmult() follow Octave guidelines, vectorize last for loop in qmult().
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 28 May 2015 18:32:47 +0200
parents 9fc020886ae9
children 7503499a252b
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rev   line source
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1 ## Copyright (C) 2000-2015 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 (isnumeric (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), ...
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80 varargin{:});
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81 else
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82 error ("gradient: first input must be an array or a function");
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83 endif
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84
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85 endfunction
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86
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87 function varargout = matrix_gradient (m, varargin)
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88 transposed = false;
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89 if (isvector (m))
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90 ## make a row vector.
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91 transposed = (columns (m) == 1);
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92 m = m(:).';
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93 endif
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94
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95 nd = ndims (m);
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96 sz = size (m);
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97 if (length (sz) > 1)
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98 tmp = sz(1); sz(1) = sz(2); sz(2) = tmp;
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99 endif
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100
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101 if (nargin > 2 && nargin != nd + 1)
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102 print_usage ();
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103 endif
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104
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105 ## cell d stores a spacing vector for each dimension
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106 d = cell (1, nd);
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107 if (nargin == 1)
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108 ## no spacing given - assume 1.0 for all dimensions
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109 for i = 1:nd
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110 d{i} = ones (sz(i) - 1, 1);
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111 endfor
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112 elseif (nargin == 2)
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113 if (isscalar (varargin{1}))
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114 ## single scalar value for all dimensions
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115 for i = 1:nd
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116 d{i} = varargin{1} * ones (sz(i) - 1, 1);
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117 endfor
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118 else
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119 ## vector for one-dimensional derivative
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120 d{1} = diff (varargin{1}(:));
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121 endif
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122 else
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123 ## have spacing value for each dimension
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124 if (length(varargin) != nd)
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125 error ("gradient: dimensions and number of spacing values do not match");
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126 endif
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127 for i = 1:nd
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128 if (isscalar (varargin{i}))
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129 d{i} = varargin{i} * ones (sz(i) - 1, 1);
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130 else
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131 d{i} = diff (varargin{i}(:));
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132 endif
55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
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133 endfor
55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
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134 endif
55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
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135
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136 m = shiftdim (m, 1);
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137 for i = 1:min (nd, nargout)
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138 mr = rows (m);
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139 mc = numel (m) / mr;
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140 Y = zeros (size (m), class (m));
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141
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142 if (mr > 1)
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143 ## Top and bottom boundary.
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144 Y(1,:) = diff (m(1:2, :)) / d{i}(1);
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145 Y(mr,:) = diff (m(mr-1:mr, :) / d{i}(mr - 1));
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146 endif
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147
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148 if (mr > 2)
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149 ## Interior points.
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150 Y(2:mr-1,:) = ((m(3:mr,:) - m(1:mr-2,:))
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151 ./ kron (d{i}(1:mr-2) + d{i}(2:mr-1), ones (1, mc)));
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152 endif
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153
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154 ## turn multi-dimensional matrix in a way, that gradient
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155 ## along x-direction is calculated first then y, z, ...
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156
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157 if (i == 1)
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158 varargout{i} = shiftdim (Y, nd - 1);
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159 m = shiftdim (m, nd - 1);
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160 elseif (i == 2)
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161 varargout{i} = Y;
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162 m = shiftdim (m, 2);
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163 else
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164 varargout{i} = shiftdim (Y, nd - i + 1);
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165 m = shiftdim (m, 1);
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166 endif
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167 endfor
55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
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168
55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
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169 if (transposed)
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170 varargout{1} = varargout{1}.';
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171 endif
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172 endfunction
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173
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174 function varargout = handle_gradient (f, p0, varargin)
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175 ## Input checking
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176 p0_size = size (p0);
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177
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178 if (numel (p0_size) != 2)
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179 error ("gradient: the second input argument should either be a vector or a matrix");
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180 endif
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181
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182 if (any (p0_size == 1))
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183 p0 = p0(:);
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184 dim = 1;
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185 num_points = numel (p0);
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186 else
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187 num_points = p0_size (1);
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188 dim = p0_size (2);
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189 endif
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190
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191 if (length (varargin) == 0)
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192 delta = 1;
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193 elseif (length (varargin) == 1 || length (varargin) == dim)
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194 try
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195 delta = [varargin{:}];
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196 catch
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197 error ("gradient: spacing parameters must be scalars or a vector");
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198 end_try_catch
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199 else
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200 error ("gradient: incorrect number of spacing parameters");
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201 endif
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202
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203 if (isscalar (delta))
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204 delta = repmat (delta, 1, dim);
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205 elseif (! isvector (delta))
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206 error ("gradient: spacing values must be scalars or a vector");
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207 endif
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208
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209 ## Calculate the gradient
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210 p0 = mat2cell (p0, num_points, ones (1, dim));
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211 varargout = cell (1, dim);
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212 for d = 1:dim
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213 s = delta(d);
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214 df_dx = (f (p0{1:d-1}, p0{d}+s, p0{d+1:end})
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215 - f (p0{1:d-1}, p0{d}-s, p0{d+1:end})) ./ (2*s);
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216 if (dim == 1)
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217 varargout{d} = reshape (df_dx, p0_size);
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218 else
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219 varargout{d} = df_dx;
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220 endif
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221 endfor
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222 endfunction
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223
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224
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225 %!test
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226 %! data = [1, 2, 4, 2];
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227 %! dx = gradient (data);
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228 %! dx2 = gradient (data, 0.25);
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229 %! dx3 = gradient (data, [0.25, 0.5, 1, 3]);
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230 %! assert (dx, [1, 3/2, 0, -2]);
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231 %! assert (dx2, [4, 6, 0, -8]);
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232 %! assert (dx3, [4, 4, 0, -1]);
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233 %! assert (size_equal (data, dx));
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234
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235 %!test
06cebb6c5dde Fix calculation of gradient for dims>2. Vector arguments are interpreted as
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diff changeset
236 %! [Y,X,Z,U] = ndgrid (2:2:8,1:5,4:4:12,3:5:30);
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237 %! [dX,dY,dZ,dU] = gradient (X);
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238 %! assert (all (dX(:) == 1));
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239 %! assert (all (dY(:) == 0));
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diff changeset
240 %! assert (all (dZ(:) == 0));
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diff changeset
241 %! assert (all (dU(:) == 0));
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242 %! [dX,dY,dZ,dU] = gradient (Y);
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243 %! assert (all (dX(:) == 0));
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244 %! assert (all (dY(:) == 2));
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245 %! assert (all (dZ(:) == 0));
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diff changeset
246 %! assert (all (dU(:) == 0));
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247 %! [dX,dY,dZ,dU] = gradient (Z);
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248 %! assert (all (dX(:) == 0));
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249 %! assert (all (dY(:) == 0));
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diff changeset
250 %! assert (all (dZ(:) == 4));
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diff changeset
251 %! assert (all (dU(:) == 0));
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252 %! [dX,dY,dZ,dU] = gradient (U);
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253 %! assert (all (dX(:) == 0));
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254 %! assert (all (dY(:) == 0));
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255 %! assert (all (dZ(:) == 0));
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diff changeset
256 %! assert (all (dU(:) == 5));
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257 %! assert (size_equal (dX, dY, dZ, dU, X, Y, Z, U));
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258 %! [dX,dY,dZ,dU] = gradient (U, 5.0);
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259 %! assert (all (dU(:) == 1));
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260 %! [dX,dY,dZ,dU] = gradient (U, 1.0, 2.0, 3.0, 2.5);
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261 %! assert (all (dU(:) == 2));
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262
b297b86f4ad9 gradient.m: Add support for computing the gradient of a function handle
sh@sh-t400
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263 %!test
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264 %! [Y,X,Z,U] = ndgrid (2:2:8,1:5,4:4:12,3:5:30);
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265 %! [dX,dY,dZ,dU] = gradient (X+j*X);
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266 %! assert (all (dX(:) == 1+1j));
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267 %! assert (all (dY(:) == 0));
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268 %! assert (all (dZ(:) == 0));
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269 %! assert (all (dU(:) == 0));
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270 %! [dX,dY,dZ,dU] = gradient (Y-j*Y);
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271 %! assert (all (dX(:) == 0));
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272 %! assert (all (dY(:) == 2-j*2));
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273 %! assert (all (dZ(:) == 0));
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274 %! assert (all (dU(:) == 0));
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275 %! [dX,dY,dZ,dU] = gradient (Z+j*1);
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276 %! assert (all (dX(:) == 0));
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277 %! assert (all (dY(:) == 0));
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278 %! assert (all (dZ(:) == 4));
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279 %! assert (all (dU(:) == 0));
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280 %! [dX,dY,dZ,dU] = gradient (U-j*1);
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281 %! assert (all (dX(:) == 0));
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282 %! assert (all (dY(:) == 0));
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283 %! assert (all (dZ(:) == 0));
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284 %! assert (all (dU(:) == 5));
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285 %! assert (size_equal (dX, dY, dZ, dU, X, Y, Z, U));
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286 %! [dX,dY,dZ,dU] = gradient (U, 5.0);
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287 %! assert (all (dU(:) == 1));
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288 %! [dX,dY,dZ,dU] = gradient (U, 1.0, 2.0, 3.0, 2.5);
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289 %! assert (all (dU(:) == 2));
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290
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291 %!test
8601
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292 %! x = 0:10;
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293 %! f = @cos;
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294 %! df_dx = @(x) -sin (x);
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295 %! assert (gradient (f, x), df_dx (x), 0.2);
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296 %! assert (gradient (f, x, 0.5), df_dx (x), 0.1);
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297
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298 %!test
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299 %! xy = reshape (1:10, 5, 2);
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300 %! f = @(x,y) sin (x) .* cos (y);
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301 %! df_dx = @(x, y) cos (x) .* cos (y);
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302 %! df_dy = @(x, y) -sin (x) .* sin (y);
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303 %! [dx, dy] = gradient (f, xy);
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304 %! assert (dx, df_dx (xy (:, 1), xy (:, 2)), 0.1)
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305 %! assert (dy, df_dy (xy (:, 1), xy (:, 2)), 0.1)
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306