Mercurial > hg > octave-lyh
annotate scripts/optimization/fsolve.m @ 10201:5c66978f3fdf
support TypicalX and AutoScaling in fsolve/fminunc, don't autoscale by default
| author | Jaroslav Hajek <highegg@gmail.com> |
|---|---|
| date | Tue, 26 Jan 2010 11:55:18 +0100 |
| parents | dc88a0b6472c |
| children | 95c3e38098bf |
| rev | line source |
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| 8590 | 1 ## Copyright (C) 2008, 2009 VZLU Prague, a.s. |
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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 ## Author: Jaroslav Hajek <highegg@gmail.com> |
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20 |
| 8466 | 21 ## -*- texinfo -*- |
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22 ## @deftypefn {Function File} {} fsolve (@var{fcn}, @var{x0}, @var{options}) |
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23 ## @deftypefnx {Function File} {[@var{x}, @var{fvec}, @var{info}, @var{output}, @var{fjac}] =} fsolve (@var{fcn}, @dots{}) |
| 8513 | 24 ## Solve a system of nonlinear equations defined by the function @var{fcn}. |
| 8466 | 25 ## @var{fcn} should accepts a vector (array) defining the unknown variables, |
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26 ## and return a vector of left-hand sides of the equations. Right-hand sides |
| 8466 | 27 ## are defined to be zeros. |
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28 ## In other words, this function attempts to determine a vector @var{x} such |
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29 ## that @code{@var{fcn} (@var{x})} gives (approximately) all zeros. |
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30 ## @var{x0} determines a starting guess. The shape of @var{x0} is preserved |
| 8466 | 31 ## in all calls to @var{fcn}, but otherwise it is treated as a column vector. |
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32 ## @var{options} is a structure specifying additional options. |
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33 ## Currently, @code{fsolve} recognizes these options: |
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34 ## @code{"FunValCheck"}, @code{"OutputFcn"}, @code{"TolX"}, |
| 8590 | 35 ## @code{"TolFun"}, @code{"MaxIter"}, @code{"MaxFunEvals"}, |
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36 ## @code{"Jacobian"}, @code{"Updating"}, @code{"ComplexEqn"} |
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37 ## @code{"TypicalX"}, @code{"AutoScaling"} and @code{"FinDiffType"}. |
| 8466 | 38 ## |
| 8513 | 39 ## If @code{"Jacobian"} is @code{"on"}, it specifies that @var{fcn}, |
| 40 ## called with 2 output arguments, also returns the Jacobian matrix | |
| 41 ## of right-hand sides at the requested point. @code{"TolX"} specifies | |
| 42 ## the termination tolerance in the unknown variables, while | |
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43 ## @code{"TolFun"} is a tolerance for equations. Default is @code{1e-7} |
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44 ## for both @code{"TolX"} and @code{"TolFun"}. |
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45 ## |
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46 ## If @code{"AutoScaling"} is on, the variables will be automatically scaled |
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47 ## according to the column norms of the (estimated) Jacobian. As a result, |
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48 ## TolF becomes scaling-independent. By default, this option is off, because |
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49 ## it may sometimes deliver unexpected (though mathematically correct) results. |
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50 ## |
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51 ## If @code{"Updating"} is "on", the function will attempt to use Broyden |
| 8590 | 52 ## updates to update the Jacobian, in order to reduce the amount of jacobian |
| 53 ## calculations. | |
| 54 ## If your user function always calculates the Jacobian (regardless of number | |
| 55 ## of output arguments), this option provides no advantage and should be set to | |
| 56 ## false. | |
| 57 ## | |
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58 ## @code{"ComplexEqn"} is @code{"on"}, @code{fsolve} will attempt to solve |
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59 ## complex equations in complex variables, assuming that the equations possess a |
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60 ## complex derivative (i.e., are holomorphic). If this is not what you want, |
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61 ## should unpack the real and imaginary parts of the system to get a real |
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62 ## system. |
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63 ## |
| 8466 | 64 ## For description of the other options, see @code{optimset}. |
| 65 ## | |
| 66 ## On return, @var{fval} contains the value of the function @var{fcn} | |
| 67 ## evaluated at @var{x}, and @var{info} may be one of the following values: | |
| 68 ## | |
| 69 ## @table @asis | |
| 70 ## @item 1 | |
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71 ## Converged to a solution point. Relative residual error is less than specified |
| 8466 | 72 ## by TolFun. |
| 73 ## @item 2 | |
| 74 ## Last relative step size was less that TolX. | |
| 75 ## @item 3 | |
| 76 ## Last relative decrease in residual was less than TolF. | |
| 77 ## @item 0 | |
| 78 ## Iteration limit exceeded. | |
| 79 ## @item -3 | |
| 80 ## The trust region radius became excessively small. | |
| 81 ## @end table | |
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82 ## |
| 8466 | 83 ## Note: If you only have a single nonlinear equation of one variable, using |
| 84 ## @code{fzero} is usually a much better idea. | |
| 8513 | 85 ## @seealso{fzero, optimset} |
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86 ## |
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87 ## Note about user-supplied jacobians: |
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88 ## As an inherent property of the algorithm, jacobian is always requested for a |
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89 ## solution vector whose residual vector is already known, and it is the last |
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90 ## accepted successful step. Often this will be one of the last two calls, but |
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91 ## not always. If the savings by reusing intermediate results from residual |
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92 ## calculation in jacobian calculation are significant, the best strategy is to |
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93 ## employ OutputFcn: After a vector is evaluated for residuals, if OutputFcn is |
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94 ## called with that vector, then the intermediate results should be saved for |
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95 ## future jacobian evaluation, and should be kept until a jacobian evaluation |
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96 ## is requested or until outputfcn is called with a different vector, in which |
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97 ## case they should be dropped in favor of this most recent vector. A short |
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98 ## example how this can be achieved follows: |
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99 ## |
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100 ## @example |
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101 ## function [fvec, fjac] = user_func (x, optimvalues, state) |
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102 ## persistent sav = [], sav0 = []; |
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103 ## if (nargin == 1) |
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104 ## ## evaluation call |
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105 ## if (nargout == 1) |
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106 ## sav0.x = x; # mark saved vector |
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107 ## ## calculate fvec, save results to sav0. |
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108 ## elseif (nargout == 2) |
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109 ## ## calculate fjac using sav. |
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110 ## endif |
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111 ## else |
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112 ## ## outputfcn call. |
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113 ## if (all (x == sav0.x)) |
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114 ## sav = sav0; |
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115 ## endif |
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116 ## ## maybe output iteration status, etc. |
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117 ## endif |
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118 ## endfunction |
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119 ## |
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120 ## ## @dots{}. |
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121 ## |
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122 ## fsolve (@@user_func, x0, optimset ("OutputFcn", @@user_func, @dots{})) |
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123 ## @end example |
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124 ### |
| 8466 | 125 ## @end deftypefn |
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126 |
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127 ## PKG_ADD: __all_opts__ ("fsolve"); |
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128 |
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129 function [x, fvec, info, output, fjac] = fsolve (fcn, x0, options = struct ()) |
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130 |
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131 ## Get default options if requested. |
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132 if (nargin == 1 && ischar (fcn) && strcmp (fcn, 'defaults')) |
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133 x = optimset ("MaxIter", 400, "MaxFunEvals", Inf, \ |
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134 "Jacobian", "off", "TolX", 1e-7, "TolFun", 1e-7, |
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135 "OutputFcn", [], "Updating", "on", "FunValCheck", "off", |
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136 "ComplexEqn", "off", "FinDiffType", "central", |
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137 "TypicalX", [], "AutoScaling", "off"); |
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138 return; |
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139 endif |
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140 |
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141 if (nargin < 2 || nargin > 3 || ! ismatrix (x0)) |
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142 print_usage (); |
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143 endif |
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144 |
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145 if (ischar (fcn)) |
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146 fcn = str2func (fcn, "global"); |
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147 elseif (iscell (fcn)) |
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148 fcn = @(x) make_fcn_jac (x, fcn{1}, fcn{2}); |
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149 endif |
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150 |
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151 xsiz = size (x0); |
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152 n = numel (x0); |
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153 |
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154 has_jac = strcmpi (optimget (options, "Jacobian", "off"), "on"); |
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155 cdif = strcmpi (optimget (options, "FinDiffType", "central"), "central"); |
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156 maxiter = optimget (options, "MaxIter", 400); |
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157 maxfev = optimget (options, "MaxFunEvals", Inf); |
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158 outfcn = optimget (options, "OutputFcn"); |
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159 updating = strcmpi (optimget (options, "Updating", "on"), "on"); |
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160 complexeqn = strcmpi (optimget (options, "ComplexEqn", "off"), "on"); |
| 8590 | 161 |
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162 ## Get scaling matrix using the TypicalX option. If set to "auto", the |
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163 ## scaling matrix is estimated using the jacobian. |
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164 typicalx = optimget (options, "TypicalX"); |
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165 if (isempty (typicalx)) |
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166 typicalx = ones (n, 1); |
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167 endif |
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168 autoscale = strcmpi (optimget (options, "AutoScaling", "off"), "on"); |
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169 if (! autoscale) |
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170 dg = 1 ./ typicalx; |
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171 endif |
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172 |
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173 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
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174 |
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175 if (funvalchk) |
| 8604 | 176 ## Replace fcn with a guarded version. |
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177 fcn = @(x) guarded_eval (fcn, x, complexeqn); |
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178 endif |
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179 |
| 8466 | 180 ## These defaults are rather stringent. I think that normally, user |
| 181 ## prefers accuracy to performance. | |
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182 |
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183 macheps = eps (class (x0)); |
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184 |
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185 tolx = optimget (options, "TolX", 1e-7); |
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186 tolf = optimget (options, "TolFun", 1e-7); |
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187 |
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188 factor = 1; |
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189 |
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190 niter = 1; |
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191 nfev = 1; |
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192 |
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193 x = x0(:); |
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194 info = 0; |
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195 |
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196 ## Initial evaluation. |
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197 ## Handle arbitrary shapes of x and f and remember them. |
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198 fvec = fcn (reshape (x, xsiz)); |
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199 fsiz = size (fvec); |
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200 fvec = fvec(:); |
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201 fn = norm (fvec); |
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202 m = length (fvec); |
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203 n = length (x); |
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204 |
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205 if (! isempty (outfcn)) |
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206 optimvalues.iter = niter; |
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207 optimvalues.funccount = nfev; |
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208 optimvalues.fval = fn; |
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209 optimvalues.searchdirection = zeros (n, 1); |
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210 state = 'init'; |
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211 stop = outfcn (x, optimvalues, state); |
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212 if (stop) |
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213 info = -1; |
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214 break; |
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215 endif |
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216 endif |
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217 |
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218 nsuciter = 0; |
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219 |
| 8507 | 220 ## Outer loop. |
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221 while (niter < maxiter && nfev < maxfev && ! info) |
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222 |
| 8507 | 223 ## Calculate function value and Jacobian (possibly via FD). |
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224 if (has_jac) |
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225 [fvec, fjac] = fcn (reshape (x, xsiz)); |
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226 ## If the jacobian is sparse, disable Broyden updating. |
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227 if (issparse (fjac)) |
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228 updating = false; |
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229 endif |
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230 fvec = fvec(:); |
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231 nfev ++; |
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232 else |
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233 fjac = __fdjac__ (fcn, reshape (x, xsiz), fvec, typicalx, cdif); |
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234 nfev += (1 + cdif) * length (x); |
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235 endif |
| 8590 | 236 |
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237 ## For square and overdetermined systems, we update a QR |
| 8590 | 238 ## factorization of the jacobian to avoid solving a full system in each |
| 239 ## step. In this case, we pass a triangular matrix to __dogleg__. | |
| 240 useqr = updating && m >= n && n > 10; | |
| 241 | |
| 242 if (useqr) | |
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243 ## FIXME: Currently, pivoting is mostly useless because the \ operator |
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244 ## cannot exploit the resulting props of the triangular factor. |
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245 ## Unpivoted QR is significantly faster so it doesn't seem right to pivot |
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246 ## just to get invariance. Original MINPACK didn't pivot either, at least |
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247 ## when qr updating was used. |
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248 [q, r] = qr (fjac, 0); |
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249 endif |
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250 |
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251 if (autoscale) |
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252 ## Get column norms, use them as scaling factors. |
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253 jcn = norm (fjac, 'columns').'; |
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254 if (niter == 1) |
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255 dg = jcn; |
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256 dg(dg == 0) = 1; |
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257 else |
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258 ## Rescale adaptively. |
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259 ## FIXME: the original minpack used the following rescaling strategy: |
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260 ## dg = max (dg, jcn); |
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261 ## but it seems not good if we start with a bad guess yielding jacobian |
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262 ## columns with large norms that later decrease, because the corresponding |
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263 ## variable will still be overscaled. So instead, we only give the old |
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264 ## scaling a small momentum, but do not honor it. |
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265 |
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266 dg = max (0.1*dg, jcn); |
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267 endif |
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268 endif |
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269 |
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270 if (niter == 1) |
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271 xn = norm (dg .* x); |
| 8590 | 272 ## FIXME: something better? |
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273 delta = factor * max (xn, 1); |
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274 endif |
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275 |
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276 ## It also seems that in the case of fast (and inhomogeneously) changing |
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277 ## jacobian, the Broyden updates are of little use, so maybe we could |
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278 ## skip them if a big disproportional change is expected. The question is, |
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279 ## of course, how to define the above terms :) |
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280 |
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281 lastratio = 0; |
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282 nfail = 0; |
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283 nsuc = 0; |
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284 decfac = 0.5; |
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285 |
| 8507 | 286 ## Inner loop. |
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287 while (niter <= maxiter && nfev < maxfev && ! info) |
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288 |
| 8590 | 289 ## Get trust-region model (dogleg) minimizer. |
| 290 if (useqr) | |
| 291 qtf = q'*fvec; | |
| 292 s = - __dogleg__ (r, qtf, dg, delta); | |
| 293 w = qtf + r * s; | |
| 294 else | |
| 295 s = - __dogleg__ (fjac, fvec, dg, delta); | |
| 296 w = fvec + fjac * s; | |
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297 endif |
| 8590 | 298 |
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299 sn = norm (dg .* s); |
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300 if (niter == 1) |
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301 delta = min (delta, sn); |
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302 endif |
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303 |
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304 fvec1 = fcn (reshape (x + s, xsiz)) (:); |
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305 fn1 = norm (fvec1); |
| 8590 | 306 nfev ++; |
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307 |
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308 if (fn1 < fn) |
| 8507 | 309 ## Scaled actual reduction. |
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310 actred = 1 - (fn1/fn)^2; |
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311 else |
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312 actred = -1; |
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313 endif |
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314 |
| 8507 | 315 ## Scaled predicted reduction, and ratio. |
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316 t = norm (w); |
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317 if (t < fn) |
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318 prered = 1 - (t/fn)^2; |
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319 ratio = actred / prered; |
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320 else |
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321 prered = 0; |
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322 ratio = 0; |
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323 endif |
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324 |
| 8507 | 325 ## Update delta. |
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326 if (ratio < min(max(0.1, 0.8*lastratio), 0.9)) |
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327 nsuc = 0; |
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328 nfail ++; |
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329 delta *= decfac; |
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330 decfac ^= 1.4142; |
| 8590 | 331 if (delta <= 1e1*macheps*xn) |
| 8507 | 332 ## Trust region became uselessly small. |
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333 info = -3; |
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334 break; |
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335 endif |
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336 else |
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337 lastratio = ratio; |
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338 decfac = 0.5; |
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339 nfail = 0; |
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340 nsuc ++; |
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341 if (abs (1-ratio) <= 0.1) |
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342 delta = 1.4142*sn; |
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343 elseif (ratio >= 0.5 || nsuc > 1) |
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344 delta = max (delta, 1.4142*sn); |
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345 endif |
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346 endif |
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347 |
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348 if (ratio >= 1e-4) |
| 8507 | 349 ## Successful iteration. |
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350 x += s; |
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351 xn = norm (dg .* x); |
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352 fvec = fvec1; |
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353 fn = fn1; |
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354 nsuciter ++; |
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355 endif |
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356 |
| 8857 | 357 niter ++; |
| 358 | |
| 8590 | 359 ## FIXME: should outputfcn be only called after a successful iteration? |
| 8615 | 360 if (! isempty (outfcn)) |
| 8590 | 361 optimvalues.iter = niter; |
| 362 optimvalues.funccount = nfev; | |
| 363 optimvalues.fval = fn; | |
| 364 optimvalues.searchdirection = s; | |
| 365 state = 'iter'; | |
| 366 stop = outfcn (x, optimvalues, state); | |
| 367 if (stop) | |
| 368 info = -1; | |
| 369 break; | |
| 370 endif | |
| 371 endif | |
| 372 | |
| 8466 | 373 ## Tests for termination conditions. A mysterious place, anything |
| 374 ## can happen if you change something here... | |
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375 |
| 8466 | 376 ## The rule of thumb (which I'm not sure M*b is quite following) |
| 377 ## is that for a tolerance that depends on scaling, only 0 makes | |
| 378 ## sense as a default value. But 0 usually means uselessly long | |
| 379 ## iterations, so we need scaling-independent tolerances wherever | |
| 380 ## possible. | |
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381 |
| 8466 | 382 ## FIXME -- why tolf*n*xn? If abs (e) ~ abs(x) * eps is a vector |
| 383 ## of perturbations of x, then norm (fjac*e) <= eps*n*xn, i.e. by | |
| 384 ## tolf ~ eps we demand as much accuracy as we can expect. | |
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385 if (fn <= tolf*n*xn) |
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386 info = 1; |
| 8466 | 387 ## The following tests done only after successful step. |
| 9075 | 388 elseif (ratio >= 1e-4) |
| 8466 | 389 ## This one is classic. Note that we use scaled variables again, |
| 390 ## but compare to scaled step, so nothing bad. | |
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391 if (sn <= tolx*xn) |
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392 info = 2; |
| 8466 | 393 ## Again a classic one. It seems weird to use the same tolf |
| 394 ## for two different tests, but that's what M*b manual appears | |
| 395 ## to say. | |
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396 elseif (actred < tolf) |
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397 info = 3; |
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398 endif |
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399 endif |
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400 |
| 8507 | 401 ## Criterion for recalculating jacobian. |
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402 if (! updating || nfail == 2 || nsuciter < 2) |
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403 break; |
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404 endif |
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405 |
| 8507 | 406 ## Compute the scaled Broyden update. |
| 8590 | 407 if (useqr) |
| 408 u = (fvec1 - q*w) / sn; | |
| 409 v = dg .* ((dg .* s) / sn); | |
| 410 | |
| 411 ## Update the QR factorization. | |
| 412 [q, r] = qrupdate (q, r, u, v); | |
| 413 else | |
| 414 u = (fvec1 - w); | |
| 415 v = dg .* ((dg .* s) / sn); | |
| 416 | |
| 417 ## update the jacobian | |
| 418 fjac += u * v'; | |
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419 endif |
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420 endwhile |
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421 endwhile |
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422 |
| 8507 | 423 ## Restore original shapes. |
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424 x = reshape (x, xsiz); |
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425 fvec = reshape (fvec, fsiz); |
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426 |
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427 output.iterations = niter; |
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428 output.successful = nsuciter; |
| 8590 | 429 output.funcCount = nfev; |
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430 |
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431 endfunction |
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432 |
| 8466 | 433 ## An assistant function that evaluates a function handle and checks for |
| 434 ## bad results. | |
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435 function [fx, jx] = guarded_eval (fun, x, complexeqn) |
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improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
436 if (nargout > 1) |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
437 [fx, jx] = fun (x); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
438 else |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
439 fx = fun (x); |
|
8997
187a9d9c2f04
simplifications to fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8991
diff
changeset
|
440 jx = []; |
|
8693
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
441 endif |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
442 |
|
8997
187a9d9c2f04
simplifications to fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8991
diff
changeset
|
443 if (! complexeqn && ! (isreal (fx) && isreal (jx))) |
|
8468
866492035ecf
fsolve.m, fzero.m: undo part of previous change
John W. Eaton <jwe@octave.org>
parents:
8467
diff
changeset
|
444 error ("fsolve:notreal", "fsolve: non-real value encountered"); |
|
8997
187a9d9c2f04
simplifications to fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8991
diff
changeset
|
445 elseif (complexeqn && ! (isnumeric (fx) && isnumeric(jx))) |
|
8693
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
446 error ("fsolve:notnum", "fsolve: non-numeric value encountered"); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
447 elseif (any (isnan (fx(:)))) |
|
8468
866492035ecf
fsolve.m, fzero.m: undo part of previous change
John W. Eaton <jwe@octave.org>
parents:
8467
diff
changeset
|
448 error ("fsolve:isnan", "fsolve: NaN value encountered"); |
|
8306
43795cf108d0
initial implementation of fsolve
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parents:
diff
changeset
|
449 endif |
|
43795cf108d0
initial implementation of fsolve
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parents:
diff
changeset
|
450 endfunction |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
451 |
|
10050
dc88a0b6472c
support old style jacobian for fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
9899
diff
changeset
|
452 function [fx, jx] = make_fcn_jac (x, fcn, fjac) |
|
dc88a0b6472c
support old style jacobian for fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
9899
diff
changeset
|
453 fx = fcn (x); |
|
dc88a0b6472c
support old style jacobian for fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
9899
diff
changeset
|
454 if (nargout == 2) |
|
dc88a0b6472c
support old style jacobian for fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
9899
diff
changeset
|
455 jx = fjac (x); |
|
dc88a0b6472c
support old style jacobian for fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
9899
diff
changeset
|
456 endif |
|
dc88a0b6472c
support old style jacobian for fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
9899
diff
changeset
|
457 endfunction |
|
dc88a0b6472c
support old style jacobian for fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
9899
diff
changeset
|
458 |
|
8306
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
459 %!function retval = f (p) |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
460 %! x = p(1); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
461 %! y = p(2); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
462 %! z = p(3); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
463 %! retval = zeros (3, 1); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
464 %! retval(1) = sin(x) + y**2 + log(z) - 7; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
465 %! retval(2) = 3*x + 2**y -z**3 + 1; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
466 %! retval(3) = x + y + z - 5; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
467 %!test |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
468 %! x_opt = [ 0.599054; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
469 %! 2.395931; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
470 %! 2.005014 ]; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
471 %! tol = 1.0e-5; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
472 %! [x, fval, info] = fsolve (@f, [ 0.5; 2.0; 2.5 ]); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
473 %! assert (info > 0); |
|
8383
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
474 %! assert (norm (x - x_opt, Inf) < tol); |
|
8306
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
475 %! assert (norm (fval) < tol); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
476 |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
477 %!function retval = f (p) |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
478 %! x = p(1); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
479 %! y = p(2); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
480 %! z = p(3); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
481 %! w = p(4); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
482 %! retval = zeros (4, 1); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
483 %! retval(1) = 3*x + 4*y + exp (z + w) - 1.007; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
484 %! retval(2) = 6*x - 4*y + exp (3*z + w) - 11; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
485 %! retval(3) = x^4 - 4*y^2 + 6*z - 8*w - 20; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
486 %! retval(4) = x^2 + 2*y^3 + z - w - 4; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
487 %!test |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
488 %! x_opt = [ -0.767297326653401, 0.590671081117440, 1.47190018629642, -1.52719341133957 ]; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
489 %! tol = 1.0e-5; |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
490 %! [x, fval, info] = fsolve (@f, [-1, 1, 2, -1]); |
|
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
491 %! assert (info > 0); |
|
8383
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
492 %! assert (norm (x - x_opt, Inf) < tol); |
|
8306
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
493 %! assert (norm (fval) < tol); |
|
8383
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
494 |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
495 %!function retval = f (p) |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
496 %! x = p(1); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
497 %! y = p(2); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
498 %! z = p(3); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
499 %! retval = zeros (3, 1); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
500 %! retval(1) = sin(x) + y**2 + log(z) - 7; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
501 %! retval(2) = 3*x + 2**y -z**3 + 1; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
502 %! retval(3) = x + y + z - 5; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
503 %! retval(4) = x*x + y - z*log(z) - 1.36; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
504 %!test |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
505 %! x_opt = [ 0.599054; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
506 %! 2.395931; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
507 %! 2.005014 ]; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
508 %! tol = 1.0e-5; |
| 8590 | 509 %! [x, fval, info] = fsolve (@f, [ 0.5; 2.0; 2.5 ]); |
|
8383
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
510 %! assert (info > 0); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
511 %! assert (norm (x - x_opt, Inf) < tol); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
512 %! assert (norm (fval) < tol); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
513 |
|
8395
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
514 %!function retval = f (p) |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
515 %! x = p(1); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
516 %! y = p(2); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
517 %! z = p(3); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
518 %! retval = zeros (3, 1); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
519 %! retval(1) = sin(x) + y**2 + log(z) - 7; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
520 %! retval(2) = 3*x + 2**y -z**3 + 1; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
521 %! retval(3) = x + y + z - 5; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
522 %!test |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
523 %! x_opt = [ 0.599054; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
524 %! 2.395931; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
525 %! 2.005014 ]; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
526 %! tol = 1.0e-5; |
| 8590 | 527 %! opt = optimset ("Updating", "qrp"); |
|
8395
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
528 %! [x, fval, info] = fsolve (@f, [ 0.5; 2.0; 2.5 ], opt); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
529 %! assert (info > 0); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
530 %! assert (norm (x - x_opt, Inf) < tol); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
531 %! assert (norm (fval) < tol); |
| 8590 | 532 |
| 533 %!test | |
| 534 %! b0 = 3; | |
| 535 %! a0 = 0.2; | |
| 536 %! x = 0:.5:5; | |
| 537 %! noise = 1e-5 * sin (100*x); | |
| 538 %! y = exp (-a0*x) + b0 + noise; | |
| 539 %! c_opt = [a0, b0]; | |
| 540 %! tol = 1e-5; | |
| 541 %! | |
| 542 %! [c, fval, info, output] = fsolve (@(c) (exp(-c(1)*x) + c(2) - y), [0, 0]); | |
| 543 %! assert (info > 0); | |
| 544 %! assert (norm (c - c_opt, Inf) < tol); | |
| 545 %! assert (norm (fval) < norm (noise)); | |
| 546 | |
|
8693
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
547 |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
548 %!function y = cfun (x) |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
549 %! y(1) = (1+i)*x(1)^2 - (1-i)*x(2) - 2; |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
550 %! y(2) = sqrt (x(1)*x(2)) - (1-2i)*x(3) + (3-4i); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
551 %! y(3) = x(1) * x(2) - x(3)^2 + (3+2i); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
552 |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
553 %!test |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
554 %! x_opt = [-1+i, 1-i, 2+i]; |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
555 %! x = [i, 1, 1+i]; |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
556 %! |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
557 %! [x, f, info] = fsolve (@cfun, x, optimset ("ComplexEqn", "on")); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
558 %! tol = 1e-5; |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
559 %! assert (norm (f) < tol); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
560 %! assert (norm (x - x_opt, Inf) < tol); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
561 |
|
9899
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562 ## Solve the double dogleg trust-region least-squares problem: |
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563 ## Minimize norm(r*x-b) subject to the constraint norm(d.*x) <= delta, |
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564 ## x being a convex combination of the gauss-newton and scaled gradient. |
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565 |
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566 ## TODO: error checks |
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567 ## TODO: handle singularity, or leave it up to mldivide? |
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568 |
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569 function x = __dogleg__ (r, b, d, delta) |
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570 ## Get Gauss-Newton direction. |
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571 x = r \ b; |
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572 xn = norm (d .* x); |
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573 if (xn > delta) |
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574 ## GN is too big, get scaled gradient. |
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575 s = (r' * b) ./ d; |
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576 sn = norm (s); |
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577 if (sn > 0) |
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578 ## Normalize and rescale. |
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579 s = (s / sn) ./ d; |
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580 ## Get the line minimizer in s direction. |
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581 tn = norm (r*s); |
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582 snm = (sn / tn) / tn; |
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583 if (snm < delta) |
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584 ## Get the dogleg path minimizer. |
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585 bn = norm (b); |
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586 dxn = delta/xn; snmd = snm/delta; |
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587 t = (bn/sn) * (bn/xn) * snmd; |
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588 t -= dxn * snmd^2 - sqrt ((t-dxn)^2 + (1-dxn^2)*(1-snmd^2)); |
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589 alpha = dxn*(1-snmd^2) / t; |
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590 else |
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591 alpha = 0; |
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592 endif |
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593 else |
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594 alpha = delta / xn; |
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595 snm = 0; |
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596 endif |
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597 ## Form the appropriate convex combination. |
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598 x = alpha * x + ((1-alpha) * min (snm, delta)) * s; |
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599 endif |
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600 endfunction |
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601 |