Mercurial > octave
annotate scripts/optimization/fsolve.m @ 14363:f3d52523cde1
Use Octave coding conventions in all m-file %!test blocks
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Use Octave coding conventions in all m-file %!test blocks
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Mon, 13 Feb 2012 07:29:44 -0800 |
| parents | 04dcbb8fb880 |
| children | 59aab666f2bf |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2008-2012 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}. |
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25 ## @var{fcn} should accept 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"}, |
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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 | |
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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 |
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52 ## updates to update the Jacobian, in order to reduce the amount of Jacobian |
| 8590 | 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. | |
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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: | |
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68 ## |
| 8466 | 69 ## @table @asis |
| 70 ## @item 1 | |
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71 ## Converged to a solution point. Relative residual error is less than |
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72 ## specified by TolFun. |
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73 ## |
| 8466 | 74 ## @item 2 |
| 75 ## Last relative step size was less that TolX. | |
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76 ## |
| 8466 | 77 ## @item 3 |
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78 ## Last relative decrease in residual was less than TolF. |
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79 ## |
| 8466 | 80 ## @item 0 |
| 81 ## Iteration limit exceeded. | |
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82 ## |
| 8466 | 83 ## @item -3 |
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84 ## The trust region radius became excessively small. |
| 8466 | 85 ## @end table |
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86 ## |
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87 ## Note: If you only have a single nonlinear equation of one variable, using |
| 8466 | 88 ## @code{fzero} is usually a much better idea. |
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89 ## |
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90 ## Note about user-supplied Jacobians: |
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91 ## As an inherent property of the algorithm, Jacobian is always requested for a |
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92 ## solution vector whose residual vector is already known, and it is the last |
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93 ## accepted successful step. Often this will be one of the last two calls, but |
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94 ## not always. If the savings by reusing intermediate results from residual |
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95 ## calculation in Jacobian calculation are significant, the best strategy is to |
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96 ## employ OutputFcn: After a vector is evaluated for residuals, if OutputFcn is |
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97 ## called with that vector, then the intermediate results should be saved for |
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98 ## future Jacobian evaluation, and should be kept until a Jacobian evaluation |
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99 ## is requested or until outputfcn is called with a different vector, in which |
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100 ## case they should be dropped in favor of this most recent vector. A short |
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101 ## example how this can be achieved follows: |
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102 ## |
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103 ## @example |
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104 ## function [fvec, fjac] = user_func (x, optimvalues, state) |
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105 ## persistent sav = [], sav0 = []; |
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106 ## if (nargin == 1) |
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107 ## ## evaluation call |
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108 ## if (nargout == 1) |
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109 ## sav0.x = x; # mark saved vector |
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110 ## ## calculate fvec, save results to sav0. |
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111 ## elseif (nargout == 2) |
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112 ## ## calculate fjac using sav. |
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113 ## endif |
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114 ## else |
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115 ## ## outputfcn call. |
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116 ## if (all (x == sav0.x)) |
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117 ## sav = sav0; |
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118 ## endif |
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119 ## ## maybe output iteration status, etc. |
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120 ## endif |
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121 ## endfunction |
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122 ## |
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123 ## ## @dots{} |
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124 ## |
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125 ## fsolve (@@user_func, x0, optimset ("OutputFcn", @@user_func, @dots{})) |
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126 ## @end example |
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127 ## @seealso{fzero, optimset} |
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129 |
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130 ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup. |
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131 ## PKG_ADD: [~] = __all_opts__ ("fsolve"); |
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132 |
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133 function [x, fvec, info, output, fjac] = fsolve (fcn, x0, options = struct ()) |
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134 |
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135 ## Get default options if requested. |
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136 if (nargin == 1 && ischar (fcn) && strcmp (fcn, 'defaults')) |
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137 x = optimset ("MaxIter", 400, "MaxFunEvals", Inf, \ |
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138 "Jacobian", "off", "TolX", 1e-7, "TolFun", 1e-7, |
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139 "OutputFcn", [], "Updating", "on", "FunValCheck", "off", |
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140 "ComplexEqn", "off", "FinDiffType", "central", |
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141 "TypicalX", [], "AutoScaling", "off"); |
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142 return; |
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143 endif |
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144 |
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145 if (nargin < 2 || nargin > 3 || ! ismatrix (x0)) |
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146 print_usage (); |
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147 endif |
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148 |
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149 if (ischar (fcn)) |
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150 fcn = str2func (fcn, "global"); |
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151 elseif (iscell (fcn)) |
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152 fcn = @(x) make_fcn_jac (x, fcn{1}, fcn{2}); |
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153 endif |
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154 |
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155 xsiz = size (x0); |
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156 n = numel (x0); |
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157 |
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158 has_jac = strcmpi (optimget (options, "Jacobian", "off"), "on"); |
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159 cdif = strcmpi (optimget (options, "FinDiffType", "central"), "central"); |
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160 maxiter = optimget (options, "MaxIter", 400); |
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161 maxfev = optimget (options, "MaxFunEvals", Inf); |
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162 outfcn = optimget (options, "OutputFcn"); |
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163 updating = strcmpi (optimget (options, "Updating", "on"), "on"); |
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164 complexeqn = strcmpi (optimget (options, "ComplexEqn", "off"), "on"); |
| 8590 | 165 |
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166 ## Get scaling matrix using the TypicalX option. If set to "auto", the |
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167 ## scaling matrix is estimated using the Jacobian. |
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168 typicalx = optimget (options, "TypicalX"); |
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169 if (isempty (typicalx)) |
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170 typicalx = ones (n, 1); |
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171 endif |
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172 autoscale = strcmpi (optimget (options, "AutoScaling", "off"), "on"); |
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173 if (! autoscale) |
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174 dg = 1 ./ typicalx; |
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175 endif |
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176 |
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177 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
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178 |
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179 if (funvalchk) |
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181 fcn = @(x) guarded_eval (fcn, x, complexeqn); |
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182 endif |
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183 |
| 8466 | 184 ## These defaults are rather stringent. I think that normally, user |
| 185 ## prefers accuracy to performance. | |
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186 |
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187 macheps = eps (class (x0)); |
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188 |
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189 tolx = optimget (options, "TolX", 1e-7); |
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190 tolf = optimget (options, "TolFun", 1e-7); |
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191 |
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192 factor = 1; |
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193 |
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194 niter = 1; |
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195 nfev = 1; |
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196 |
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197 x = x0(:); |
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198 info = 0; |
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199 |
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200 ## Initial evaluation. |
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201 ## Handle arbitrary shapes of x and f and remember them. |
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202 fvec = fcn (reshape (x, xsiz)); |
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203 fsiz = size (fvec); |
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204 fvec = fvec(:); |
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205 fn = norm (fvec); |
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206 m = length (fvec); |
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207 n = length (x); |
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208 |
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209 if (! isempty (outfcn)) |
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210 optimvalues.iter = niter; |
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211 optimvalues.funccount = nfev; |
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212 optimvalues.fval = fn; |
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213 optimvalues.searchdirection = zeros (n, 1); |
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214 state = 'init'; |
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215 stop = outfcn (x, optimvalues, state); |
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216 if (stop) |
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217 info = -1; |
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218 break; |
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219 endif |
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220 endif |
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221 |
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222 nsuciter = 0; |
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223 |
| 8507 | 224 ## Outer loop. |
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225 while (niter < maxiter && nfev < maxfev && ! info) |
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226 |
| 8507 | 227 ## Calculate function value and Jacobian (possibly via FD). |
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228 if (has_jac) |
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229 [fvec, fjac] = fcn (reshape (x, xsiz)); |
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230 ## If the Jacobian is sparse, disable Broyden updating. |
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231 if (issparse (fjac)) |
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232 updating = false; |
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233 endif |
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234 fvec = fvec(:); |
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235 nfev ++; |
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236 else |
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237 fjac = __fdjac__ (fcn, reshape (x, xsiz), fvec, typicalx, cdif); |
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238 nfev += (1 + cdif) * length (x); |
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239 endif |
| 8590 | 240 |
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241 ## For square and overdetermined systems, we update a QR |
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242 ## factorization of the Jacobian to avoid solving a full system in each |
| 8590 | 243 ## step. In this case, we pass a triangular matrix to __dogleg__. |
| 244 useqr = updating && m >= n && n > 10; | |
| 245 | |
| 246 if (useqr) | |
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247 ## FIXME: Currently, pivoting is mostly useless because the \ operator |
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248 ## cannot exploit the resulting props of the triangular factor. |
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249 ## Unpivoted QR is significantly faster so it doesn't seem right to pivot |
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250 ## just to get invariance. Original MINPACK didn't pivot either, at least |
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251 ## when qr updating was used. |
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252 [q, r] = qr (fjac, 0); |
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253 endif |
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254 |
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255 if (autoscale) |
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256 ## Get column norms, use them as scaling factors. |
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257 jcn = norm (fjac, 'columns').'; |
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258 if (niter == 1) |
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259 dg = jcn; |
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260 dg(dg == 0) = 1; |
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261 else |
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262 ## Rescale adaptively. |
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263 ## FIXME: the original minpack used the following rescaling strategy: |
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264 ## dg = max (dg, jcn); |
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265 ## but it seems not good if we start with a bad guess yielding Jacobian |
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266 ## columns with large norms that later decrease, because the corresponding |
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267 ## variable will still be overscaled. So instead, we only give the old |
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268 ## scaling a small momentum, but do not honor it. |
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269 |
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270 dg = max (0.1*dg, jcn); |
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271 endif |
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272 endif |
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273 |
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274 if (niter == 1) |
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275 xn = norm (dg .* x); |
| 8590 | 276 ## FIXME: something better? |
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277 delta = factor * max (xn, 1); |
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278 endif |
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279 |
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280 ## It also seems that in the case of fast (and inhomogeneously) changing |
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281 ## Jacobian, the Broyden updates are of little use, so maybe we could |
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282 ## skip them if a big disproportional change is expected. The question is, |
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283 ## of course, how to define the above terms :) |
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284 |
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285 lastratio = 0; |
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286 nfail = 0; |
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287 nsuc = 0; |
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288 decfac = 0.5; |
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289 |
| 8507 | 290 ## Inner loop. |
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291 while (niter <= maxiter && nfev < maxfev && ! info) |
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292 |
| 8590 | 293 ## Get trust-region model (dogleg) minimizer. |
| 294 if (useqr) | |
| 295 qtf = q'*fvec; | |
| 296 s = - __dogleg__ (r, qtf, dg, delta); | |
| 297 w = qtf + r * s; | |
| 298 else | |
| 299 s = - __dogleg__ (fjac, fvec, dg, delta); | |
| 300 w = fvec + fjac * s; | |
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301 endif |
| 8590 | 302 |
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303 sn = norm (dg .* s); |
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304 if (niter == 1) |
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305 delta = min (delta, sn); |
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306 endif |
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307 |
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308 fvec1 = fcn (reshape (x + s, xsiz)) (:); |
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309 fn1 = norm (fvec1); |
| 8590 | 310 nfev ++; |
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311 |
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312 if (fn1 < fn) |
| 8507 | 313 ## Scaled actual reduction. |
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314 actred = 1 - (fn1/fn)^2; |
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315 else |
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316 actred = -1; |
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317 endif |
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318 |
| 8507 | 319 ## Scaled predicted reduction, and ratio. |
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320 t = norm (w); |
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321 if (t < fn) |
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322 prered = 1 - (t/fn)^2; |
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323 ratio = actred / prered; |
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324 else |
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325 prered = 0; |
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326 ratio = 0; |
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327 endif |
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328 |
| 8507 | 329 ## Update delta. |
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330 if (ratio < min(max(0.1, 0.8*lastratio), 0.9)) |
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331 nsuc = 0; |
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332 nfail ++; |
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333 delta *= decfac; |
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334 decfac ^= 1.4142; |
| 8590 | 335 if (delta <= 1e1*macheps*xn) |
| 8507 | 336 ## Trust region became uselessly small. |
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337 info = -3; |
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338 break; |
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339 endif |
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340 else |
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341 lastratio = ratio; |
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342 decfac = 0.5; |
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343 nfail = 0; |
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344 nsuc ++; |
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345 if (abs (1-ratio) <= 0.1) |
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346 delta = 1.4142*sn; |
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347 elseif (ratio >= 0.5 || nsuc > 1) |
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348 delta = max (delta, 1.4142*sn); |
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349 endif |
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350 endif |
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351 |
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352 if (ratio >= 1e-4) |
| 8507 | 353 ## Successful iteration. |
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354 x += s; |
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355 xn = norm (dg .* x); |
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356 fvec = fvec1; |
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357 fn = fn1; |
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358 nsuciter ++; |
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359 endif |
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360 |
| 8857 | 361 niter ++; |
| 362 | |
| 8590 | 363 ## FIXME: should outputfcn be only called after a successful iteration? |
| 8615 | 364 if (! isempty (outfcn)) |
| 8590 | 365 optimvalues.iter = niter; |
| 366 optimvalues.funccount = nfev; | |
| 367 optimvalues.fval = fn; | |
| 368 optimvalues.searchdirection = s; | |
| 369 state = 'iter'; | |
| 370 stop = outfcn (x, optimvalues, state); | |
| 371 if (stop) | |
| 372 info = -1; | |
| 373 break; | |
| 374 endif | |
| 375 endif | |
| 376 | |
| 8466 | 377 ## Tests for termination conditions. A mysterious place, anything |
| 378 ## can happen if you change something here... | |
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379 |
| 8466 | 380 ## The rule of thumb (which I'm not sure M*b is quite following) |
| 381 ## is that for a tolerance that depends on scaling, only 0 makes | |
| 382 ## sense as a default value. But 0 usually means uselessly long | |
| 383 ## iterations, so we need scaling-independent tolerances wherever | |
| 384 ## possible. | |
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385 |
| 8466 | 386 ## FIXME -- why tolf*n*xn? If abs (e) ~ abs(x) * eps is a vector |
| 387 ## of perturbations of x, then norm (fjac*e) <= eps*n*xn, i.e. by | |
| 388 ## tolf ~ eps we demand as much accuracy as we can expect. | |
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389 if (fn <= tolf*n*xn) |
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390 info = 1; |
| 10549 | 391 ## The following tests done only after successful step. |
| 9075 | 392 elseif (ratio >= 1e-4) |
| 8466 | 393 ## This one is classic. Note that we use scaled variables again, |
| 10549 | 394 ## but compare to scaled step, so nothing bad. |
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395 if (sn <= tolx*xn) |
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396 info = 2; |
| 8466 | 397 ## Again a classic one. It seems weird to use the same tolf |
| 10549 | 398 ## for two different tests, but that's what M*b manual appears |
| 399 ## to say. | |
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400 elseif (actred < tolf) |
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401 info = 3; |
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402 endif |
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403 endif |
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404 |
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405 ## Criterion for recalculating Jacobian. |
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406 if (! updating || nfail == 2 || nsuciter < 2) |
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407 break; |
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408 endif |
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409 |
| 8507 | 410 ## Compute the scaled Broyden update. |
| 8590 | 411 if (useqr) |
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412 u = (fvec1 - q*w) / sn; |
| 8590 | 413 v = dg .* ((dg .* s) / sn); |
| 414 | |
| 415 ## Update the QR factorization. | |
| 416 [q, r] = qrupdate (q, r, u, v); | |
| 417 else | |
| 418 u = (fvec1 - w); | |
| 419 v = dg .* ((dg .* s) / sn); | |
| 420 | |
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421 ## update the Jacobian |
| 8590 | 422 fjac += u * v'; |
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423 endif |
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424 endwhile |
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425 endwhile |
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426 |
| 8507 | 427 ## Restore original shapes. |
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428 x = reshape (x, xsiz); |
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429 fvec = reshape (fvec, fsiz); |
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430 |
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431 output.iterations = niter; |
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432 output.successful = nsuciter; |
| 8590 | 433 output.funcCount = nfev; |
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434 |
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435 endfunction |
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436 |
| 8466 | 437 ## An assistant function that evaluates a function handle and checks for |
| 438 ## bad results. | |
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439 function [fx, jx] = guarded_eval (fun, x, complexeqn) |
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440 if (nargout > 1) |
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441 [fx, jx] = fun (x); |
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442 else |
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443 fx = fun (x); |
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444 jx = []; |
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445 endif |
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446 |
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447 if (! complexeqn && ! (isreal (fx) && isreal (jx))) |
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448 error ("fsolve:notreal", "fsolve: non-real value encountered"); |
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449 elseif (complexeqn && ! (isnumeric (fx) && isnumeric(jx))) |
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450 error ("fsolve:notnum", "fsolve: non-numeric value encountered"); |
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451 elseif (any (isnan (fx(:)))) |
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452 error ("fsolve:isnan", "fsolve: NaN value encountered"); |
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453 endif |
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454 endfunction |
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455 |
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456 function [fx, jx] = make_fcn_jac (x, fcn, fjac) |
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457 fx = fcn (x); |
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458 if (nargout == 2) |
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459 jx = fjac (x); |
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460 endif |
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461 endfunction |
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462 |
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463 %!function retval = __f (p) |
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464 %! x = p(1); |
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465 %! y = p(2); |
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466 %! z = p(3); |
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467 %! retval = zeros (3, 1); |
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468 %! retval(1) = sin (x) + y^2 + log (z) - 7; |
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469 %! retval(2) = 3*x + 2^y -z^3 + 1; |
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470 %! retval(3) = x + y + z - 5; |
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471 %!endfunction |
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472 %!test |
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473 %! x_opt = [ 0.599054; |
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474 %! 2.395931; |
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475 %! 2.005014 ]; |
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476 %! tol = 1.0e-5; |
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477 %! [x, fval, info] = fsolve (@__f, [ 0.5; 2.0; 2.5 ]); |
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478 %! assert (info > 0); |
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479 %! assert (norm (x - x_opt, Inf) < tol); |
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480 %! assert (norm (fval) < tol); |
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481 |
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482 %!function retval = __f (p) |
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483 %! x = p(1); |
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484 %! y = p(2); |
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485 %! z = p(3); |
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486 %! w = p(4); |
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487 %! retval = zeros (4, 1); |
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488 %! retval(1) = 3*x + 4*y + exp (z + w) - 1.007; |
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489 %! retval(2) = 6*x - 4*y + exp (3*z + w) - 11; |
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490 %! retval(3) = x^4 - 4*y^2 + 6*z - 8*w - 20; |
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491 %! retval(4) = x^2 + 2*y^3 + z - w - 4; |
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492 %!endfunction |
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493 %!test |
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494 %! x_opt = [ -0.767297326653401, 0.590671081117440, ... |
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495 %! 1.47190018629642, -1.52719341133957 ]; |
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496 %! tol = 1.0e-5; |
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497 %! [x, fval, info] = fsolve (@__f, [-1, 1, 2, -1]); |
|
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498 %! assert (info > 0); |
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499 %! assert (norm (x - x_opt, Inf) < tol); |
|
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500 %! assert (norm (fval) < tol); |
|
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501 |
|
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502 %!function retval = __f (p) |
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503 %! x = p(1); |
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504 %! y = p(2); |
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505 %! z = p(3); |
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506 %! retval = zeros (3, 1); |
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507 %! retval(1) = sin (x) + y^2 + log (z) - 7; |
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508 %! retval(2) = 3*x + 2^y -z^3 + 1; |
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509 %! retval(3) = x + y + z - 5; |
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510 %! retval(4) = x*x + y - z*log (z) - 1.36; |
|
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511 %!endfunction |
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512 %!test |
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513 %! x_opt = [ 0.599054; |
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514 %! 2.395931; |
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515 %! 2.005014 ]; |
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516 %! tol = 1.0e-5; |
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517 %! [x, fval, info] = fsolve (@__f, [ 0.5; 2.0; 2.5 ]); |
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518 %! assert (info > 0); |
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519 %! assert (norm (x - x_opt, Inf) < tol); |
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520 %! assert (norm (fval) < tol); |
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521 |
|
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522 %!function retval = __f (p) |
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523 %! x = p(1); |
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524 %! y = p(2); |
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525 %! z = p(3); |
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526 %! retval = zeros (3, 1); |
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527 %! retval(1) = sin (x) + y^2 + log (z) - 7; |
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528 %! retval(2) = 3*x + 2^y -z^3 + 1; |
|
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529 %! retval(3) = x + y + z - 5; |
|
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530 %!endfunction |
|
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531 %!test |
|
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532 %! x_opt = [ 0.599054; |
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533 %! 2.395931; |
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534 %! 2.005014 ]; |
|
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535 %! tol = 1.0e-5; |
| 8590 | 536 %! opt = optimset ("Updating", "qrp"); |
|
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537 %! [x, fval, info] = fsolve (@__f, [ 0.5; 2.0; 2.5 ], opt); |
|
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538 %! assert (info > 0); |
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539 %! assert (norm (x - x_opt, Inf) < tol); |
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540 %! assert (norm (fval) < tol); |
| 8590 | 541 |
| 542 %!test | |
| 543 %! b0 = 3; | |
| 544 %! a0 = 0.2; | |
| 545 %! x = 0:.5:5; | |
| 546 %! noise = 1e-5 * sin (100*x); | |
| 547 %! y = exp (-a0*x) + b0 + noise; | |
| 548 %! c_opt = [a0, b0]; | |
| 549 %! tol = 1e-5; | |
|
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550 %! |
| 8590 | 551 %! [c, fval, info, output] = fsolve (@(c) (exp(-c(1)*x) + c(2) - y), [0, 0]); |
| 552 %! assert (info > 0); | |
| 553 %! assert (norm (c - c_opt, Inf) < tol); | |
| 554 %! assert (norm (fval) < norm (noise)); | |
| 555 | |
|
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556 %!function y = cfun (x) |
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557 %! y(1) = (1+i)*x(1)^2 - (1-i)*x(2) - 2; |
|
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558 %! y(2) = sqrt (x(1)*x(2)) - (1-2i)*x(3) + (3-4i); |
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559 %! y(3) = x(1) * x(2) - x(3)^2 + (3+2i); |
|
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|
560 %!endfunction |
|
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|
561 |
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562 %!test |
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563 %! x_opt = [-1+i, 1-i, 2+i]; |
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564 %! x = [i, 1, 1+i]; |
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565 %! |
|
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|
566 %! [x, f, info] = fsolve (@cfun, x, optimset ("ComplexEqn", "on")); |
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567 %! tol = 1e-5; |
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|
568 %! assert (norm (f) < tol); |
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|
569 %! assert (norm (x - x_opt, Inf) < tol); |
|
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|
570 |
|
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571 ## Solve the double dogleg trust-region least-squares problem: |
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572 ## Minimize norm(r*x-b) subject to the constraint norm(d.*x) <= delta, |
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573 ## x being a convex combination of the gauss-newton and scaled gradient. |
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|
574 |
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575 ## TODO: error checks |
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576 ## TODO: handle singularity, or leave it up to mldivide? |
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|
577 |
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|
578 function x = __dogleg__ (r, b, d, delta) |
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579 ## Get Gauss-Newton direction. |
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580 x = r \ b; |
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581 xn = norm (d .* x); |
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582 if (xn > delta) |
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583 ## GN is too big, get scaled gradient. |
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584 s = (r' * b) ./ d; |
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585 sn = norm (s); |
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586 if (sn > 0) |
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587 ## Normalize and rescale. |
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588 s = (s / sn) ./ d; |
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589 ## Get the line minimizer in s direction. |
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590 tn = norm (r*s); |
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591 snm = (sn / tn) / tn; |
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592 if (snm < delta) |
| 10549 | 593 ## Get the dogleg path minimizer. |
|
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594 bn = norm (b); |
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595 dxn = delta/xn; snmd = snm/delta; |
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596 t = (bn/sn) * (bn/xn) * snmd; |
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597 t -= dxn * snmd^2 - sqrt ((t-dxn)^2 + (1-dxn^2)*(1-snmd^2)); |
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598 alpha = dxn*(1-snmd^2) / t; |
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599 else |
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600 alpha = 0; |
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601 endif |
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602 else |
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603 alpha = delta / xn; |
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604 snm = 0; |
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605 endif |
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606 ## Form the appropriate convex combination. |
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607 x = alpha * x + ((1-alpha) * min (snm, delta)) * s; |
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608 endif |
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609 endfunction |
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|
610 |