Mercurial > hg > octave-lyh
annotate scripts/optimization/fsolve.m @ 8990:349d75161672
more cosmetic adjustments to fsolve
| author | Jaroslav Hajek <highegg@gmail.com> |
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| date | Tue, 17 Mar 2009 13:38:16 +0100 |
| parents | 315828058e0d |
| children | c235a59d30a4 |
| 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 -*- |
| 8513 | 22 ## @deftypefn{Function File} {} fsolve (@var{fcn}, @var{x0}, @var{options}) |
| 8466 | 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, |
| 26 ## and return a vector of left-hand sides of the equations. Right-hand sides | |
| 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. |
| 8466 | 30 ## @var{x0} determines a starting guess. The shape of @var{x0} is preserved |
| 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"} and @code{"ComplexEqn"}. |
| 8466 | 37 ## |
| 8513 | 38 ## If @code{"Jacobian"} is @code{"on"}, it specifies that @var{fcn}, |
| 39 ## called with 2 output arguments, also returns the Jacobian matrix | |
| 40 ## of right-hand sides at the requested point. @code{"TolX"} specifies | |
| 41 ## the termination tolerance in the unknown variables, while | |
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42 ## @code{"TolFun"} is a tolerance for equations. Default is @code{1e-7} |
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43 ## for both @code{"TolX"} and @code{"TolFun"}. |
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44 ## If @code{"Updating"} is "on", the function will attempt to use Broyden |
| 8590 | 45 ## updates to update the Jacobian, in order to reduce the amount of jacobian |
| 46 ## calculations. | |
| 47 ## If your user function always calculates the Jacobian (regardless of number | |
| 48 ## of output arguments), this option provides no advantage and should be set to | |
| 49 ## false. | |
| 50 ## | |
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51 ## @code{"ComplexEqn"} is @code{"on"}, @code{fsolve} will attempt to solve |
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52 ## complex equations in complex variables, assuming that the equations posess a |
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53 ## complex derivative (i.e. are holomorphic). If this is not what you want, |
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54 ## should unpack the real and imaginary parts of the system to get a real |
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55 ## system. |
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56 ## |
| 8466 | 57 ## For description of the other options, see @code{optimset}. |
| 58 ## | |
| 59 ## On return, @var{fval} contains the value of the function @var{fcn} | |
| 60 ## evaluated at @var{x}, and @var{info} may be one of the following values: | |
| 61 ## | |
| 62 ## @table @asis | |
| 63 ## @item 1 | |
| 64 ## Converged to a solution point. Relative residual error is less than specified | |
| 65 ## by TolFun. | |
| 66 ## @item 2 | |
| 67 ## Last relative step size was less that TolX. | |
| 68 ## @item 3 | |
| 69 ## Last relative decrease in residual was less than TolF. | |
| 70 ## @item 0 | |
| 71 ## Iteration limit exceeded. | |
| 72 ## @item -3 | |
| 73 ## The trust region radius became excessively small. | |
| 74 ## @end table | |
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75 ## |
| 8466 | 76 ## Note: If you only have a single nonlinear equation of one variable, using |
| 77 ## @code{fzero} is usually a much better idea. | |
| 8513 | 78 ## @seealso{fzero, optimset} |
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79 ## |
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80 ## Note about user-supplied jacobians: |
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81 ## As an inherent property of the algorithm, jacobian is always requested for a |
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82 ## solution vector whose residual vector is already known, and it is the last |
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83 ## accepted successful step. Often this will be one of the last two calls, but |
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84 ## not always. If the savings by reusing intermediate results from residual |
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85 ## calculation in jacobian calculation are significant, the best strategy is to |
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86 ## employ OutputFcn: After a vector is evaluated for residuals, if OutputFcn is |
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87 ## called with that vector, then the intermediate results should be saved for |
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88 ## future jacobian evaluation, and should be kept until a jacobian evaluation |
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89 ## is requested or until outputfcn is called with a different vector, in which |
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90 ## case they should be dropped in favor of this most recent vector. A short |
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91 ## example how this can be achieved follows: |
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92 ## |
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93 ## @example |
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94 ## function [fvec, fjac] = my_optim_func (x, optimvalues, state) |
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95 ## persistent sav = [], sav0 = []; |
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96 ## if (nargin == 1) |
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97 ## ## evaluation call |
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98 ## if (nargout == 1) |
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99 ## sav0.x = x; # mark saved vector |
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100 ## ## calculate fvec, save results to sav0. |
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101 ## elseif (nargout == 2) |
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102 ## ## calculate fjac using sav. |
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103 ## endif |
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104 ## else |
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105 ## ## outputfcn call. |
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106 ## if (all (x == sav0.x)) |
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107 ## sav = sav0; |
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108 ## endif |
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109 ## ## maybe output iteration status etc. |
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110 ## endif |
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111 ## endfunction |
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112 ## |
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113 ## ## .... |
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114 ## |
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115 ## fsolve (@@my_optim_func, x0, optimset ("OutputFcn", @@my_optim_func, @dots{})) |
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116 ## @end example |
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117 ### |
| 8466 | 118 ## @end deftypefn |
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119 |
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120 ## PKG_ADD: __all_opts__ ("fsolve"); |
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121 |
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122 function [x, fvec, info, output, fjac] = fsolve (fcn, x0, options = struct ()) |
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123 |
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124 ## Get default options if requested. |
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125 if (nargin == 1 && ischar (fcn) && strcmp (fcn, 'defaults')) |
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126 x = optimset ("MaxIter", 400, "MaxFunEvals", Inf, \ |
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127 "Jacobian", "off", "TolX", 1.5e-8, "TolFun", 1.5e-8, |
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128 "OutputFcn", [], "Updating", "on", "FunValCheck", "off", |
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129 "ComplexEqn", "off"); |
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130 return; |
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131 endif |
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132 |
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133 if (nargin < 2 || nargin > 3 || ! ismatrix (x0)) |
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134 print_usage (); |
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135 endif |
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136 |
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137 if (ischar (fcn)) |
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138 fcn = str2func (fcn); |
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139 endif |
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140 |
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141 xsiz = size (x0); |
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142 n = numel (x0); |
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143 |
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144 has_jac = strcmpi (optimget (options, "Jacobian", "off"), "on"); |
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145 maxiter = optimget (options, "MaxIter", 400); |
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146 maxfev = optimget (options, "MaxFunEvals", Inf); |
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147 outfcn = optimget (options, "OutputFcn"); |
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148 updating = strcmpi (optimget (options, "Updating", "on"), "on"); |
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149 complexeqn = strcmpi (optimget (options, "ComplexEqn", "off"), "on"); |
| 8590 | 150 |
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151 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
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152 |
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153 if (funvalchk) |
| 8604 | 154 ## Replace fcn with a guarded version. |
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155 fcn = @(x) guarded_eval (fcn, x, complexeqn); |
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156 endif |
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157 |
| 8466 | 158 ## These defaults are rather stringent. I think that normally, user |
| 159 ## prefers accuracy to performance. | |
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160 |
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161 macheps = eps (class (x0)); |
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162 |
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163 tolx = optimget (options, "TolX", sqrt (macheps)); |
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164 tolf = optimget (options, "TolFun", sqrt (macheps)); |
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165 |
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166 factor = 100; |
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167 |
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168 niter = 1; |
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169 nfev = 1; |
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170 |
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171 x = x0(:); |
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172 info = 0; |
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173 |
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174 ## Initial evaluation. |
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175 ## Handle arbitrary shapes of x and f and remember them. |
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176 fvec = fcn (reshape (x, xsiz)); |
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177 fsiz = size (fvec); |
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178 fvec = fvec(:); |
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179 fn = norm (fvec); |
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180 m = length (fvec); |
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181 n = length (x); |
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182 |
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183 if (! isempty (outfcn)) |
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184 optimvalues.iter = niter; |
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185 optimvalues.funccount = nfev; |
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186 optimvalues.fval = fn; |
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187 optimvalues.searchdirection = zeros (n, 1); |
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188 state = 'init'; |
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189 stop = outfcn (x, optimvalues, state); |
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190 if (stop) |
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191 info = -1; |
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192 break; |
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193 endif |
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194 endif |
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195 |
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196 nsuciter = 0; |
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197 |
| 8507 | 198 ## Outer loop. |
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199 while (niter < maxiter && nfev < maxfev && ! info) |
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200 |
| 8507 | 201 ## Calculate function value and Jacobian (possibly via FD). |
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202 if (has_jac) |
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203 [fvec, fjac] = fcn (reshape (x, xsiz)); |
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204 ## If the jacobian is sparse, disable Broyden updating. |
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205 if (issparse (fjac)) |
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206 updating = false; |
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207 endif |
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208 fvec = fvec(:); |
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209 nfev ++; |
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210 else |
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211 fjac = __fdjac__ (fcn, reshape (x, xsiz), fvec); |
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212 nfev += length (x); |
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213 endif |
| 8590 | 214 |
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215 ## For square and overdetermined systems, we update a QR |
| 8590 | 216 ## factorization of the jacobian to avoid solving a full system in each |
| 217 ## step. In this case, we pass a triangular matrix to __dogleg__. | |
| 218 useqr = updating && m >= n && n > 10; | |
| 219 | |
| 220 if (useqr) | |
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221 ## FIXME: Currently, pivoting is mostly useless because the \ operator |
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222 ## cannot exploit the resulting props of the triangular factor. |
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223 ## Unpivoted QR is significantly faster so it doesn't seem right to pivot |
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224 ## just to get invariance. Original MINPACK didn't pivot either, at least |
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225 ## when qr updating was used. |
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226 [q, r] = qr (fjac, 0); |
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227 endif |
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228 |
| 8590 | 229 ## Get column norms, use them as scaling factors. |
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230 jcn = norm (fjac, 'columns').'; |
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231 if (niter == 1) |
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232 dg = jcn; |
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233 dg(dg == 0) = 1; |
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234 xn = norm (dg .* x); |
| 8590 | 235 ## FIXME: something better? |
| 236 delta = max (factor * xn, 1); | |
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237 endif |
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238 |
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239 ## Rescale adaptively. |
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240 ## FIXME: the original minpack used the following rescaling strategy: |
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241 ## dg = max (dg, jcn); |
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242 ## but it seems not good if we start with a bad guess yielding jacobian |
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243 ## columns with large norms that later decrease, because the corresponding |
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244 ## variable will still be overscaled. So instead, we only give the old |
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245 ## scaling a small momentum, but do not honor it. |
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246 |
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247 dg = max (0.1*dg, jcn); |
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248 |
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249 ## It also seems that in the case of fast (and inhomogeneously) changing |
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250 ## jacobian, the Broyden updates are of little use, so maybe we could |
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251 ## skip them if a big disproportional change is expected. The question is, |
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252 ## of course, how to define the above terms :) |
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253 |
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254 lastratio = 0; |
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255 nfail = 0; |
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256 nsuc = 0; |
| 8507 | 257 ## Inner loop. |
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258 while (niter <= maxiter && nfev < maxfev && ! info) |
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259 |
| 8590 | 260 if (useqr) |
| 261 tr_mat = r; | |
| 262 tr_vec = q'*fvec; | |
| 263 else | |
| 264 tr_mat = fjac; | |
| 265 tr_vec = fvec; | |
| 266 endif | |
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267 |
| 8590 | 268 ## Get trust-region model (dogleg) minimizer. |
| 269 if (useqr) | |
| 270 qtf = q'*fvec; | |
| 271 s = - __dogleg__ (r, qtf, dg, delta); | |
| 272 w = qtf + r * s; | |
| 273 else | |
| 274 s = - __dogleg__ (fjac, fvec, dg, delta); | |
| 275 w = fvec + fjac * s; | |
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276 endif |
| 8590 | 277 |
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278 sn = norm (dg .* s); |
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279 if (niter == 1) |
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280 delta = min (delta, sn); |
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281 endif |
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282 |
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283 fvec1 = fcn (reshape (x + s, xsiz)) (:); |
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284 fn1 = norm (fvec1); |
| 8590 | 285 nfev ++; |
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286 |
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287 if (fn1 < fn) |
| 8507 | 288 ## Scaled actual reduction. |
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289 actred = 1 - (fn1/fn)^2; |
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290 else |
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291 actred = -1; |
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292 endif |
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293 |
| 8507 | 294 ## Scaled predicted reduction, and ratio. |
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295 t = norm (w); |
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296 if (t < fn) |
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297 prered = 1 - (t/fn)^2; |
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298 ratio = actred / prered; |
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299 else |
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300 prered = 0; |
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301 ratio = 0; |
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302 endif |
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303 |
| 8507 | 304 ## Update delta. |
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305 if (ratio < min(max(0.1, lastratio), 0.9)) |
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306 nsuc = 0; |
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307 nfail ++; |
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308 delta *= 0.5; |
| 8590 | 309 if (delta <= 1e1*macheps*xn) |
| 8507 | 310 ## Trust region became uselessly small. |
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311 info = -3; |
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312 break; |
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313 endif |
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314 else |
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315 lastratio = ratio; |
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316 nfail = 0; |
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317 nsuc ++; |
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318 if (abs (1-ratio) <= 0.1) |
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319 delta = 2*sn; |
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320 elseif (ratio >= 0.5 || nsuc > 1) |
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321 delta = max (delta, 2*sn); |
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322 endif |
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323 endif |
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324 |
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325 if (ratio >= 1e-4) |
| 8507 | 326 ## Successful iteration. |
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327 x += s; |
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328 xn = norm (dg .* x); |
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329 fvec = fvec1; |
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330 fn = fn1; |
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331 nsuciter ++; |
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332 endif |
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333 |
| 8857 | 334 niter ++; |
| 335 | |
| 8590 | 336 ## FIXME: should outputfcn be only called after a successful iteration? |
| 8615 | 337 if (! isempty (outfcn)) |
| 8590 | 338 optimvalues.iter = niter; |
| 339 optimvalues.funccount = nfev; | |
| 340 optimvalues.fval = fn; | |
| 341 optimvalues.searchdirection = s; | |
| 342 state = 'iter'; | |
| 343 stop = outfcn (x, optimvalues, state); | |
| 344 if (stop) | |
| 345 info = -1; | |
| 346 break; | |
| 347 endif | |
| 348 endif | |
| 349 | |
| 8466 | 350 ## Tests for termination conditions. A mysterious place, anything |
| 351 ## can happen if you change something here... | |
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352 |
| 8466 | 353 ## The rule of thumb (which I'm not sure M*b is quite following) |
| 354 ## is that for a tolerance that depends on scaling, only 0 makes | |
| 355 ## sense as a default value. But 0 usually means uselessly long | |
| 356 ## iterations, so we need scaling-independent tolerances wherever | |
| 357 ## possible. | |
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358 |
| 8466 | 359 ## FIXME -- why tolf*n*xn? If abs (e) ~ abs(x) * eps is a vector |
| 360 ## of perturbations of x, then norm (fjac*e) <= eps*n*xn, i.e. by | |
| 361 ## tolf ~ eps we demand as much accuracy as we can expect. | |
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362 if (fn <= tolf*n*xn) |
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363 info = 1; |
| 8466 | 364 ## The following tests done only after successful step. |
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365 elseif (actred > 0) |
| 8466 | 366 ## This one is classic. Note that we use scaled variables again, |
| 367 ## but compare to scaled step, so nothing bad. | |
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368 if (sn <= tolx*xn) |
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369 info = 2; |
| 8466 | 370 ## Again a classic one. It seems weird to use the same tolf |
| 371 ## for two different tests, but that's what M*b manual appears | |
| 372 ## to say. | |
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373 elseif (actred < tolf) |
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374 info = 3; |
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375 endif |
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376 endif |
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377 |
| 8507 | 378 ## Criterion for recalculating jacobian. |
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379 if (! updating || nfail == 2 || nsuciter < 2) |
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380 break; |
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381 endif |
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382 |
| 8507 | 383 ## Compute the scaled Broyden update. |
| 8590 | 384 if (useqr) |
| 385 u = (fvec1 - q*w) / sn; | |
| 386 v = dg .* ((dg .* s) / sn); | |
| 387 | |
| 388 ## Update the QR factorization. | |
| 389 [q, r] = qrupdate (q, r, u, v); | |
| 390 else | |
| 391 u = (fvec1 - w); | |
| 392 v = dg .* ((dg .* s) / sn); | |
| 393 | |
| 394 ## update the jacobian | |
| 395 fjac += u * v'; | |
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396 endif |
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397 endwhile |
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398 endwhile |
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399 |
| 8507 | 400 ## Restore original shapes. |
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401 x = reshape (x, xsiz); |
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402 fvec = reshape (fvec, fsiz); |
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403 |
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404 output.iterations = niter; |
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405 output.successful = nsuciter; |
| 8590 | 406 output.funcCount = nfev; |
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407 |
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408 endfunction |
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409 |
| 8466 | 410 ## An assistant function that evaluates a function handle and checks for |
| 411 ## bad results. | |
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412 function [fx, jx] = guarded_eval (fun, x, complexeqn) |
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413 if (nargout > 1) |
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414 [fx, jx] = fun (x); |
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415 else |
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416 fx = fun (x); |
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417 jx = [] |
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418 endif |
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419 |
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420 if (! complexeqn && ! (all (isreal (fx(:))) && all (isreal (jx(:))))) |
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421 error ("fsolve:notreal", "fsolve: non-real value encountered"); |
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422 elseif (complexeqn && ! (all (isnumeric (fx(:))) && all (isnumeric(jx(:))))) |
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423 error ("fsolve:notnum", "fsolve: non-numeric value encountered"); |
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424 elseif (any (isnan (fx(:)))) |
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425 error ("fsolve:isnan", "fsolve: NaN value encountered"); |
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426 endif |
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427 endfunction |
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428 |
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429 %!function retval = f (p) |
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430 %! x = p(1); |
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431 %! y = p(2); |
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432 %! z = p(3); |
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433 %! retval = zeros (3, 1); |
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434 %! retval(1) = sin(x) + y**2 + log(z) - 7; |
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435 %! retval(2) = 3*x + 2**y -z**3 + 1; |
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436 %! retval(3) = x + y + z - 5; |
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437 %!test |
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438 %! x_opt = [ 0.599054; |
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439 %! 2.395931; |
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440 %! 2.005014 ]; |
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441 %! tol = 1.0e-5; |
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442 %! [x, fval, info] = fsolve (@f, [ 0.5; 2.0; 2.5 ]); |
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443 %! assert (info > 0); |
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444 %! assert (norm (x - x_opt, Inf) < tol); |
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445 %! assert (norm (fval) < tol); |
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446 |
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447 %!function retval = f (p) |
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448 %! x = p(1); |
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449 %! y = p(2); |
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450 %! z = p(3); |
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451 %! w = p(4); |
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452 %! retval = zeros (4, 1); |
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453 %! retval(1) = 3*x + 4*y + exp (z + w) - 1.007; |
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454 %! retval(2) = 6*x - 4*y + exp (3*z + w) - 11; |
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455 %! retval(3) = x^4 - 4*y^2 + 6*z - 8*w - 20; |
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456 %! retval(4) = x^2 + 2*y^3 + z - w - 4; |
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457 %!test |
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458 %! x_opt = [ -0.767297326653401, 0.590671081117440, 1.47190018629642, -1.52719341133957 ]; |
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459 %! tol = 1.0e-5; |
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460 %! [x, fval, info] = fsolve (@f, [-1, 1, 2, -1]); |
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461 %! assert (info > 0); |
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462 %! assert (norm (x - x_opt, Inf) < tol); |
|
8306
43795cf108d0
initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
diff
changeset
|
463 %! assert (norm (fval) < tol); |
|
8383
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
464 |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
465 %!function retval = f (p) |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
466 %! x = p(1); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
467 %! y = p(2); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
468 %! z = p(3); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
469 %! retval = zeros (3, 1); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
470 %! retval(1) = sin(x) + y**2 + log(z) - 7; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
471 %! retval(2) = 3*x + 2**y -z**3 + 1; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
472 %! retval(3) = x + y + z - 5; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
473 %! 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
|
474 %!test |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
475 %! x_opt = [ 0.599054; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
476 %! 2.395931; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
477 %! 2.005014 ]; |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
478 %! tol = 1.0e-5; |
| 8590 | 479 %! [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
|
480 %! assert (info > 0); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
481 %! assert (norm (x - x_opt, Inf) < tol); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
482 %! assert (norm (fval) < tol); |
|
a762d9daa700
allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8306
diff
changeset
|
483 |
|
8395
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
484 %!function retval = f (p) |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
485 %! x = p(1); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
486 %! y = p(2); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
487 %! z = p(3); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
488 %! retval = zeros (3, 1); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
489 %! retval(1) = sin(x) + y**2 + log(z) - 7; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
490 %! retval(2) = 3*x + 2**y -z**3 + 1; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
491 %! retval(3) = x + y + z - 5; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
492 %!test |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
493 %! x_opt = [ 0.599054; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
494 %! 2.395931; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
495 %! 2.005014 ]; |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
496 %! tol = 1.0e-5; |
| 8590 | 497 %! opt = optimset ("Updating", "qrp"); |
|
8395
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
498 %! [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
|
499 %! assert (info > 0); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
500 %! assert (norm (x - x_opt, Inf) < tol); |
|
88fd356b0d95
optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
8383
diff
changeset
|
501 %! assert (norm (fval) < tol); |
| 8590 | 502 |
| 503 %!test | |
| 504 %! b0 = 3; | |
| 505 %! a0 = 0.2; | |
| 506 %! x = 0:.5:5; | |
| 507 %! noise = 1e-5 * sin (100*x); | |
| 508 %! y = exp (-a0*x) + b0 + noise; | |
| 509 %! c_opt = [a0, b0]; | |
| 510 %! tol = 1e-5; | |
| 511 %! | |
| 512 %! [c, fval, info, output] = fsolve (@(c) (exp(-c(1)*x) + c(2) - y), [0, 0]); | |
| 513 %! assert (info > 0); | |
| 514 %! assert (norm (c - c_opt, Inf) < tol); | |
| 515 %! assert (norm (fval) < norm (noise)); | |
| 516 | |
|
8693
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
517 |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
518 %!function y = cfun (x) |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
519 %! 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
|
520 %! 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
|
521 %! 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
|
522 |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
523 %!test |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
524 %! x_opt = [-1+i, 1-i, 2+i]; |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
525 %! x = [i, 1, 1+i]; |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
526 %! |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
527 %! [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
|
528 %! tol = 1e-5; |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
529 %! assert (norm (f) < tol); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
530 %! assert (norm (x - x_opt, Inf) < tol); |
|
e5ffb52c9c61
improve fsolve and add ComplexEqn option
Jaroslav Hajek <highegg@gmail.com>
parents:
8661
diff
changeset
|
531 |