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optionally allow pivoting in fsolve
author Jaroslav Hajek <highegg@gmail.com>
date Thu, 11 Dec 2008 07:51:40 +0100
parents a762d9daa700
children 4d008d9f0ccf
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1 ## Copyright (C) 2008 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
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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{})
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24 # Solves a system of nonlinear equations defined by the function @var{fcn}.
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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
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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
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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. Currently, fsolve
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33 # recognizes these options: FunValCheck, OutputFcn, TolX, TolFun, MaxIter,
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34 # MaxFunEvals and Jacobian.
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35 #
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36 # If Jacobian is 'on', it specifies that @var{fcn}, called with 2 output arguments,
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37 # also returns the Jacobian matrix of right-hand sides at the requested point.
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38 # TolX specifies the termination tolerance in the unknown variables, while
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39 # TolFun is a tolerance for equations. Default is @code{1e1*eps}
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40 # for TolX and @code{1e2*eps} for TolFun.
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41 # For description of the other options, see @code{optimset}.
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42 #
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43 # On return, @var{fval} contains the value of the function @var{fcn}
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44 # evaluated at @var{x}, and @var{info} may be one of the following values:
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45 #
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46 # @table @asis
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47 # @item 1
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48 # Converged to a solution point. Relative residual error is less than specified
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49 # by TolFun.
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50 # @item 2
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51 # Last relative step size was less that TolX.
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52 # @item 3
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53 # Last relative decrease in residual was less than TolF.
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54 # @item 0
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55 # Iteration limit exceeded.
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56 # @item -3
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57 # The trust region radius became excessively small.
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58 # @end table
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59 #
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60 # Note: If you only have a single nonlinear equation of one variable, using
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61 # @code{fzero} is usually a much better idea.
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62 # @seealso{fzero,optimset}
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63 # @end deftypefn
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64
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65 function [x, fvec, info, output, fjac] = fsolve (fcn, x0, options)
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66
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67 if (nargin < 3)
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68 options = struct ();
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69 endif
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70
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71 xsiz = size (x0);
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72 n = numel (x0);
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73
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74 has_jac = strcmp (optimget (options, "Jacobian", "off"), "on");
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75 maxiter = optimget (options, "MaxIter", Inf);
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76 maxfev = optimget (options, "MaxFunEvals", Inf);
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77 outfcn = optimget (options, "OutputFcn");
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78 pivoting = optimget (options, "Pivoting", false);
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79 funvalchk = strcmp (optimget (options, "FunValCheck", "off"), "on");
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80
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81 if (funvalchk)
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82 # replace fun with a guarded version
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83 fun = @(x) guarded_eval (fun, x);
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84 endif
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85
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86 # These defaults are rather stringent. I think that normally, user prefers
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87 # accuracy to performance.
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88
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89 macheps = eps (class (x0));
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90
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91 tolx = optimget (options, "TolX", 1e1*macheps);
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92 tolf = optimget (options, "TolFun",1e2*macheps);
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93
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94 factor = 100;
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95 # FIXME: TypicalX corresponds to user scaling (???)
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96 autodg = true;
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97
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98 niter = 1; nfev = 0;
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99
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100 x = x0(:);
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101 info = 0;
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102
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103 # outer loop
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104 while (niter < maxiter && nfev < maxfev && ! info)
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105
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106 # calc func value and jacobian (possibly via FD)
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107 # handle arbitrary shapes of x and f and remember them
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108 if (has_jac)
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109 [fvec, fjac] = fcn (reshape (x, xsiz));
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110 nfev ++;
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111 else
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112 [fvec, fjac] = __fdjac__ (fcn, reshape (x, xsiz));
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113 nfev += 1 + length (x);
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114 endif
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115 fsiz = size (fvec);
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116 fvec = fvec(:);
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117 fn = norm (fvec);
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118 m = length (fvec);
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119 n = length (x);
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120 if (m < n)
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121 error ("fsolve:under", "cannot solve underdetermined systems");
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122 elseif (m > n && niter == 1)
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123 if (isempty (optimget (options, "TolFun")))
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124 warning (["An overdetermined system cannot usually be solved exactly. ", ...
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125 "Consider specifying the TolFun option."]);
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126 endif
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127 endif
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128
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129 # get QR factorization of the jacobian
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130 if (pivoting)
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131 [q, r, p] = qr (fjac);
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132 else
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133 [q, r] = qr (fjac);
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134 endif
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135
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136 # get column norms, use them as scaling factor
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137 jcn = norm (fjac, 'columns').';
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138 if (niter == 1)
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139 if (autodg)
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140 dg = jcn;
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141 dg(dg == 0) = 1;
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142 endif
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143 xn = norm (dg .* x);
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144 delta = factor * xn;
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145 endif
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146
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147 # rescale if necessary
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148 if (autodg)
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149 dg = max (dg, jcn);
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150 endif
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151
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152 nfail = 0;
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153 nsuc = 0;
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154 # inner loop
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155 while (niter <= maxiter && nfev < maxfev && ! info)
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156
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157 qtf = q'*fvec;
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158
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159 # get TR model (dogleg) minimizer
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160 # in case of an overdetermined system, get lsq solution
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161 s = - __dogleg__ (r(1:n,:), qtf(1:n), dg, delta);
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162 if (pivoting)
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163 s = p * s;
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164 endif
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165 sn = norm (dg .* s);
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166
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167 if (niter == 1)
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168 delta = min (delta, sn);
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169 endif
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170
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171 fvec1 = fcn (reshape (x + s, xsiz)) (:);
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172 fn1 = norm (fvec1);
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173
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174 if (fn1 < fn)
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175 # scaled actual reduction
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176 actred = 1 - (fn1/fn)^2;
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177 else
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178 actred = -1;
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179 endif
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180
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181 # scaled predicted reduction, and ratio
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182 if (pivoting)
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183 w = qtf + r*(p'*s);
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184 else
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185 w = qtf + r*s;
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186 endif
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187 t = norm (w);
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188 if (t < fn)
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189 prered = 1 - (t/fn)^2;
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190 ratio = actred / prered;
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191 else
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192 prered = 0;
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193 ratio = 0;
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194 endif
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195
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196 # update delta
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197 if (ratio < 0.1)
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198 nsuc = 0;
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199 nfail ++;
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200 delta *= 0.5;
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201 if (delta <= sqrt (macheps)*xn)
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202 # trust region became uselessly small
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203 info = -3;
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204 break;
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205 endif
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206 else
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207 nfail = 0;
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208 nsuc ++;
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209 if (abs (1-ratio) <= 0.1)
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210 delta = 2*sn;
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211 elseif (ratio >= 0.5 || nsuc > 1)
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212 delta = max (delta, 2*sn);
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213 endif
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214 endif
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215
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216 if (ratio >= 1e-4)
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217 # successful iteration
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218 x += s;
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219 xn = norm (dg .* x);
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220 fvec = fvec1;
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221 fn = fn1;
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222 niter ++;
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223 endif
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224
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225 # Tests for termination conditions. A mysterious place, anything can
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226 # happen if you change something here...
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227
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228 # The rule of thumb (which I'm not sure M*b is quite following) is that
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229 # for a tolerance that depends on scaling, only 0 makes sense as a
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230 # default value. But 0 usually means uselessly long iterations,
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231 # so we need scaling-independent tolerances wherever possible.
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232
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233 # XXX: why tolf*n*xn? If abs (e) ~ abs(x) * eps is a vector of
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234 # perturbations of x, then norm (fjac*e) <= eps*n*xn, i.e. by tolf ~
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235 # eps we demand as much accuracy as we can expect.
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236 if (fn <= tolf*n*xn)
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237 info = 1;
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238 # The following tests done only after successful step.
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239 elseif (actred > 0)
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240 # This one is classic. Note that we use scaled variables again, but
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241 # compare to scaled step, so nothing bad.
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242 if (sn <= tolx*xn)
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243 info = 2;
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244 # Again a classic one. It seems weird to use the same tolf for two
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245 # different tests, but that's what M*b manual appears to say.
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246 elseif (actred < tolf)
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247 info = 3;
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248 endif
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249 endif
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250
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251 # criterion for recalculating jacobian
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252 if (nfail == 2)
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253 break;
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254 endif
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255
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256 # compute the scaled Broyden update
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257 u = (fvec1 - q*w) / sn;
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258 v = dg .* ((dg .* s) / sn);
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259 if (pivoting)
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260 v = p'*v;
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261 endif
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262
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263 # update the QR factorization
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264 [q, r] = qrupdate (q, r, u, v);
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265
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266 endwhile
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267 endwhile
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268
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269 # restore original shapes
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270 x = reshape (x, xsiz);
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271 fvec = reshape (fvec, fsiz);
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272
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273 output.iterations = niter;
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274 output.funcCount = niter + 2;
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275
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276 endfunction
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277
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278 # an assistant function that evaluates a function handle and checks for bad
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279 # results.
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280 function fx = guarded_eval (fun, x)
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281 fx = fun (x);
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282 if (! all (isreal (fx)))
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283 error ("fsolve:notreal", "fsolve: non-real value encountered");
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284 elseif (any (isnan (fx)))
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285 error ("fsolve:isnan", "fsolve: NaN value encountered");
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286 endif
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287 endfunction
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288
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289 %!function retval = f (p)
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290 %! x = p(1);
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291 %! y = p(2);
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292 %! z = p(3);
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293 %! retval = zeros (3, 1);
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294 %! retval(1) = sin(x) + y**2 + log(z) - 7;
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295 %! retval(2) = 3*x + 2**y -z**3 + 1;
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296 %! retval(3) = x + y + z - 5;
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297 %!test
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298 %! x_opt = [ 0.599054;
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299 %! 2.395931;
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300 %! 2.005014 ];
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301 %! tol = 1.0e-5;
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302 %! [x, fval, info] = fsolve (@f, [ 0.5; 2.0; 2.5 ]);
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303 %! assert (info > 0);
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304 %! assert (norm (x - x_opt, Inf) < tol);
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305 %! assert (norm (fval) < tol);
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306
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307 %!function retval = f (p)
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308 %! x = p(1);
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309 %! y = p(2);
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310 %! z = p(3);
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311 %! w = p(4);
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312 %! retval = zeros (4, 1);
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313 %! retval(1) = 3*x + 4*y + exp (z + w) - 1.007;
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314 %! retval(2) = 6*x - 4*y + exp (3*z + w) - 11;
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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315 %! retval(3) = x^4 - 4*y^2 + 6*z - 8*w - 20;
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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316 %! retval(4) = x^2 + 2*y^3 + z - w - 4;
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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317 %!test
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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318 %! x_opt = [ -0.767297326653401, 0.590671081117440, 1.47190018629642, -1.52719341133957 ];
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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319 %! tol = 1.0e-5;
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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320 %! [x, fval, info] = fsolve (@f, [-1, 1, 2, -1]);
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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321 %! assert (info > 0);
8383
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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322 %! assert (norm (x - x_opt, Inf) < tol);
8306
43795cf108d0 initial implementation of fsolve
Jaroslav Hajek <highegg@gmail.com>
parents:
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323 %! assert (norm (fval) < tol);
8383
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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324
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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325 %!function retval = f (p)
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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326 %! x = p(1);
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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327 %! y = p(2);
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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328 %! z = p(3);
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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329 %! retval = zeros (3, 1);
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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330 %! retval(1) = sin(x) + y**2 + log(z) - 7;
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
331 %! retval(2) = 3*x + 2**y -z**3 + 1;
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
332 %! retval(3) = x + y + z - 5;
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
333 %! 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
334 %!test
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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335 %! x_opt = [ 0.599054;
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
336 %! 2.395931;
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
337 %! 2.005014 ];
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
338 %! tol = 1.0e-5;
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
339 %! [x, fval, info] = fsolve (@f, [ 0.5; 2.0; 2.5 ], optimset ("TolFun", 1e-6));
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
340 %! assert (info > 0);
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
341 %! assert (norm (x - x_opt, Inf) < tol);
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
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342 %! assert (norm (fval) < tol);
a762d9daa700 allow overdetermined systems in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8306
diff changeset
343
8395
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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344 %!function retval = f (p)
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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345 %! x = p(1);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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346 %! y = p(2);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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347 %! z = p(3);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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348 %! retval = zeros (3, 1);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
349 %! retval(1) = sin(x) + y**2 + log(z) - 7;
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
350 %! retval(2) = 3*x + 2**y -z**3 + 1;
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
351 %! retval(3) = x + y + z - 5;
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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352 %!test
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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353 %! x_opt = [ 0.599054;
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
354 %! 2.395931;
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
355 %! 2.005014 ];
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
356 %! tol = 1.0e-5;
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
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357 %! opt = optimset ("Pivoting", true);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
358 %! [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
359 %! assert (info > 0);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
360 %! assert (norm (x - x_opt, Inf) < tol);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
361 %! assert (norm (fval) < tol);
88fd356b0d95 optionally allow pivoting in fsolve
Jaroslav Hajek <highegg@gmail.com>
parents: 8383
diff changeset
362