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annotate scripts/optimization/fminbnd.m @ 21099:52af4092f863

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For optimization scripts, correctly choose tolerance (eps) based on class of fun and X0. * fminbnd.m, fminsearch.m, fminunc.m, fsolve.m: Use 'isa (..., "single")' to check both initial guess X0 and result of function evaluation to decide whether eps should be single or double.
author Rik <rik@octave.org>
date Mon, 18 Jan 2016 10:00:56 -0800
parents 516bb87ea72e
children 6c2fd62db1f7
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
19697
4197fc428c7d maint: Update copyright notices for 2015.
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1 ## Copyright (C) 2008-2015 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 {} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} fminbnd (@var{fun}, @var{a}, @var{b}, @var{options})
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23 ## Find a minimum point of a univariate function.
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24 ##
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25 ## @var{fun} should be a function handle or name. @var{a}, @var{b} specify a
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26 ## starting interval. @var{options} is a structure specifying additional
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27 ## options. Currently, @code{fminbnd} recognizes these options:
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28 ## @qcode{"FunValCheck"}, @qcode{"OutputFcn"}, @qcode{"TolX"},
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29 ## @qcode{"MaxIter"}, @qcode{"MaxFunEvals"}. For a description of these
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30 ## options, see @ref{XREFoptimset,,optimset}.
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31 ##
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32 ## On exit, the function returns @var{x}, the approximate minimum point and
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33 ## @var{fval}, the function value thereof.
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34 ##
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35 ## @var{info} is an exit flag that can have these values:
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36 ##
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37 ## @itemize
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38 ## @item 1
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39 ## The algorithm converged to a solution.
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40 ##
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41 ## @item 0
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42 ## Maximum number of iterations or function evaluations has been exhausted.
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43 ##
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44 ## @item -1
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45 ## The algorithm has been terminated from user output function.
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46 ## @end itemize
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47 ##
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48 ## Notes: The search for a minimum is restricted to be in the interval bound by
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49 ## @var{a} and @var{b}. If you only have an initial point to begin searching
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50 ## from you will need to use an unconstrained minimization algorithm such as
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51 ## @code{fminunc} or @code{fminsearch}. @code{fminbnd} internally uses a
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52 ## Golden Section search strategy.
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53 ## @seealso{fzero, fminunc, fminsearch, optimset}
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54 ## @end deftypefn
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55
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56 ## This is patterned after opt/fmin.f from Netlib, which in turn is taken from
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57 ## Richard Brent: Algorithms For Minimization Without Derivatives,
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58 ## Prentice-Hall (1973)
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59
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60 ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup.
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61 ## PKG_ADD: [~] = __all_opts__ ("fminbnd");
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62
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63 function [x, fval, info, output] = fminbnd (fun, xmin, xmax, options = struct ())
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64
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65 ## Get default options if requested.
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66 if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults'))
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67 x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 1e-8,
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68 "OutputFcn", [], "FunValCheck", "off");
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69 return;
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70 endif
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71
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72 if (nargin < 2 || nargin > 4)
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73 print_usage ();
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74 endif
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75
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76 if (xmin > xmax)
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77 error ("Octave:invalid-input-arg",
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78 "fminbnd: the lower bound cannot be greater than the upper one");
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79 endif
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80
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81 if (ischar (fun))
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82 fun = str2func (fun, "global");
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83 endif
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84
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85 displ = optimget (options, "Display", "notify");
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86 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on");
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87 outfcn = optimget (options, "OutputFcn");
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88 tolx = optimget (options, "TolX", 1e-8);
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89 maxiter = optimget (options, "MaxIter", Inf);
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90 maxfev = optimget (options, "MaxFunEvals", Inf);
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91
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92 if (funvalchk)
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93 ## Replace fun with a guarded version.
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94 fun = @(x) guarded_eval (fun, x);
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95 endif
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96
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97 ## The default exit flag if exceeded number of iterations.
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98 info = 0;
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99 niter = 0;
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100 nfev = 0;
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101
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102 c = 0.5*(3 - sqrt (5));
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103 a = xmin; b = xmax;
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104 v = a + c*(b-a);
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105 w = x = v;
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106 e = 0;
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107 fv = fw = fval = fun (x);
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108 nfev += 1;
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109
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110 if (isa (xmin, "single") || isa (xmax, "single") || isa (fval, "single"))
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111 sqrteps = eps ("single");
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112 else
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113 sqrteps = eps ("double");
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114 endif
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115
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116 ## Only for display purposes.
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117 iter(1).funccount = nfev;
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118 iter(1).x = x;
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119 iter(1).fx = fval;
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120
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121 while (niter < maxiter && nfev < maxfev)
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122 xm = 0.5*(a+b);
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123 ## FIXME: the golden section search can actually get closer than sqrt(eps)
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124 ## sometimes. Sometimes not, it depends on the function. This is the
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125 ## strategy from the Netlib code. Something smarter would be good.
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126 tol = 2 * sqrteps * abs (x) + tolx / 3;
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127 if (abs (x - xm) <= (2*tol - 0.5*(b-a)))
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128 info = 1;
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129 break;
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130 endif
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131
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132 if (abs (e) > tol)
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133 dogs = false;
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134 ## Try inverse parabolic step.
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135 iter(niter+1).procedure = "parabolic";
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136
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137 r = (x - w)*(fval - fv);
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138 q = (x - v)*(fval - fw);
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139 p = (x - v)*q - (x - w)*r;
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140 q = 2*(q - r);
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141 p *= -sign (q);
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142 q = abs (q);
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143 r = e;
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144 e = d;
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145
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146 if (abs (p) < abs (0.5*q*r) && p > q*(a-x) && p < q*(b-x))
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147 ## The parabolic step is acceptable.
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148 d = p / q;
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149 u = x + d;
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150
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151 ## f must not be evaluated too close to ax or bx.
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152 if (min (u-a, b-u) < 2*tol)
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153 d = tol * (sign (xm - x) + (xm == x));
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154 endif
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155 else
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156 dogs = true;
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157 endif
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158 else
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159 dogs = true;
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160 endif
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161 if (dogs)
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162 ## Default to golden section step.
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163
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164 ## WARNING: This is also the "initial" procedure following
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165 ## MATLAB nomenclature. After the loop we'll fix the string
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166 ## for the first step.
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167 iter(niter+1).procedure = "golden";
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168
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169 e = ifelse (x >= xm, a - x, b - x);
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170 d = c * e;
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171 endif
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172
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173 ## f must not be evaluated too close to x.
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174 u = x + max (abs (d), tol) * (sign (d) + (d == 0));
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175 fu = fun (u);
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176
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418ae0cb752f Replace ++,-- with in-place operators for performance.
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177 niter += 1;
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178
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179 iter(niter).funccount = nfev++;
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180 iter(niter).x = u;
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181 iter(niter).fx = fu;
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182
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183 ## update a, b, v, w, and x
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184
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61dab64aa5a6 fminbnd.m: Correctly handle certain functions with odd discontinuities (bug #36923)
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185 if (fu < fval)
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186 if (u < x)
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187 b = x;
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188 else
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189 a = x;
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190 endif
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191 v = w; fv = fw;
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192 w = x; fw = fval;
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193 x = u; fval = fu;
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194 else
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195 ## The following if-statement was originally executed even if fu == fval.
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196 if (u < x)
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197 a = u;
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198 else
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199 b = u;
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200 endif
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201 if (fu <= fw || w == x)
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202 v = w; fv = fw;
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203 w = u; fw = fu;
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204 elseif (fu <= fv || v == x || v == w)
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205 v = u;
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206 fv = fu;
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207 endif
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208 endif
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209
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210 ## If there's an output function, use it now.
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211 if (outfcn)
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212 optv.funccount = nfev;
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213 optv.fval = fval;
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214 optv.iteration = niter;
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215 if (outfcn (x, optv, "iter"))
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216 info = -1;
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217 break;
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218 endif
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219 endif
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220 endwhile
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221
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222 ## Fix the first step procedure.
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223 iter(1).procedure = "initial";
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224
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225 ## Handle the "Display" option
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226 switch (displ)
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227 case "iter"
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228 print_formatted_table (iter);
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229 print_exit_msg (info, struct("TolX", tolx, "fx", fval));
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230 case "notify"
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231 if (info == 0)
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232 print_exit_msg (info, struct("fx",fval));
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233 endif
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234 case "final"
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235 print_exit_msg (info, struct("TolX", tolx, "fx", fval));
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236 case "off"
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237 "skip";
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238 otherwise
20727
a5949b3d2332 Preface warning() messages with name of function when possible.
Rik <rik@octave.org>
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diff changeset
239 warning ("fminbnd: unknown option for Display: '%s'", displ);
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240 endswitch
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241
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242 output.iterations = niter;
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243 output.funcCount = nfev;
035ac548a67e implement fminbnd
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244 output.bracket = [a, b];
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diff changeset
245 ## FIXME: bracketf possibly unavailable.
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246
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247 endfunction
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248
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249 ## An assistant function that evaluates a function handle and checks for
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250 ## bad results.
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251 function fx = guarded_eval (fun, x)
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252 fx = fun (x);
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253 fx = fx(1);
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diff changeset
254 if (! isreal (fx))
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diff changeset
255 error ("fminbnd:notreal", "fminbnd: non-real value encountered");
10296
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diff changeset
256 elseif (isnan (fx))
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diff changeset
257 error ("fminbnd:isnan", "fminbnd: NaN value encountered");
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258 endif
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259 endfunction
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260
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261 ## A hack for printing a formatted table
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262 function print_formatted_table (table)
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263 printf ("\n Func-count x f(x) Procedure\n");
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264 for row=table
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265 printf("%5.5s %7.7s %8.8s\t%s\n",
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diff changeset
266 int2str (row.funccount), num2str (row.x,"%.5f"),
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diff changeset
267 num2str (row.fx,"%.6f"), row.procedure);
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268 endfor
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269 printf ("\n");
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270 endfunction
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271
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272 ## Print either a success termination message or bad news
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273 function print_exit_msg (info, opt=struct())
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diff changeset
274 printf ("");
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diff changeset
275 switch (info)
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276 case 1
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277 printf ("Optimization terminated:\n");
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278 printf (" the current x satisfies the termination criteria using OPTIONS.TolX of %e\n", opt.TolX);
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279 case 0
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280 printf ("Exiting: Maximum number of iterations has been exceeded\n");
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281 printf (" - increase MaxIter option.\n");
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282 printf (" Current function value: %.6f\n", opt.fx);
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diff changeset
283 case -1
17336
b81b9d079515 Use '##' for comments which stand alone on a line.
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parents: 17281
diff changeset
284 "FIXME"; # FIXME: what's the message MATLAB prints for this case?
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Júlio Hoffimann <julio.hoffimann@gmail.com>
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285 otherwise
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Rik <rik@octave.org>
parents: 20165
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286 error ("fminbnd: internal error, info return code was %d", info);
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287 endswitch
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
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288 printf ("\n");
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
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289 endfunction
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
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290
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Rik <octave@nomad.inbox5.com>
parents: 14138
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291
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Jaroslav Hajek <highegg@gmail.com>
parents:
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292 %!shared opt0
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Jaroslav Hajek <highegg@gmail.com>
parents:
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293 %! opt0 = optimset ("tolx", 0);
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Rik <octave@nomad.inbox5.com>
parents: 14138
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294 %!assert (fminbnd (@cos, pi/2, 3*pi/2, opt0), pi, 10*sqrt (eps))
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Rik <octave@nomad.inbox5.com>
parents: 14138
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295 %!assert (fminbnd (@(x) (x - 1e-3)^4, -1, 1, opt0), 1e-3, 10e-3*sqrt (eps))
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Rik <octave@nomad.inbox5.com>
parents: 14138
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296 %!assert (fminbnd (@(x) abs (x-1e7), 0, 1e10, opt0), 1e7, 10e7*sqrt (eps))
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Rik <octave@nomad.inbox5.com>
parents: 14138
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297 %!assert (fminbnd (@(x) x^2 + sin (2*pi*x), 0.4, 1, opt0), fzero (@(x) 2*x + 2*pi*cos (2*pi*x), [0.4, 1], opt0), sqrt (eps))
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John W. Eaton <jwe@octave.org>
parents: 19443
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298 %!assert (fminbnd (@(x) x > 0.3, 0, 1) < 0.3)
db92e7e28e1f strip trailing whitespace from most source files
John W. Eaton <jwe@octave.org>
parents: 19443
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299 %!assert (fminbnd (@(x) sin (x), 0, 0), 0, eps)
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Rik <octave@nomad.inbox5.com>
parents: 14138
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300
19443
0f79fa9b3a8c fmindbnd.m: Check input range is low to high (bug #43219).
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 18290
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301 %!error <lower bound cannot be greater> fminbnd (@(x) sin (x), 0, -pi)
0f79fa9b3a8c fmindbnd.m: Check input range is low to high (bug #43219).
Massimiliano Fasi <massimiliano.fasi@gmail.com>
parents: 18290
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302