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annotate scripts/optimization/fminbnd.m @ 31548:c8ad083a5802 stable

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maint: Clean up m-files before Octave 8.1 release. * external.txi, oop.txi, Table.h, documentation.cc, gui-preferences-ed.h, lo-specfun.cc, range.tst : Eliminate triple newlines. * Map.m, MemoizedFunction.m, delaunayn.m, inputParser.m, __publish_latex_output__.m, publish.m, unpack.m, fminbnd.m, __add_default_menu__.m, gammainc.m, gallery.m, hadamard.m, weboptions.m: Add newline after keyword "function" or before keyword "endfunction" for readability. * getaudiodata.m, pkg.m : Add semicolon to end of line for error() statement. * movegui.m: Combine mutliple calls to set() into one for performance. * __unimplemented__.m (missing_functions): Remove missing functions that have been implemented. * __vectorize__.m, check_default_input.m, betaincinv.m, gammaincinv.m: Remove semicolon at end of line with "function" declaration. * weboptions.m: Remove semicolon at end of line with "if" keyword. * integrate_adaptive.m, factor.m: Use keyword "endif" rather than bare "end".
author Rik <rik@octave.org>
date Fri, 25 Nov 2022 21:23:54 -0800
parents e1788b1a315f
children 597f3ee61a48
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 2008-2022 The Octave Project Developers
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4 ##
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5 ## See the file COPYRIGHT.md in the top-level directory of this
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6 ## distribution or <https://octave.org/copyright/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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11 ## under the terms of the GNU General Public License as published by
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12 ## the Free Software Foundation, either version 3 of the License, or
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13 ## (at your option) any later version.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 ## GNU General Public License for more details.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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21 ## along with Octave; see the file COPYING. If not, see
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22 ## <https://www.gnu.org/licenses/>.
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23 ##
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24 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {@var{x} =} fminbnd (@var{fcn}, @var{a}, @var{b})
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28 ## @deftypefnx {} {@var{x} =} fminbnd (@var{fcn}, @var{a}, @var{b}, @var{options})
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29 ## @deftypefnx {} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} fminbnd (@dots{})
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30 ## Find a minimum point of a univariate function.
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31 ##
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32 ## @var{fcn} is a function handle, inline function, or string containing the
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33 ## name of the function to evaluate.
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34 ##
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35 ## The starting interval is specified by @var{a} (left boundary) and @var{b}
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36 ## (right boundary). The endpoints must be finite.
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37 ##
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38 ## @var{options} is a structure specifying additional parameters which
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39 ## control the algorithm. Currently, @code{fminbnd} recognizes these options:
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40 ## @qcode{"Display"}, @qcode{"FunValCheck"}, @qcode{"MaxFunEvals"},
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41 ## @qcode{"MaxIter"}, @qcode{"OutputFcn"}, @qcode{"TolX"}.
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42 ##
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43 ## @qcode{"MaxFunEvals"} proscribes the maximum number of function evaluations
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44 ## before optimization is halted. The default value is 500.
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45 ## The value must be a positive integer.
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46 ##
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47 ## @qcode{"MaxIter"} proscribes the maximum number of algorithm iterations
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48 ## before optimization is halted. The default value is 500.
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49 ## The value must be a positive integer.
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50 ##
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51 ## @qcode{"TolX"} specifies the termination tolerance for the solution @var{x}.
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52 ## The default is @code{1e-4}.
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53 ##
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54 ## For a description of the other options,
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55 ## @pxref{XREFoptimset,,@code{optimset}}.
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56 ## To initialize an options structure with default values for @code{fminbnd}
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57 ## use @code{options = optimset ("fminbnd")}.
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58 ##
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59 ## On exit, the function returns @var{x}, the approximate minimum point, and
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60 ## @var{fval}, the function evaluated @var{x}.
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61 ##
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62 ## The third output @var{info} reports whether the algorithm succeeded and may
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63 ## take one of the following values:
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64 ##
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65 ## @itemize
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66 ## @item 1
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67 ## The algorithm converged to a solution.
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68 ##
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69 ## @item 0
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70 ## Iteration limit (either @code{MaxIter} or @code{MaxFunEvals}) exceeded.
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71 ##
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72 ## @item -1
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73 ## The algorithm was terminated by a user @code{OutputFcn}.
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74 ## @end itemize
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75 ##
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76 ## Programming Notes: The search for a minimum is restricted to be in the
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77 ## finite interval bound by @var{a} and @var{b}. If you have only one initial
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78 ## point to begin searching from then you will need to use an unconstrained
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79 ## minimization algorithm such as @code{fminunc} or @code{fminsearch}.
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80 ## @code{fminbnd} internally uses a Golden Section search strategy.
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81 ## @seealso{fzero, fminunc, fminsearch, optimset}
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82 ## @end deftypefn
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83
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84 ## This is patterned after opt/fmin.f from Netlib, which in turn is taken from
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85 ## Richard Brent: Algorithms For Minimization Without Derivatives,
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86 ## Prentice-Hall (1973)
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87
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88 ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup.
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89 ## PKG_ADD: [~] = __all_opts__ ("fminbnd");
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90
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91 function [x, fval, info, output] = fminbnd (fcn, a, b, options = struct ())
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92
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93 ## Get default options if requested.
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94 if (nargin == 1 && ischar (fcn) && strcmp (fcn, "defaults"))
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95 x = struct ("Display", "notify", "FunValCheck", "off",
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96 "MaxFunEvals", 500, "MaxIter", 500,
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97 "OutputFcn", [], "TolX", 1e-4);
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98 return;
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99 endif
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100
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101 if (nargin < 2)
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102 print_usage ();
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103 endif
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104
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105 if (a > b)
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106 error ("Octave:invalid-input-arg",
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107 "fminbnd: the lower bound cannot be greater than the upper one");
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108 endif
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109
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110 if (ischar (fcn))
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111 fcn = str2func (fcn);
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112 endif
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113
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114 displ = optimget (options, "Display", "notify");
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115 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on");
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116 outfcn = optimget (options, "OutputFcn");
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117 tolx = optimget (options, "TolX", 1e-4);
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118 maxiter = optimget (options, "MaxIter", 500);
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119 maxfev = optimget (options, "MaxFunEvals", 500);
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120
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121 if (funvalchk)
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122 ## Replace fcn with a guarded version.
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123 fcn = @(x) guarded_eval (fcn, x);
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124 endif
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125
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126 ## The default exit flag if exceeded number of iterations.
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127 info = 0;
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diff changeset
128 niter = 0;
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129 nfev = 0;
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diff changeset
130
14868
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diff changeset
131 c = 0.5*(3 - sqrt (5));
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diff changeset
132 v = a + c*(b-a);
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diff changeset
133 w = x = v;
035ac548a67e implement fminbnd
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parents:
diff changeset
134 e = 0;
30893
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Rik <rik@octave.org>
parents: 30564
diff changeset
135 fv = fw = fval = fcn (x);
20735
418ae0cb752f Replace ++,-- with in-place operators for performance.
Rik <rik@octave.org>
parents: 20727
diff changeset
136 nfev += 1;
10296
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137
22765
01aae08a0105 maint: Rename variables to match documentation in m-files.
Rik <rik@octave.org>
parents: 22755
diff changeset
138 if (isa (a, "single") || isa (b, "single") || isa (fval, "single"))
21099
52af4092f863 For optimization scripts, correctly choose tolerance (eps) based on class of fun and X0.
Rik <rik@octave.org>
parents: 20852
diff changeset
139 sqrteps = eps ("single");
52af4092f863 For optimization scripts, correctly choose tolerance (eps) based on class of fun and X0.
Rik <rik@octave.org>
parents: 20852
diff changeset
140 else
52af4092f863 For optimization scripts, correctly choose tolerance (eps) based on class of fun and X0.
Rik <rik@octave.org>
parents: 20852
diff changeset
141 sqrteps = eps ("double");
52af4092f863 For optimization scripts, correctly choose tolerance (eps) based on class of fun and X0.
Rik <rik@octave.org>
parents: 20852
diff changeset
142 endif
52af4092f863 For optimization scripts, correctly choose tolerance (eps) based on class of fun and X0.
Rik <rik@octave.org>
parents: 20852
diff changeset
143
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144 ## Only for display purposes.
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diff changeset
145 iter(1).funccount = nfev;
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Júlio Hoffimann <julio.hoffimann@gmail.com>
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diff changeset
146 iter(1).x = x;
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Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
147 iter(1).fx = fval;
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parents: 14895
diff changeset
148
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149 while (niter < maxiter && nfev < maxfev)
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150 xm = 0.5*(a+b);
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parents: 14552
diff changeset
151 ## FIXME: the golden section search can actually get closer than sqrt(eps)
5d3a684236b0 maint: Use Octave coding conventions for cuddling parentheses in scripts directory
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diff changeset
152 ## sometimes. Sometimes not, it depends on the function. This is the
21099
52af4092f863 For optimization scripts, correctly choose tolerance (eps) based on class of fun and X0.
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parents: 20852
diff changeset
153 ## strategy from the Netlib code. Something smarter would be good.
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154 tol = 2 * sqrteps * abs (x) + tolx / 3;
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parents:
diff changeset
155 if (abs (x - xm) <= (2*tol - 0.5*(b-a)))
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diff changeset
156 info = 1;
035ac548a67e implement fminbnd
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157 break;
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parents:
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158 endif
035ac548a67e implement fminbnd
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159
035ac548a67e implement fminbnd
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parents:
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160 if (abs (e) > tol)
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161 dogs = false;
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162 ## Try inverse parabolic step.
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diff changeset
163 iter(niter+1).procedure = "parabolic";
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164
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165 r = (x - w)*(fval - fv);
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166 q = (x - v)*(fval - fw);
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167 p = (x - v)*q - (x - w)*r;
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168 q = 2*(q - r);
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diff changeset
169 p *= -sign (q);
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170 q = abs (q);
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171 r = e;
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172 e = d;
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173
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diff changeset
174 if (abs (p) < abs (0.5*q*r) && p > q*(a-x) && p < q*(b-x))
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175 ## The parabolic step is acceptable.
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176 d = p / q;
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diff changeset
177 u = x + d;
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178
035ac548a67e implement fminbnd
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179 ## f must not be evaluated too close to ax or bx.
10392
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diff changeset
180 if (min (u-a, b-u) < 2*tol)
10296
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181 d = tol * (sign (xm - x) + (xm == x));
035ac548a67e implement fminbnd
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182 endif
035ac548a67e implement fminbnd
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183 else
035ac548a67e implement fminbnd
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diff changeset
184 dogs = true;
035ac548a67e implement fminbnd
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parents:
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185 endif
035ac548a67e implement fminbnd
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parents:
diff changeset
186 else
035ac548a67e implement fminbnd
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parents:
diff changeset
187 dogs = true;
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188 endif
035ac548a67e implement fminbnd
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parents:
diff changeset
189 if (dogs)
035ac548a67e implement fminbnd
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190 ## Default to golden section step.
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diff changeset
191
25740
b1e0e58971f9 fminbnd.m: Update solver options to be Matlab compatible.
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diff changeset
192 ## WARNING: This is also the "initial" procedure following MATLAB
b1e0e58971f9 fminbnd.m: Update solver options to be Matlab compatible.
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193 ## nomenclature. After the loop we'll fix the string for the first step.
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diff changeset
194 iter(niter+1).procedure = "golden";
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diff changeset
195
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196 e = ifelse (x >= xm, a - x, b - x);
035ac548a67e implement fminbnd
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diff changeset
197 d = c * e;
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198 endif
035ac548a67e implement fminbnd
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parents:
diff changeset
199
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diff changeset
200 ## f must not be evaluated too close to x.
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diff changeset
201 u = x + max (abs (d), tol) * (sign (d) + (d == 0));
30893
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diff changeset
202 fu = fcn (u);
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diff changeset
203
20735
418ae0cb752f Replace ++,-- with in-place operators for performance.
Rik <rik@octave.org>
parents: 20727
diff changeset
204 niter += 1;
11587
c792872f8942 all script files: untabify and strip trailing whitespace
John W. Eaton <jwe@octave.org>
parents: 11523
diff changeset
205
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diff changeset
206 iter(niter).funccount = nfev++;
242e9efd4315 Added Display option for fminbnd()
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diff changeset
207 iter(niter).x = u;
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
208 iter(niter).fx = fu;
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parents:
diff changeset
209
21562
6c2fd62db1f7 maint: Eliminate accidental double spaces in code.
Rik <rik@octave.org>
parents: 21099
diff changeset
210 ## update a, b, v, w, and x
10296
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parents:
diff changeset
211
18290
61dab64aa5a6 fminbnd.m: Correctly handle certain functions with odd discontinuities (bug #36923)
Rik <rik@octave.org>
parents: 17744
diff changeset
212 if (fu < fval)
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diff changeset
213 if (u < x)
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Júlio Hoffimann <julio.hoffimann@gmail.com>
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diff changeset
214 b = x;
242e9efd4315 Added Display option for fminbnd()
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diff changeset
215 else
242e9efd4315 Added Display option for fminbnd()
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diff changeset
216 a = x;
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diff changeset
217 endif
242e9efd4315 Added Display option for fminbnd()
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diff changeset
218 v = w; fv = fw;
242e9efd4315 Added Display option for fminbnd()
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diff changeset
219 w = x; fw = fval;
242e9efd4315 Added Display option for fminbnd()
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diff changeset
220 x = u; fval = fu;
242e9efd4315 Added Display option for fminbnd()
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diff changeset
221 else
242e9efd4315 Added Display option for fminbnd()
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diff changeset
222 ## The following if-statement was originally executed even if fu == fval.
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
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diff changeset
223 if (u < x)
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
224 a = u;
242e9efd4315 Added Display option for fminbnd()
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parents: 14895
diff changeset
225 else
242e9efd4315 Added Display option for fminbnd()
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parents: 14895
diff changeset
226 b = u;
242e9efd4315 Added Display option for fminbnd()
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parents: 14895
diff changeset
227 endif
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
228 if (fu <= fw || w == x)
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
229 v = w; fv = fw;
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
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diff changeset
230 w = u; fw = fu;
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
231 elseif (fu <= fv || v == x || v == w)
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
232 v = u;
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
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diff changeset
233 fv = fu;
242e9efd4315 Added Display option for fminbnd()
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diff changeset
234 endif
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
235 endif
10296
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diff changeset
236
035ac548a67e implement fminbnd
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parents:
diff changeset
237 ## If there's an output function, use it now.
29302
a121ce5a515f fminbnd.m: do not ignore "OutputFcn" (bug #59901)
Kai T. Ohlhus <k.ohlhus@gmail.com>
parents: 27978
diff changeset
238 if (! isempty (outfcn))
10296
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diff changeset
239 optv.funccount = nfev;
035ac548a67e implement fminbnd
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parents:
diff changeset
240 optv.fval = fval;
035ac548a67e implement fminbnd
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parents:
diff changeset
241 optv.iteration = niter;
035ac548a67e implement fminbnd
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parents:
diff changeset
242 if (outfcn (x, optv, "iter"))
035ac548a67e implement fminbnd
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parents:
diff changeset
243 info = -1;
035ac548a67e implement fminbnd
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parents:
diff changeset
244 break;
035ac548a67e implement fminbnd
Jaroslav Hajek <highegg@gmail.com>
parents:
diff changeset
245 endif
035ac548a67e implement fminbnd
Jaroslav Hajek <highegg@gmail.com>
parents:
diff changeset
246 endif
035ac548a67e implement fminbnd
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parents:
diff changeset
247 endwhile
035ac548a67e implement fminbnd
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parents:
diff changeset
248
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diff changeset
249 ## Fix the first step procedure.
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diff changeset
250 iter(1).procedure = "initial";
242e9efd4315 Added Display option for fminbnd()
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diff changeset
251
242e9efd4315 Added Display option for fminbnd()
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diff changeset
252 ## Handle the "Display" option
16933
e39f00a32dc7 maint: Use parentheses around condition for switch(),while(),if() statements.
Rik <rik@octave.org>
parents: 16772
diff changeset
253 switch (displ)
15706
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diff changeset
254 case "iter"
242e9efd4315 Added Display option for fminbnd()
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diff changeset
255 print_formatted_table (iter);
25740
b1e0e58971f9 fminbnd.m: Update solver options to be Matlab compatible.
Rik <rik@octave.org>
parents: 25054
diff changeset
256 print_exit_msg (info, struct ("TolX", tolx, "fx", fval));
15706
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diff changeset
257 case "notify"
242e9efd4315 Added Display option for fminbnd()
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diff changeset
258 if (info == 0)
25740
b1e0e58971f9 fminbnd.m: Update solver options to be Matlab compatible.
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diff changeset
259 print_exit_msg (info, struct ("fx",fval));
15706
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diff changeset
260 endif
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diff changeset
261 case "final"
25740
b1e0e58971f9 fminbnd.m: Update solver options to be Matlab compatible.
Rik <rik@octave.org>
parents: 25054
diff changeset
262 print_exit_msg (info, struct ("TolX", tolx, "fx", fval));
15706
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diff changeset
263 case "off"
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
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diff changeset
264 "skip";
242e9efd4315 Added Display option for fminbnd()
Júlio Hoffimann <julio.hoffimann@gmail.com>
parents: 14895
diff changeset
265 otherwise
20727
a5949b3d2332 Preface warning() messages with name of function when possible.
Rik <rik@octave.org>
parents: 20726
diff changeset
266 warning ("fminbnd: unknown option for Display: '%s'", displ);
15706
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diff changeset
267 endswitch
242e9efd4315 Added Display option for fminbnd()
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diff changeset
268
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parents:
diff changeset
269 output.iterations = niter;
035ac548a67e implement fminbnd
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parents:
diff changeset
270 output.funcCount = nfev;
25740
b1e0e58971f9 fminbnd.m: Update solver options to be Matlab compatible.
Rik <rik@octave.org>
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271 output.algorithm = "golden section search, parabolic interpolation";
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272 output.bracket = [a, b];
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273 ## FIXME: bracketf possibly unavailable.
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274
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275 endfunction
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276
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277 ## A helper function that evaluates a function and checks for bad results.
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278 function fx = guarded_eval (fcn, x)
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279
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280 fx = fcn (x);
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281 fx = fx(1);
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282 if (! isreal (fx))
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283 error ("Octave:fmindbnd:notreal", "fminbnd: non-real value encountered");
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284 elseif (isnan (fx))
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285 error ("Octave:fmindbnd:isnan", "fminbnd: NaN value encountered");
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286 endif
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287
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288 endfunction
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289
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290 ## A hack for printing a formatted table
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291 function print_formatted_table (table)
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292
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293 printf ("\n Fcn-count x f(x) Procedure\n");
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294 for row=table
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295 printf ("%5.5s %7.7s %8.8s\t%s\n",
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296 int2str (row.funccount), num2str (row.x,"%.5f"),
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297 num2str (row.fx,"%.6f"), row.procedure);
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298 endfor
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299 printf ("\n");
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300
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301 endfunction
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302
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303 ## Print either a success termination message or bad news
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304 function print_exit_msg (info, opt=struct ())
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305
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306 printf ("");
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307 switch (info)
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308 case 1
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309 printf ("Optimization terminated:\n");
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310 printf (" the current x satisfies the termination criteria using OPTIONS.TolX of %e\n", opt.TolX);
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311 case 0
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312 printf ("Exiting: Maximum number of iterations has been exceeded\n");
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313 printf (" - increase MaxIter option.\n");
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314 printf (" Current function value: %.6f\n", opt.fx);
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315 case -1
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316 "FIXME"; # FIXME: what's the message MATLAB prints for this case?
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317 otherwise
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318 error ("fminbnd: internal error, info return code was %d", info);
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319 endswitch
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320 printf ("\n");
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321
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322 endfunction
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323
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324
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325 %!shared opt0
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326 %! opt0 = optimset ("tolx", 0);
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327 %!assert (fminbnd (@cos, pi/2, 3*pi/2, opt0), pi, 10*sqrt (eps))
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328 %!assert (fminbnd (@(x) (x - 1e-3)^4, -1, 1, opt0), 1e-3, 10e-3*sqrt (eps))
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329 %!assert (fminbnd (@(x) abs (x-1e7), 0, 1e10, opt0), 1e7, 10e7*sqrt (eps))
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330 %!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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331 %!assert (fminbnd (@(x) x > 0.3, 0, 1) < 0.3)
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332 %!assert (fminbnd (@(x) sin (x), 0, 0), 0, eps)
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333
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diff changeset
334 %!error <lower bound cannot be greater> fminbnd (@(x) sin (x), 0, -pi)