Mercurial > hg > octave-lyh
annotate scripts/optimization/fminbnd.m @ 17521:c3a3532e3d98
linsolve.m: Add new function for Matlab compatibility.
* scripts/linear-algebra/linsolve.m: New function.
* scripts/linear-algebra/module.mk: Add linsolve.m to build system.
* NEWS: Announce new function.
| author | Nir Krakauer < nkrakauer@ccny.cuny.edu> |
|---|---|
| date | Thu, 26 Sep 2013 08:30:26 -0700 |
| parents | b81b9d079515 |
| children |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2008-2012 VZLU Prague, a.s. |
| 10296 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7 ## the Free Software Foundation; either version 3 of the License, or (at | |
| 8 ## your option) any later version. | |
| 9 ## | |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 16 ## along with Octave; see the file COPYING. If not, see | |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 18 ## | |
| 19 ## Author: Jaroslav Hajek <highegg@gmail.com> | |
| 20 | |
| 21 ## -*- texinfo -*- | |
| 22 ## @deftypefn {Function File} {[@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 ## |
| 10296 | 32 ## On exit, the function returns @var{x}, the approximate minimum point |
| 33 ## and @var{fval}, the function value thereof. | |
| 34 ## @var{info} is an exit flag that can have these values: | |
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35 ## |
| 10296 | 36 ## @itemize |
| 37 ## @item 1 | |
| 38 ## The algorithm converged to a solution. | |
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39 ## |
| 10296 | 40 ## @item 0 |
| 41 ## Maximum number of iterations or function evaluations has been exhausted. | |
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42 ## |
| 10296 | 43 ## @item -1 |
| 44 ## The algorithm has been terminated from user output function. | |
| 45 ## @end itemize | |
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46 ## |
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47 ## Notes: The search for a minimum is restricted to be in the interval |
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48 ## bound by @var{a} and @var{b}. If you only have an initial point |
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49 ## to begin searching from you will need to use an unconstrained |
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50 ## minimization algorithm such as @code{fminunc} or @code{fminsearch}. |
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51 ## @code{fminbnd} internally uses a Golden Section search strategy. |
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52 ## @seealso{fzero, fminunc, fminsearch, optimset} |
| 10296 | 53 ## @end deftypefn |
| 54 | |
| 55 ## This is patterned after opt/fmin.f from Netlib, which in turn is taken from | |
| 56 ## Richard Brent: Algorithms For Minimization Without Derivatives, Prentice-Hall (1973) | |
| 57 | |
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58 ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup. |
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59 ## PKG_ADD: [~] = __all_opts__ ("fminbnd"); |
| 10296 | 60 |
| 61 function [x, fval, info, output] = fminbnd (fun, xmin, xmax, options = struct ()) | |
| 62 | |
| 63 ## Get default options if requested. | |
| 64 if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults')) | |
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65 x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 1e-8, |
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66 "OutputFcn", [], "FunValCheck", "off"); |
| 10296 | 67 return; |
| 68 endif | |
| 69 | |
| 70 if (nargin < 2 || nargin > 4) | |
| 71 print_usage (); | |
| 72 endif | |
| 73 | |
| 74 if (ischar (fun)) | |
| 75 fun = str2func (fun, "global"); | |
| 76 endif | |
| 77 | |
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78 displ = optimget (options, "Display", "notify"); |
| 10296 | 79 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
| 80 outfcn = optimget (options, "OutputFcn"); | |
| 81 tolx = optimget (options, "TolX", 1e-8); | |
| 82 maxiter = optimget (options, "MaxIter", Inf); | |
| 83 maxfev = optimget (options, "MaxFunEvals", Inf); | |
| 84 | |
| 85 if (funvalchk) | |
| 86 ## Replace fun with a guarded version. | |
| 87 fun = @(x) guarded_eval (fun, x); | |
| 88 endif | |
| 89 | |
| 90 ## The default exit flag if exceeded number of iterations. | |
| 91 info = 0; | |
| 92 niter = 0; | |
| 93 nfev = 0; | |
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94 sqrteps = eps (class (xmin + xmax)); |
| 10296 | 95 |
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96 c = 0.5*(3 - sqrt (5)); |
| 10296 | 97 a = xmin; b = xmax; |
| 98 v = a + c*(b-a); | |
| 99 w = x = v; | |
| 100 e = 0; | |
| 101 fv = fw = fval = fun (x); | |
| 102 nfev++; | |
| 103 | |
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104 ## Only for display purposes. |
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105 iter(1).funccount = nfev; |
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106 iter(1).x = x; |
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107 iter(1).fx = fval; |
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108 |
| 10296 | 109 while (niter < maxiter && nfev < maxfev) |
| 110 xm = 0.5*(a+b); | |
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111 ## FIXME: the golden section search can actually get closer than sqrt(eps) |
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112 ## sometimes. Sometimes not, it depends on the function. This is the |
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113 ## strategy from the Netlib code. Something yet smarter would be good. |
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114 tol = 2 * sqrteps * abs (x) + tolx / 3; |
| 10296 | 115 if (abs (x - xm) <= (2*tol - 0.5*(b-a))) |
| 116 info = 1; | |
| 117 break; | |
| 118 endif | |
| 119 | |
| 120 if (abs (e) > tol) | |
| 121 dogs = false; | |
| 122 ## Try inverse parabolic step. | |
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123 iter(niter+1).procedure = "parabolic"; |
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124 |
| 10296 | 125 r = (x - w)*(fval - fv); |
| 126 q = (x - v)*(fval - fw); | |
| 127 p = (x - v)*q - (x - w)*r; | |
| 128 q = 2*(q - r); | |
| 129 p *= -sign (q); | |
| 130 q = abs (q); | |
| 131 r = e; | |
| 132 e = d; | |
| 133 | |
| 134 if (abs (p) < abs (0.5*q*r) && p > q*(a-x) && p < q*(b-x)) | |
| 135 ## The parabolic step is acceptable. | |
| 136 d = p / q; | |
| 137 u = x + d; | |
| 138 | |
| 139 ## f must not be evaluated too close to ax or bx. | |
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140 if (min (u-a, b-u) < 2*tol) |
| 10296 | 141 d = tol * (sign (xm - x) + (xm == x)); |
| 142 endif | |
| 143 else | |
| 144 dogs = true; | |
| 145 endif | |
| 146 else | |
| 147 dogs = true; | |
| 148 endif | |
| 149 if (dogs) | |
| 150 ## Default to golden section step. | |
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151 |
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152 ## WARNING: This is also the "initial" procedure following |
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153 ## MATLAB nomenclature. After the loop we'll fix the string |
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154 ## for the first step. |
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155 iter(niter+1).procedure = "golden"; |
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156 |
| 10296 | 157 e = ifelse (x >= xm, a - x, b - x); |
| 158 d = c * e; | |
| 159 endif | |
| 160 | |
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161 ## f must not be evaluated too close to x. |
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162 u = x + max (abs (d), tol) * (sign (d) + (d == 0)); |
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163 fu = fun (u); |
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164 |
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165 niter++; |
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166 |
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167 iter(niter).funccount = nfev++; |
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168 iter(niter).x = u; |
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169 iter(niter).fx = fu; |
| 10296 | 170 |
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171 ## update a, b, v, w, and x |
| 10296 | 172 |
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173 if (fu <= fval) |
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174 if (u < x) |
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175 b = x; |
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176 else |
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177 a = x; |
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178 endif |
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179 v = w; fv = fw; |
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180 w = x; fw = fval; |
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181 x = u; fval = fu; |
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182 else |
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183 ## The following if-statement was originally executed even if fu == fval. |
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184 if (u < x) |
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185 a = u; |
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186 else |
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187 b = u; |
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188 endif |
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189 if (fu <= fw || w == x) |
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190 v = w; fv = fw; |
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191 w = u; fw = fu; |
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192 elseif (fu <= fv || v == x || v == w) |
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193 v = u; |
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194 fv = fu; |
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195 endif |
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196 endif |
| 10296 | 197 |
| 198 ## If there's an output function, use it now. | |
| 199 if (outfcn) | |
| 200 optv.funccount = nfev; | |
| 201 optv.fval = fval; | |
| 202 optv.iteration = niter; | |
| 203 if (outfcn (x, optv, "iter")) | |
| 204 info = -1; | |
| 205 break; | |
| 206 endif | |
| 207 endif | |
| 208 endwhile | |
| 209 | |
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210 ## Fix the first step procedure. |
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211 iter(1).procedure = "initial"; |
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212 |
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213 ## Handle the "Display" option |
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214 switch (displ) |
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215 case "iter" |
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216 print_formatted_table (iter); |
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217 print_exit_msg (info, struct("TolX", tolx, "fx", fval)); |
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218 case "notify" |
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219 if (info == 0) |
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220 print_exit_msg (info, struct("fx",fval)); |
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221 endif |
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222 case "final" |
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223 print_exit_msg (info, struct("TolX", tolx, "fx", fval)); |
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224 case "off" |
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225 "skip"; |
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226 otherwise |
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227 warning ("unknown option for Display: '%s'", displ); |
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228 endswitch |
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229 |
| 10296 | 230 output.iterations = niter; |
| 231 output.funcCount = nfev; | |
| 232 output.bracket = [a, b]; | |
| 233 ## FIXME: bracketf possibly unavailable. | |
| 234 | |
| 235 endfunction | |
| 236 | |
| 237 ## An assistant function that evaluates a function handle and checks for | |
| 238 ## bad results. | |
| 239 function fx = guarded_eval (fun, x) | |
| 240 fx = fun (x); | |
| 241 fx = fx(1); | |
| 242 if (! isreal (fx)) | |
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243 error ("fminbnd:notreal", "fminbnd: non-real value encountered"); |
| 10296 | 244 elseif (isnan (fx)) |
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245 error ("fminbnd:isnan", "fminbnd: NaN value encountered"); |
| 10296 | 246 endif |
| 247 endfunction | |
| 248 | |
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249 ## A hack for printing a formatted table |
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250 function print_formatted_table (table) |
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251 printf ("\n Func-count x f(x) Procedure\n"); |
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252 for row=table |
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253 printf("%5.5s %7.7s %8.8s\t%s\n", |
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254 int2str (row.funccount), num2str (row.x,"%.5f"), |
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255 num2str (row.fx,"%.6f"), row.procedure); |
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256 endfor |
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257 printf ("\n"); |
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258 endfunction |
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259 |
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260 ## Print either a success termination message or bad news |
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261 function print_exit_msg (info, opt=struct()) |
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262 printf (""); |
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263 switch (info) |
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264 case 1 |
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265 printf ("Optimization terminated:\n"); |
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266 printf (" the current x satisfies the termination criteria using OPTIONS.TolX of %e\n", opt.TolX); |
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267 case 0 |
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268 printf ("Exiting: Maximum number of iterations has been exceeded\n"); |
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269 printf (" - increase MaxIter option.\n"); |
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270 printf (" Current function value: %.6f\n", opt.fx); |
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271 case -1 |
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272 "FIXME"; # FIXME: what's the message MATLAB prints for this case? |
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273 otherwise |
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274 error ("internal error - fminbnd() is bug, sorry!"); |
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275 endswitch |
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276 printf ("\n"); |
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277 endfunction |
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278 |
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279 |
| 10296 | 280 %!shared opt0 |
| 281 %! opt0 = optimset ("tolx", 0); | |
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282 %!assert (fminbnd (@cos, pi/2, 3*pi/2, opt0), pi, 10*sqrt (eps)) |
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283 %!assert (fminbnd (@(x) (x - 1e-3)^4, -1, 1, opt0), 1e-3, 10e-3*sqrt (eps)) |
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284 %!assert (fminbnd (@(x) abs (x-1e7), 0, 1e10, opt0), 1e7, 10e7*sqrt (eps)) |
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285 %!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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286 |