Mercurial > hg > octave-lyh
annotate scripts/optimization/fzero.m @ 14526:e12945668746
doc: Stop constant rebuild of spline images in txt format.
* splineimages.m: Fix typo in output filename generation for txt format.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Wed, 04 Apr 2012 19:28:16 -0700 |
| parents | f3d52523cde1 |
| children | 86854d032a37 |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2008-2012 VZLU Prague, a.s. |
| 8305 | 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 | |
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21 ## -*- texinfo -*- |
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22 ## @deftypefn {Function File} {} fzero (@var{fun}, @var{x0}) |
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23 ## @deftypefnx {Function File} {} fzero (@var{fun}, @var{x0}, @var{options}) |
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24 ## @deftypefnx {Function File} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} fzero (@dots{}) |
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25 ## Find a zero of a univariate function. |
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26 ## |
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27 ## @var{fun} is a function handle, inline function, or string |
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28 ## containing the name of the function to evaluate. |
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29 ## @var{x0} should be a two-element vector specifying two points which |
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30 ## bracket a zero. In other words, there must be a change in sign of the |
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31 ## function between @var{x0}(1) and @var{x0}(2). More mathematically, the |
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32 ## following must hold |
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33 ## |
| 10470 | 34 ## @example |
| 35 ## sign (@var{fun}(@var{x0}(1))) * sign (@var{fun}(@var{x0}(2))) <= 0 | |
| 36 ## @end example | |
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37 ## |
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38 ## If @var{x0} is a single scalar then several nearby and distant |
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39 ## values are probed in an attempt to obtain a valid bracketing. If this |
| 10470 | 40 ## is not successful, the function fails. |
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41 ## @var{options} is a structure specifying additional options. |
| 10470 | 42 ## Currently, @code{fzero} |
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43 ## recognizes these options: @code{"FunValCheck"}, @code{"OutputFcn"}, |
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44 ## @code{"TolX"}, @code{"MaxIter"}, @code{"MaxFunEvals"}. |
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45 ## For a description of these options, see @ref{doc-optimset,,optimset}. |
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46 ## |
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47 ## On exit, the function returns @var{x}, the approximate zero point |
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48 ## and @var{fval}, the function value thereof. |
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49 ## @var{info} is an exit flag that can have these values: |
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50 ## |
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51 ## @itemize |
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52 ## @item 1 |
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53 ## The algorithm converged to a solution. |
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54 ## |
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55 ## @item 0 |
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56 ## Maximum number of iterations or function evaluations has been reached. |
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57 ## |
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58 ## @item -1 |
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59 ## The algorithm has been terminated from user output function. |
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60 ## |
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61 ## @item -5 |
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62 ## The algorithm may have converged to a singular point. |
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63 ## @end itemize |
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64 ## |
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65 ## @var{output} is a structure containing runtime information about the |
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66 ## @code{fzero} algorithm. Fields in the structure are: |
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67 ## |
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68 ## @itemize |
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69 ## @item iterations |
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70 ## Number of iterations through loop. |
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71 ## |
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72 ## @item nfev |
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73 ## Number of function evaluations. |
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74 ## |
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75 ## @item bracketx |
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76 ## A two-element vector with the final bracketing of the zero along the x-axis. |
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77 ## |
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78 ## @item brackety |
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79 ## A two-element vector with the final bracketing of the zero along the y-axis. |
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80 ## @end itemize |
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81 ## @seealso{optimset, fsolve} |
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82 ## @end deftypefn |
| 8305 | 83 |
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84 ## This is essentially the ACM algorithm 748: Enclosing Zeros of |
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85 ## Continuous Functions due to Alefeld, Potra and Shi, ACM Transactions |
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86 ## on Mathematical Software, Vol. 21, No. 3, September 1995. Although |
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87 ## the workflow should be the same, the structure of the algorithm has |
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88 ## been transformed non-trivially; instead of the authors' approach of |
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89 ## sequentially calling building blocks subprograms we implement here a |
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90 ## FSM version using one interior point determination and one bracketing |
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91 ## per iteration, thus reducing the number of temporary variables and |
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92 ## simplifying the algorithm structure. Further, this approach reduces |
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93 ## the need for external functions and error handling. The algorithm has |
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94 ## also been slightly modified. |
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95 |
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96 ## PKG_ADD: ## Discard result to avoid polluting workspace with ans at startup. |
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97 ## PKG_ADD: [~] = __all_opts__ ("fzero"); |
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98 |
| 8305 | 99 function [x, fval, info, output] = fzero (fun, x0, options = struct ()) |
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100 |
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101 ## Get default options if requested. |
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102 if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults')) |
| 10296 | 103 x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 1e-8, \ |
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104 "OutputFcn", [], "FunValCheck", "off"); |
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105 return; |
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106 endif |
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107 |
| 8305 | 108 if (nargin < 2 || nargin > 3) |
| 109 print_usage (); | |
| 110 endif | |
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111 |
| 8305 | 112 if (ischar (fun)) |
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113 fun = str2func (fun, "global"); |
| 8305 | 114 endif |
| 115 | |
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116 ## TODO |
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117 ## displev = optimget (options, "Display", "notify"); |
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118 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
| 8305 | 119 outfcn = optimget (options, "OutputFcn"); |
| 10296 | 120 tolx = optimget (options, "TolX", 1e-8); |
| 8305 | 121 maxiter = optimget (options, "MaxIter", Inf); |
| 122 maxfev = optimget (options, "MaxFunEvals", Inf); | |
| 123 | |
| 124 persistent mu = 0.5; | |
| 125 | |
| 126 if (funvalchk) | |
| 8507 | 127 ## Replace fun with a guarded version. |
| 8305 | 128 fun = @(x) guarded_eval (fun, x); |
| 129 endif | |
| 130 | |
| 8507 | 131 ## The default exit flag if exceeded number of iterations. |
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132 info = 0; |
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133 niter = 0; |
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134 nfev = 0; |
| 8305 | 135 |
| 136 x = fval = a = fa = b = fb = NaN; | |
| 10296 | 137 eps = eps (class (x0)); |
| 8305 | 138 |
| 8507 | 139 ## Prepare... |
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140 a = x0(1); |
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141 fa = fun (a); |
| 8305 | 142 nfev = 1; |
| 143 if (length (x0) > 1) | |
| 144 b = x0(2); | |
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145 fb = fun (b); |
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146 nfev += 1; |
| 8305 | 147 else |
| 8507 | 148 ## Try to get b. |
| 8305 | 149 if (a == 0) |
| 150 aa = 1; | |
| 151 else | |
| 152 aa = a; | |
| 153 endif | |
| 154 for b = [0.9*aa, 1.1*aa, aa-1, aa+1, 0.5*aa 1.5*aa, -aa, 2*aa, -10*aa, 10*aa] | |
| 155 fb = fun (b); nfev += 1; | |
| 156 if (sign (fa) * sign (fb) <= 0) | |
| 157 break; | |
| 158 endif | |
| 159 endfor | |
| 160 endif | |
| 161 | |
| 162 if (b < a) | |
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163 u = a; |
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164 a = b; |
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165 b = u; |
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166 |
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167 fu = fa; |
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168 fa = fb; |
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169 fb = fu; |
| 8305 | 170 endif |
| 171 | |
| 172 if (! (sign (fa) * sign (fb) <= 0)) | |
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173 error ("fzero:bracket", "fzero: not a valid initial bracketing"); |
| 8305 | 174 endif |
| 175 | |
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176 slope0 = (fb - fa) / (b - a); |
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177 |
| 10470 | 178 if (fa == 0) |
| 179 b = a; | |
| 180 fb = fa; | |
| 181 elseif (fb == 0) | |
| 182 a = b; | |
| 183 fa = fb; | |
| 184 endif | |
| 185 | |
| 8305 | 186 itype = 1; |
| 187 | |
| 188 if (abs (fa) < abs (fb)) | |
| 189 u = a; fu = fa; | |
| 190 else | |
| 191 u = b; fu = fb; | |
| 192 endif | |
| 193 | |
| 194 d = e = u; | |
| 195 fd = fe = fu; | |
| 196 mba = mu*(b - a); | |
| 197 while (niter < maxiter && nfev < maxfev) | |
| 198 switch (itype) | |
| 199 case 1 | |
| 8507 | 200 ## The initial test. |
| 8305 | 201 if (b - a <= 2*(2 * abs (u) * eps + tolx)) |
| 202 x = u; fval = fu; | |
| 203 info = 1; | |
| 204 break; | |
| 205 endif | |
| 206 if (abs (fa) <= 1e3*abs (fb) && abs (fb) <= 1e3*abs (fa)) | |
| 8507 | 207 ## Secant step. |
| 8305 | 208 c = u - (a - b) / (fa - fb) * fu; |
| 209 else | |
| 8507 | 210 ## Bisection step. |
| 8305 | 211 c = 0.5*(a + b); |
| 212 endif | |
| 213 d = u; fd = fu; | |
| 214 itype = 5; | |
| 215 case {2, 3} | |
| 216 l = length (unique ([fa, fb, fd, fe])); | |
| 217 if (l == 4) | |
| 8507 | 218 ## Inverse cubic interpolation. |
| 8305 | 219 q11 = (d - e) * fd / (fe - fd); |
| 220 q21 = (b - d) * fb / (fd - fb); | |
| 221 q31 = (a - b) * fa / (fb - fa); | |
| 222 d21 = (b - d) * fd / (fd - fb); | |
| 223 d31 = (a - b) * fb / (fb - fa); | |
| 224 q22 = (d21 - q11) * fb / (fe - fb); | |
| 225 q32 = (d31 - q21) * fa / (fd - fa); | |
| 226 d32 = (d31 - q21) * fd / (fd - fa); | |
| 227 q33 = (d32 - q22) * fa / (fe - fa); | |
| 228 c = a + q31 + q32 + q33; | |
| 229 endif | |
| 230 if (l < 4 || sign (c - a) * sign (c - b) > 0) | |
| 8507 | 231 ## Quadratic interpolation + newton. |
| 8305 | 232 a0 = fa; |
| 233 a1 = (fb - fa)/(b - a); | |
| 234 a2 = ((fd - fb)/(d - b) - a1) / (d - a); | |
| 8507 | 235 ## Modification 1: this is simpler and does not seem to be worse. |
| 8305 | 236 c = a - a0/a1; |
| 237 if (a2 != 0) | |
| 238 c = a - a0/a1; | |
| 239 for i = 1:itype | |
| 240 pc = a0 + (a1 + a2*(c - b))*(c - a); | |
| 241 pdc = a1 + a2*(2*c - a - b); | |
| 242 if (pdc == 0) | |
| 243 c = a - a0/a1; | |
| 244 break; | |
| 245 endif | |
| 246 c -= pc/pdc; | |
| 247 endfor | |
| 248 endif | |
| 249 endif | |
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250 itype += 1; |
| 8305 | 251 case 4 |
| 8507 | 252 ## Double secant step. |
| 8305 | 253 c = u - 2*(b - a)/(fb - fa)*fu; |
| 8507 | 254 ## Bisect if too far. |
| 8305 | 255 if (abs (c - u) > 0.5*(b - a)) |
| 256 c = 0.5 * (b + a); | |
| 257 endif | |
| 258 itype = 5; | |
| 259 case 5 | |
| 8507 | 260 ## Bisection step. |
| 8305 | 261 c = 0.5 * (b + a); |
| 262 itype = 2; | |
| 263 endswitch | |
| 264 | |
| 8507 | 265 ## Don't let c come too close to a or b. |
| 8305 | 266 delta = 2*0.7*(2 * abs (u) * eps + tolx); |
| 267 if ((b - a) <= 2*delta) | |
| 268 c = (a + b)/2; | |
| 269 else | |
| 270 c = max (a + delta, min (b - delta, c)); | |
| 271 endif | |
| 272 | |
| 8507 | 273 ## Calculate new point. |
| 8305 | 274 x = c; |
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275 fval = fc = fun (c); |
| 8305 | 276 niter ++; nfev ++; |
| 277 | |
| 8507 | 278 ## Modification 2: skip inverse cubic interpolation if |
| 279 ## nonmonotonicity is detected. | |
| 8305 | 280 if (sign (fc - fa) * sign (fc - fb) >= 0) |
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281 ## The new point broke monotonicity. |
| 8507 | 282 ## Disable inverse cubic. |
| 8305 | 283 fe = fc; |
| 284 else | |
| 285 e = d; fe = fd; | |
| 286 endif | |
| 287 | |
| 8507 | 288 ## Bracketing. |
| 8305 | 289 if (sign (fa) * sign (fc) < 0) |
| 290 d = b; fd = fb; | |
| 291 b = c; fb = fc; | |
| 292 elseif (sign (fb) * sign (fc) < 0) | |
| 293 d = a; fd = fa; | |
| 294 a = c; fa = fc; | |
| 295 elseif (fc == 0) | |
| 296 a = b = c; fa = fb = fc; | |
| 297 info = 1; | |
| 298 break; | |
| 299 else | |
| 8507 | 300 ## This should never happen. |
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301 error ("fzero:bracket", "fzero: zero point is not bracketed"); |
| 8305 | 302 endif |
| 303 | |
| 8507 | 304 ## If there's an output function, use it now. |
| 8305 | 305 if (outfcn) |
| 10296 | 306 optv.funccount = nfev; |
| 8305 | 307 optv.fval = fval; |
| 308 optv.iteration = niter; | |
| 309 if (outfcn (x, optv, "iter")) | |
| 310 info = -1; | |
| 311 break; | |
| 312 endif | |
| 313 endif | |
| 314 | |
| 315 if (abs (fa) < abs (fb)) | |
| 316 u = a; fu = fa; | |
| 317 else | |
| 318 u = b; fu = fb; | |
| 319 endif | |
| 320 if (b - a <= 2*(2 * abs (u) * eps + tolx)) | |
| 321 info = 1; | |
| 322 break; | |
| 323 endif | |
| 324 | |
| 8507 | 325 ## Skip bisection step if successful reduction. |
| 8305 | 326 if (itype == 5 && (b - a) <= mba) |
| 327 itype = 2; | |
| 328 endif | |
| 329 if (itype == 2) | |
| 330 mba = mu * (b - a); | |
| 331 endif | |
| 332 endwhile | |
| 333 | |
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334 ## Check solution for a singularity by examining slope |
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335 if (info == 1) |
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336 if ((b - a) != 0 && abs ((fb - fa)/(b - a) / slope0) > max (1e6, 0.5/(eps+tolx))) |
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337 info = -5; |
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338 endif |
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339 endif |
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340 |
| 8305 | 341 output.iterations = niter; |
| 10296 | 342 output.funcCount = nfev; |
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343 output.bracketx = [a, b]; |
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344 output.brackety = [fa, fb]; |
| 8305 | 345 |
| 346 endfunction | |
| 347 | |
| 8507 | 348 ## An assistant function that evaluates a function handle and checks for |
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349 ## bad results. |
| 8305 | 350 function fx = guarded_eval (fun, x) |
| 351 fx = fun (x); | |
| 352 fx = fx(1); | |
| 353 if (! isreal (fx)) | |
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354 error ("fzero:notreal", "fzero: non-real value encountered"); |
| 8305 | 355 elseif (isnan (fx)) |
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356 error ("fzero:isnan", "fzero: NaN value encountered"); |
| 8305 | 357 endif |
| 358 endfunction | |
| 359 | |
| 10296 | 360 %!shared opt0 |
| 361 %! opt0 = optimset ("tolx", 0); | |
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362 %!assert (fzero (@cos, [0, 3], opt0), pi/2, 10*eps) |
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363 %!assert (fzero (@(x) x^(1/3) - 1e-8, [0,1], opt0), 1e-24, 1e-22*eps) |
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364 |