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annotate scripts/optimization/fzero.m @ 10687:a8ce6bdecce5
Improve documentation strings.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Tue, 08 Jun 2010 20:22:38 -0700 |
| parents | 9500a66118dc |
| children | 693e22af08ae |
| rev | line source |
|---|---|
| 8920 | 1 ## Copyright (C) 2008, 2009 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} should be a function handle or name. @var{x0} should be a |
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28 ## two-element vector specifying two points which bracket a zero. In |
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29 ## other words, there must be a change in sign of the function between |
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30 ## @var{x0}(1) and @var{x0}(2). More mathematically, the following must hold |
| 10470 | 31 ## @example |
| 32 ## sign (@var{fun}(@var{x0}(1))) * sign (@var{fun}(@var{x0}(2))) <= 0 | |
| 33 ## @end example | |
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34 ## If @var{x0} is a single scalar then several nearby and distant |
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35 ## values are probed in an attempt to obtain a valid bracketing. If this |
| 10470 | 36 ## is not successful, the function fails. |
| 37 ## @var{options} is a structure specifying additional options. | |
| 38 ## Currently, @code{fzero} | |
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39 ## recognizes these options: @code{"FunValCheck"}, @code{"OutputFcn"}, |
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40 ## @code{"TolX"}, @code{"MaxIter"}, @code{"MaxFunEvals"}. |
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41 ## For a description of these options, see @ref{doc-optimset,,optimset}. |
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42 ## |
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43 ## On exit, the function returns @var{x}, the approximate zero point |
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44 ## and @var{fval}, the function value thereof. |
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45 ## @var{info} is an exit flag that can have these values: |
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46 ## |
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47 ## @itemize |
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48 ## @item 1 |
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49 ## The algorithm converged to a solution. |
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50 ## @item 0 |
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51 ## Maximum number of iterations or function evaluations has been reached. |
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52 ## @item -1 |
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53 ## The algorithm has been terminated from user output function. |
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54 ## @item -5 |
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55 ## The algorithm may have converged to a singular point. |
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56 ## @end itemize |
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57 ## |
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58 ## @var{output} is a structure containing runtime information about the |
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59 ## @code{fzero} algorithm. Fields in the structure are: |
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60 ## |
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61 ## @itemize |
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62 ## @item iterations |
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63 ## Number of iterations through loop. |
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64 ## @item nfev |
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65 ## Number of function evaluations. |
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66 ## @item bracketx |
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67 ## A two-element vector with the final bracketing of the zero along the x-axis. |
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68 ## @item brackety |
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69 ## A two-element vector with the final bracketing of the zero along the y-axis. |
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70 ## @end itemize |
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71 ## @seealso{optimset, fsolve} |
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72 ## @end deftypefn |
| 8305 | 73 |
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74 ## This is essentially the ACM algorithm 748: Enclosing Zeros of |
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75 ## Continuous Functions due to Alefeld, Potra and Shi, ACM Transactions |
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76 ## on Mathematical Software, Vol. 21, No. 3, September 1995. Although |
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77 ## the workflow should be the same, the structure of the algorithm has |
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78 ## been transformed non-trivially; instead of the authors' approach of |
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79 ## sequentially calling building blocks subprograms we implement here a |
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80 ## FSM version using one interior point determination and one bracketing |
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81 ## per iteration, thus reducing the number of temporary variables and |
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82 ## simplifying the algorithm structure. Further, this approach reduces |
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83 ## the need for external functions and error handling. The algorithm has |
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84 ## also been slightly modified. |
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85 |
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86 ## PKG_ADD: __all_opts__ ("fzero"); |
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87 |
| 8305 | 88 function [x, fval, info, output] = fzero (fun, x0, options = struct ()) |
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89 |
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90 ## Get default options if requested. |
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91 if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults')) |
| 10296 | 92 x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 1e-8, \ |
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93 "OutputFcn", [], "FunValCheck", "off"); |
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94 return; |
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95 endif |
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96 |
| 8305 | 97 if (nargin < 2 || nargin > 3) |
| 98 print_usage (); | |
| 99 endif | |
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100 |
| 8305 | 101 if (ischar (fun)) |
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102 fun = str2func (fun, "global"); |
| 8305 | 103 endif |
| 104 | |
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105 ## TODO |
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106 ## displev = optimget (options, "Display", "notify"); |
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107 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
| 8305 | 108 outfcn = optimget (options, "OutputFcn"); |
| 10296 | 109 tolx = optimget (options, "TolX", 1e-8); |
| 8305 | 110 maxiter = optimget (options, "MaxIter", Inf); |
| 111 maxfev = optimget (options, "MaxFunEvals", Inf); | |
| 112 | |
| 113 persistent mu = 0.5; | |
| 114 | |
| 115 if (funvalchk) | |
| 8507 | 116 ## Replace fun with a guarded version. |
| 8305 | 117 fun = @(x) guarded_eval (fun, x); |
| 118 endif | |
| 119 | |
| 8507 | 120 ## The default exit flag if exceeded number of iterations. |
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121 info = 0; |
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122 niter = 0; |
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123 nfev = 0; |
| 8305 | 124 |
| 125 x = fval = a = fa = b = fb = NaN; | |
| 10296 | 126 eps = eps (class (x0)); |
| 8305 | 127 |
| 8507 | 128 ## Prepare... |
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129 a = x0(1); |
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130 fa = fun (a); |
| 8305 | 131 nfev = 1; |
| 132 if (length (x0) > 1) | |
| 133 b = x0(2); | |
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134 fb = fun (b); |
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135 nfev += 1; |
| 8305 | 136 else |
| 8507 | 137 ## Try to get b. |
| 8305 | 138 if (a == 0) |
| 139 aa = 1; | |
| 140 else | |
| 141 aa = a; | |
| 142 endif | |
| 143 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] | |
| 144 fb = fun (b); nfev += 1; | |
| 145 if (sign (fa) * sign (fb) <= 0) | |
| 146 break; | |
| 147 endif | |
| 148 endfor | |
| 149 endif | |
| 150 | |
| 151 if (b < a) | |
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152 u = a; |
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153 a = b; |
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154 b = u; |
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155 |
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156 fu = fa; |
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157 fa = fb; |
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158 fb = fu; |
| 8305 | 159 endif |
| 160 | |
| 161 if (! (sign (fa) * sign (fb) <= 0)) | |
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162 error ("fzero:bracket", "fzero: not a valid initial bracketing"); |
| 8305 | 163 endif |
| 164 | |
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165 slope0 = (fb - fa) / (b - a); |
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166 |
| 10470 | 167 if (fa == 0) |
| 168 b = a; | |
| 169 fb = fa; | |
| 170 elseif (fb == 0) | |
| 171 a = b; | |
| 172 fa = fb; | |
| 173 endif | |
| 174 | |
| 8305 | 175 itype = 1; |
| 176 | |
| 177 if (abs (fa) < abs (fb)) | |
| 178 u = a; fu = fa; | |
| 179 else | |
| 180 u = b; fu = fb; | |
| 181 endif | |
| 182 | |
| 183 d = e = u; | |
| 184 fd = fe = fu; | |
| 185 mba = mu*(b - a); | |
| 186 while (niter < maxiter && nfev < maxfev) | |
| 187 switch (itype) | |
| 188 case 1 | |
| 8507 | 189 ## The initial test. |
| 8305 | 190 if (b - a <= 2*(2 * abs (u) * eps + tolx)) |
| 191 x = u; fval = fu; | |
| 192 info = 1; | |
| 193 break; | |
| 194 endif | |
| 195 if (abs (fa) <= 1e3*abs (fb) && abs (fb) <= 1e3*abs (fa)) | |
| 8507 | 196 ## Secant step. |
| 8305 | 197 c = u - (a - b) / (fa - fb) * fu; |
| 198 else | |
| 8507 | 199 ## Bisection step. |
| 8305 | 200 c = 0.5*(a + b); |
| 201 endif | |
| 202 d = u; fd = fu; | |
| 203 itype = 5; | |
| 204 case {2, 3} | |
| 205 l = length (unique ([fa, fb, fd, fe])); | |
| 206 if (l == 4) | |
| 8507 | 207 ## Inverse cubic interpolation. |
| 8305 | 208 q11 = (d - e) * fd / (fe - fd); |
| 209 q21 = (b - d) * fb / (fd - fb); | |
| 210 q31 = (a - b) * fa / (fb - fa); | |
| 211 d21 = (b - d) * fd / (fd - fb); | |
| 212 d31 = (a - b) * fb / (fb - fa); | |
| 213 q22 = (d21 - q11) * fb / (fe - fb); | |
| 214 q32 = (d31 - q21) * fa / (fd - fa); | |
| 215 d32 = (d31 - q21) * fd / (fd - fa); | |
| 216 q33 = (d32 - q22) * fa / (fe - fa); | |
| 217 c = a + q31 + q32 + q33; | |
| 218 endif | |
| 219 if (l < 4 || sign (c - a) * sign (c - b) > 0) | |
| 8507 | 220 ## Quadratic interpolation + newton. |
| 8305 | 221 a0 = fa; |
| 222 a1 = (fb - fa)/(b - a); | |
| 223 a2 = ((fd - fb)/(d - b) - a1) / (d - a); | |
| 8507 | 224 ## Modification 1: this is simpler and does not seem to be worse. |
| 8305 | 225 c = a - a0/a1; |
| 226 if (a2 != 0) | |
| 227 c = a - a0/a1; | |
| 228 for i = 1:itype | |
| 229 pc = a0 + (a1 + a2*(c - b))*(c - a); | |
| 230 pdc = a1 + a2*(2*c - a - b); | |
| 231 if (pdc == 0) | |
| 232 c = a - a0/a1; | |
| 233 break; | |
| 234 endif | |
| 235 c -= pc/pdc; | |
| 236 endfor | |
| 237 endif | |
| 238 endif | |
| 239 itype += 1; | |
| 240 case 4 | |
| 8507 | 241 ## Double secant step. |
| 8305 | 242 c = u - 2*(b - a)/(fb - fa)*fu; |
| 8507 | 243 ## Bisect if too far. |
| 8305 | 244 if (abs (c - u) > 0.5*(b - a)) |
| 245 c = 0.5 * (b + a); | |
| 246 endif | |
| 247 itype = 5; | |
| 248 case 5 | |
| 8507 | 249 ## Bisection step. |
| 8305 | 250 c = 0.5 * (b + a); |
| 251 itype = 2; | |
| 252 endswitch | |
| 253 | |
| 8507 | 254 ## Don't let c come too close to a or b. |
| 8305 | 255 delta = 2*0.7*(2 * abs (u) * eps + tolx); |
| 256 if ((b - a) <= 2*delta) | |
| 257 c = (a + b)/2; | |
| 258 else | |
| 259 c = max (a + delta, min (b - delta, c)); | |
| 260 endif | |
| 261 | |
| 8507 | 262 ## Calculate new point. |
| 8305 | 263 x = c; |
| 264 fval = fc = fun (c); | |
| 265 niter ++; nfev ++; | |
| 266 | |
| 8507 | 267 ## Modification 2: skip inverse cubic interpolation if |
| 268 ## nonmonotonicity is detected. | |
| 8305 | 269 if (sign (fc - fa) * sign (fc - fb) >= 0) |
| 8507 | 270 ## The new point broke monotonicity. |
| 271 ## Disable inverse cubic. | |
| 8305 | 272 fe = fc; |
| 273 else | |
| 274 e = d; fe = fd; | |
| 275 endif | |
| 276 | |
| 8507 | 277 ## Bracketing. |
| 8305 | 278 if (sign (fa) * sign (fc) < 0) |
| 279 d = b; fd = fb; | |
| 280 b = c; fb = fc; | |
| 281 elseif (sign (fb) * sign (fc) < 0) | |
| 282 d = a; fd = fa; | |
| 283 a = c; fa = fc; | |
| 284 elseif (fc == 0) | |
| 285 a = b = c; fa = fb = fc; | |
| 286 info = 1; | |
| 287 break; | |
| 288 else | |
| 8507 | 289 ## This should never happen. |
|
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290 error ("fzero:bracket", "fzero: zero point is not bracketed"); |
| 8305 | 291 endif |
| 292 | |
| 8507 | 293 ## If there's an output function, use it now. |
| 8305 | 294 if (outfcn) |
| 10296 | 295 optv.funccount = nfev; |
| 8305 | 296 optv.fval = fval; |
| 297 optv.iteration = niter; | |
| 298 if (outfcn (x, optv, "iter")) | |
| 299 info = -1; | |
| 300 break; | |
| 301 endif | |
| 302 endif | |
| 303 | |
| 304 if (abs (fa) < abs (fb)) | |
| 305 u = a; fu = fa; | |
| 306 else | |
| 307 u = b; fu = fb; | |
| 308 endif | |
| 309 if (b - a <= 2*(2 * abs (u) * eps + tolx)) | |
| 310 info = 1; | |
| 311 break; | |
| 312 endif | |
| 313 | |
| 8507 | 314 ## Skip bisection step if successful reduction. |
| 8305 | 315 if (itype == 5 && (b - a) <= mba) |
| 316 itype = 2; | |
| 317 endif | |
| 318 if (itype == 2) | |
| 319 mba = mu * (b - a); | |
| 320 endif | |
| 321 endwhile | |
| 322 | |
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323 ## Check solution for a singularity by examining slope |
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324 if (info == 1) |
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325 if ((b - a) != 0 && abs ((fb - fa)/(b - a) / slope0) > max (1e6, 0.5/(eps+tolx))) |
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326 info = -5; |
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327 endif |
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328 endif |
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329 |
| 8305 | 330 output.iterations = niter; |
| 10296 | 331 output.funcCount = nfev; |
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332 output.bracketx = [a, b]; |
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333 output.brackety = [fa, fb]; |
| 8305 | 334 |
| 335 endfunction | |
| 336 | |
| 8507 | 337 ## An assistant function that evaluates a function handle and checks for |
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338 ## bad results. |
| 8305 | 339 function fx = guarded_eval (fun, x) |
| 340 fx = fun (x); | |
| 341 fx = fx(1); | |
| 342 if (! isreal (fx)) | |
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343 error ("fzero:notreal", "fzero: non-real value encountered"); |
| 8305 | 344 elseif (isnan (fx)) |
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345 error ("fzero:isnan", "fzero: NaN value encountered"); |
| 8305 | 346 endif |
| 347 endfunction | |
| 348 | |
| 10296 | 349 %!shared opt0 |
| 350 %! opt0 = optimset ("tolx", 0); | |
| 351 %!assert(fzero(@cos, [0, 3], opt0), pi/2, 10*eps) | |
| 352 %!assert(fzero(@(x) x^(1/3) - 1e-8, [0,1], opt0), 1e-24, 1e-22*eps) |