Mercurial > hg > octave-nkf
annotate scripts/optimization/fzero.m @ 10544:9961fc022d9d
fix assignment to non-existing variables and octave_value::assign
| author | Jaroslav Hajek <highegg@gmail.com> |
|---|---|
| date | Fri, 23 Apr 2010 11:23:43 +0200 |
| parents | 9500a66118dc |
| children | a8ce6bdecce5 |
| 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} {[@var{x}, @var{fval}, @var{info}, @var{output}] =} fzero (@var{fun}, @var{x0}, @var{options}) |
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23 ## Find a zero point of a univariate function. @var{fun} should be a function |
| 10470 | 24 ## handle or name. @var{x0} should be a two-element vector specifying the initial |
| 25 ## bracketing. It should hold | |
| 26 ## @example | |
| 27 ## sign (@var{fun}(@var{x0}(1))) * sign (@var{fun}(@var{x0}(2))) <= 0 | |
| 28 ## @end example | |
| 29 ## If only a single scalar is given as @var{x0}, several nearby and distant | |
| 30 ## values are probed in an attempt to obtain a valid bracketing. If this | |
| 31 ## is not successful, the function fails. | |
| 32 ## @var{options} is a structure specifying additional options. | |
| 33 ## Currently, @code{fzero} | |
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34 ## recognizes these options: @code{"FunValCheck"}, @code{"OutputFcn"}, |
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35 ## @code{"TolX"}, @code{"MaxIter"}, @code{"MaxFunEvals"}. |
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36 ## For description of these options, see @ref{doc-optimset,,optimset}. |
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37 ## |
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38 ## On exit, the function returns @var{x}, the approximate zero point |
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39 ## and @var{fval}, the function value thereof. |
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40 ## @var{info} is an exit flag that can have these values: |
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41 ## |
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42 ## @itemize |
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43 ## @item 1 |
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44 ## The algorithm converged to a solution. |
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45 ## @item 0 |
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46 ## Maximum number of iterations or function evaluations has been exhausted. |
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47 ## @item -1 |
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48 ## The algorithm has been terminated from user output function. |
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49 ## @item -5 |
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50 ## The algorithm may have converged to a singular point. |
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51 ## @end itemize |
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52 ## @seealso{optimset, fsolve} |
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53 ## @end deftypefn |
| 8305 | 54 |
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55 ## This is essentially the ACM algorithm 748: Enclosing Zeros of |
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56 ## Continuous Functions due to Alefeld, Potra and Shi, ACM Transactions |
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57 ## on Mathematical Software, Vol. 21, No. 3, September 1995. Although |
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58 ## the workflow should be the same, the structure of the algorithm has |
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59 ## been transformed non-trivially; instead of the authors' approach of |
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60 ## sequentially calling building blocks subprograms we implement here a |
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61 ## FSM version using one interior point determination and one bracketing |
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62 ## per iteration, thus reducing the number of temporary variables and |
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63 ## simplifying the algorithm structure. Further, this approach reduces |
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64 ## the need for external functions and error handling. The algorithm has |
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65 ## also been slightly modified. |
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66 |
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67 ## PKG_ADD: __all_opts__ ("fzero"); |
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68 |
| 8305 | 69 function [x, fval, info, output] = fzero (fun, x0, options = struct ()) |
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70 |
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71 ## Get default options if requested. |
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72 if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults')) |
| 10296 | 73 x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 1e-8, \ |
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74 "OutputFcn", [], "FunValCheck", "off"); |
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75 return; |
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76 endif |
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77 |
| 8305 | 78 if (nargin < 2 || nargin > 3) |
| 79 print_usage (); | |
| 80 endif | |
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81 |
| 8305 | 82 if (ischar (fun)) |
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83 fun = str2func (fun, "global"); |
| 8305 | 84 endif |
| 85 | |
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86 ## TODO |
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87 ## displev = optimget (options, "Display", "notify"); |
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88 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
| 8305 | 89 outfcn = optimget (options, "OutputFcn"); |
| 10296 | 90 tolx = optimget (options, "TolX", 1e-8); |
| 8305 | 91 maxiter = optimget (options, "MaxIter", Inf); |
| 92 maxfev = optimget (options, "MaxFunEvals", Inf); | |
| 93 | |
| 94 persistent mu = 0.5; | |
| 95 | |
| 96 if (funvalchk) | |
| 8507 | 97 ## Replace fun with a guarded version. |
| 8305 | 98 fun = @(x) guarded_eval (fun, x); |
| 99 endif | |
| 100 | |
| 8507 | 101 ## The default exit flag if exceeded number of iterations. |
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102 info = 0; |
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103 niter = 0; |
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104 nfev = 0; |
| 8305 | 105 |
| 106 x = fval = a = fa = b = fb = NaN; | |
| 10296 | 107 eps = eps (class (x0)); |
| 8305 | 108 |
| 8507 | 109 ## Prepare... |
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110 a = x0(1); |
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111 fa = fun (a); |
| 8305 | 112 nfev = 1; |
| 113 if (length (x0) > 1) | |
| 114 b = x0(2); | |
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115 fb = fun (b); |
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116 nfev += 1; |
| 8305 | 117 else |
| 8507 | 118 ## Try to get b. |
| 8305 | 119 if (a == 0) |
| 120 aa = 1; | |
| 121 else | |
| 122 aa = a; | |
| 123 endif | |
| 124 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] | |
| 125 fb = fun (b); nfev += 1; | |
| 126 if (sign (fa) * sign (fb) <= 0) | |
| 127 break; | |
| 128 endif | |
| 129 endfor | |
| 130 endif | |
| 131 | |
| 132 if (b < a) | |
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133 u = a; |
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134 a = b; |
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135 b = u; |
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136 |
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137 fu = fa; |
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138 fa = fb; |
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139 fb = fu; |
| 8305 | 140 endif |
| 141 | |
| 142 if (! (sign (fa) * sign (fb) <= 0)) | |
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143 error ("fzero:bracket", "fzero: not a valid initial bracketing"); |
| 8305 | 144 endif |
| 145 | |
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146 slope0 = (fb - fa) / (b - a); |
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147 |
| 10470 | 148 if (fa == 0) |
| 149 b = a; | |
| 150 fb = fa; | |
| 151 elseif (fb == 0) | |
| 152 a = b; | |
| 153 fa = fb; | |
| 154 endif | |
| 155 | |
| 8305 | 156 itype = 1; |
| 157 | |
| 158 if (abs (fa) < abs (fb)) | |
| 159 u = a; fu = fa; | |
| 160 else | |
| 161 u = b; fu = fb; | |
| 162 endif | |
| 163 | |
| 164 d = e = u; | |
| 165 fd = fe = fu; | |
| 166 mba = mu*(b - a); | |
| 167 while (niter < maxiter && nfev < maxfev) | |
| 168 switch (itype) | |
| 169 case 1 | |
| 8507 | 170 ## The initial test. |
| 8305 | 171 if (b - a <= 2*(2 * abs (u) * eps + tolx)) |
| 172 x = u; fval = fu; | |
| 173 info = 1; | |
| 174 break; | |
| 175 endif | |
| 176 if (abs (fa) <= 1e3*abs (fb) && abs (fb) <= 1e3*abs (fa)) | |
| 8507 | 177 ## Secant step. |
| 8305 | 178 c = u - (a - b) / (fa - fb) * fu; |
| 179 else | |
| 8507 | 180 ## Bisection step. |
| 8305 | 181 c = 0.5*(a + b); |
| 182 endif | |
| 183 d = u; fd = fu; | |
| 184 itype = 5; | |
| 185 case {2, 3} | |
| 186 l = length (unique ([fa, fb, fd, fe])); | |
| 187 if (l == 4) | |
| 8507 | 188 ## Inverse cubic interpolation. |
| 8305 | 189 q11 = (d - e) * fd / (fe - fd); |
| 190 q21 = (b - d) * fb / (fd - fb); | |
| 191 q31 = (a - b) * fa / (fb - fa); | |
| 192 d21 = (b - d) * fd / (fd - fb); | |
| 193 d31 = (a - b) * fb / (fb - fa); | |
| 194 q22 = (d21 - q11) * fb / (fe - fb); | |
| 195 q32 = (d31 - q21) * fa / (fd - fa); | |
| 196 d32 = (d31 - q21) * fd / (fd - fa); | |
| 197 q33 = (d32 - q22) * fa / (fe - fa); | |
| 198 c = a + q31 + q32 + q33; | |
| 199 endif | |
| 200 if (l < 4 || sign (c - a) * sign (c - b) > 0) | |
| 8507 | 201 ## Quadratic interpolation + newton. |
| 8305 | 202 a0 = fa; |
| 203 a1 = (fb - fa)/(b - a); | |
| 204 a2 = ((fd - fb)/(d - b) - a1) / (d - a); | |
| 8507 | 205 ## Modification 1: this is simpler and does not seem to be worse. |
| 8305 | 206 c = a - a0/a1; |
| 207 if (a2 != 0) | |
| 208 c = a - a0/a1; | |
| 209 for i = 1:itype | |
| 210 pc = a0 + (a1 + a2*(c - b))*(c - a); | |
| 211 pdc = a1 + a2*(2*c - a - b); | |
| 212 if (pdc == 0) | |
| 213 c = a - a0/a1; | |
| 214 break; | |
| 215 endif | |
| 216 c -= pc/pdc; | |
| 217 endfor | |
| 218 endif | |
| 219 endif | |
| 220 itype += 1; | |
| 221 case 4 | |
| 8507 | 222 ## Double secant step. |
| 8305 | 223 c = u - 2*(b - a)/(fb - fa)*fu; |
| 8507 | 224 ## Bisect if too far. |
| 8305 | 225 if (abs (c - u) > 0.5*(b - a)) |
| 226 c = 0.5 * (b + a); | |
| 227 endif | |
| 228 itype = 5; | |
| 229 case 5 | |
| 8507 | 230 ## Bisection step. |
| 8305 | 231 c = 0.5 * (b + a); |
| 232 itype = 2; | |
| 233 endswitch | |
| 234 | |
| 8507 | 235 ## Don't let c come too close to a or b. |
| 8305 | 236 delta = 2*0.7*(2 * abs (u) * eps + tolx); |
| 237 if ((b - a) <= 2*delta) | |
| 238 c = (a + b)/2; | |
| 239 else | |
| 240 c = max (a + delta, min (b - delta, c)); | |
| 241 endif | |
| 242 | |
| 8507 | 243 ## Calculate new point. |
| 8305 | 244 x = c; |
| 245 fval = fc = fun (c); | |
| 246 niter ++; nfev ++; | |
| 247 | |
| 8507 | 248 ## Modification 2: skip inverse cubic interpolation if |
| 249 ## nonmonotonicity is detected. | |
| 8305 | 250 if (sign (fc - fa) * sign (fc - fb) >= 0) |
| 8507 | 251 ## The new point broke monotonicity. |
| 252 ## Disable inverse cubic. | |
| 8305 | 253 fe = fc; |
| 254 else | |
| 255 e = d; fe = fd; | |
| 256 endif | |
| 257 | |
| 8507 | 258 ## Bracketing. |
| 8305 | 259 if (sign (fa) * sign (fc) < 0) |
| 260 d = b; fd = fb; | |
| 261 b = c; fb = fc; | |
| 262 elseif (sign (fb) * sign (fc) < 0) | |
| 263 d = a; fd = fa; | |
| 264 a = c; fa = fc; | |
| 265 elseif (fc == 0) | |
| 266 a = b = c; fa = fb = fc; | |
| 267 info = 1; | |
| 268 break; | |
| 269 else | |
| 8507 | 270 ## This should never happen. |
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271 error ("fzero:bracket", "fzero: zero point is not bracketed"); |
| 8305 | 272 endif |
| 273 | |
| 8507 | 274 ## If there's an output function, use it now. |
| 8305 | 275 if (outfcn) |
| 10296 | 276 optv.funccount = nfev; |
| 8305 | 277 optv.fval = fval; |
| 278 optv.iteration = niter; | |
| 279 if (outfcn (x, optv, "iter")) | |
| 280 info = -1; | |
| 281 break; | |
| 282 endif | |
| 283 endif | |
| 284 | |
| 285 if (abs (fa) < abs (fb)) | |
| 286 u = a; fu = fa; | |
| 287 else | |
| 288 u = b; fu = fb; | |
| 289 endif | |
| 290 if (b - a <= 2*(2 * abs (u) * eps + tolx)) | |
| 291 info = 1; | |
| 292 break; | |
| 293 endif | |
| 294 | |
| 8507 | 295 ## Skip bisection step if successful reduction. |
| 8305 | 296 if (itype == 5 && (b - a) <= mba) |
| 297 itype = 2; | |
| 298 endif | |
| 299 if (itype == 2) | |
| 300 mba = mu * (b - a); | |
| 301 endif | |
| 302 endwhile | |
| 303 | |
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304 ## Check solution for a singularity by examining slope |
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305 if (info == 1) |
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306 if ((b - a) != 0 && abs ((fb - fa)/(b - a) / slope0) > max (1e6, 0.5/(eps+tolx))) |
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307 info = -5; |
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308 endif |
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309 endif |
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310 |
| 8305 | 311 output.iterations = niter; |
| 10296 | 312 output.funcCount = nfev; |
| 8305 | 313 output.bracket = [a, b]; |
| 314 output.bracketf = [fa, fb]; | |
| 315 | |
| 316 endfunction | |
| 317 | |
| 8507 | 318 ## An assistant function that evaluates a function handle and checks for |
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319 ## bad results. |
| 8305 | 320 function fx = guarded_eval (fun, x) |
| 321 fx = fun (x); | |
| 322 fx = fx(1); | |
| 323 if (! isreal (fx)) | |
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324 error ("fzero:notreal", "fzero: non-real value encountered"); |
| 8305 | 325 elseif (isnan (fx)) |
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326 error ("fzero:isnan", "fzero: NaN value encountered"); |
| 8305 | 327 endif |
| 328 endfunction | |
| 329 | |
| 10296 | 330 %!shared opt0 |
| 331 %! opt0 = optimset ("tolx", 0); | |
| 332 %!assert(fzero(@cos, [0, 3], opt0), pi/2, 10*eps) | |
| 333 %!assert(fzero(@(x) x^(1/3) - 1e-8, [0,1], opt0), 1e-24, 1e-22*eps) |