Mercurial > hg > octave-lyh
annotate scripts/optimization/fzero.m @ 10297:ed88ea036716
improve docs of fzero/fminbnd
| author | Jaroslav Hajek <highegg@gmail.com> |
|---|---|
| date | Wed, 10 Feb 2010 16:15:30 +0100 |
| parents | 035ac548a67e |
| children | 7c97da90fc8f |
| 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 |
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24 ## handle or name. @var{x0} specifies a starting point. @var{options} is a |
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25 ## structure specifying additional options. Currently, @code{fzero} |
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26 ## recognizes these options: @code{"FunValCheck"}, @code{"OutputFcn"}, |
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27 ## @code{"TolX"}, @code{"MaxIter"}, @code{"MaxFunEvals"}. |
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28 ## For description of these options, see @ref{doc-optimset,,optimset}. |
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29 ## |
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30 ## On exit, the function returns @var{x}, the approximate zero point |
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31 ## and @var{fval}, the function value thereof. |
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32 ## @var{info} is an exit flag that can have these values: |
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33 ## |
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34 ## @itemize |
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35 ## @item 1 |
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36 ## The algorithm converged to a solution. |
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37 ## @item 0 |
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38 ## Maximum number of iterations or function evaluations has been exhausted. |
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39 ## @item -1 |
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40 ## The algorithm has been terminated from user output function. |
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41 ## @end itemize |
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42 ## @seealso{optimset, fsolve} |
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43 ## @end deftypefn |
| 8305 | 44 |
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45 ## This is essentially the ACM algorithm 748: Enclosing Zeros of |
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46 ## Continuous Functions due to Alefeld, Potra and Shi, ACM Transactions |
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47 ## on Mathematical Software, Vol. 21, No. 3, September 1995. Although |
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48 ## the workflow should be the same, the structure of the algorithm has |
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49 ## been transformed non-trivially; instead of the authors' approach of |
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50 ## sequentially calling building blocks subprograms we implement here a |
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51 ## FSM version using one interior point determination and one bracketing |
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52 ## per iteration, thus reducing the number of temporary variables and |
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53 ## simplifying the algorithm structure. Further, this approach reduces |
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54 ## the need for external functions and error handling. The algorithm has |
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55 ## also been slightly modified. |
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56 |
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57 ## PKG_ADD: __all_opts__ ("fzero"); |
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58 |
| 8305 | 59 function [x, fval, info, output] = fzero (fun, x0, options = struct ()) |
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60 |
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61 ## Get default options if requested. |
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62 if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults')) |
| 10296 | 63 x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 1e-8, \ |
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64 "OutputFcn", [], "FunValCheck", "off"); |
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65 return; |
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66 endif |
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67 |
| 8305 | 68 if (nargin < 2 || nargin > 3) |
| 69 print_usage (); | |
| 70 endif | |
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71 |
| 8305 | 72 if (ischar (fun)) |
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73 fun = str2func (fun, "global"); |
| 8305 | 74 endif |
| 75 | |
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76 ## TODO |
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77 ## displev = optimget (options, "Display", "notify"); |
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78 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
| 8305 | 79 outfcn = optimget (options, "OutputFcn"); |
| 10296 | 80 tolx = optimget (options, "TolX", 1e-8); |
| 8305 | 81 maxiter = optimget (options, "MaxIter", Inf); |
| 82 maxfev = optimget (options, "MaxFunEvals", Inf); | |
| 83 | |
| 84 persistent mu = 0.5; | |
| 85 | |
| 86 if (funvalchk) | |
| 8507 | 87 ## Replace fun with a guarded version. |
| 8305 | 88 fun = @(x) guarded_eval (fun, x); |
| 89 endif | |
| 90 | |
| 8507 | 91 ## The default exit flag if exceeded number of iterations. |
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92 info = 0; |
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93 niter = 0; |
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94 nfev = 0; |
| 8305 | 95 |
| 96 x = fval = a = fa = b = fb = NaN; | |
| 10296 | 97 eps = eps (class (x0)); |
| 8305 | 98 |
| 8507 | 99 ## Prepare... |
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100 a = x0(1); |
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101 fa = fun (a); |
| 8305 | 102 nfev = 1; |
| 103 if (length (x0) > 1) | |
| 104 b = x0(2); | |
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105 fb = fun (b); |
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106 nfev += 1; |
| 8305 | 107 else |
| 8507 | 108 ## Try to get b. |
| 8305 | 109 if (a == 0) |
| 110 aa = 1; | |
| 111 else | |
| 112 aa = a; | |
| 113 endif | |
| 114 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] | |
| 115 fb = fun (b); nfev += 1; | |
| 116 if (sign (fa) * sign (fb) <= 0) | |
| 117 break; | |
| 118 endif | |
| 119 endfor | |
| 120 endif | |
| 121 | |
| 122 if (b < a) | |
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123 u = a; |
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124 a = b; |
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125 b = u; |
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126 |
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127 fu = fa; |
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128 fa = fb; |
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129 fb = fu; |
| 8305 | 130 endif |
| 131 | |
| 132 if (! (sign (fa) * sign (fb) <= 0)) | |
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133 error ("fzero:bracket", "fzero: not a valid initial bracketing"); |
| 8305 | 134 endif |
| 135 | |
| 136 itype = 1; | |
| 137 | |
| 138 if (abs (fa) < abs (fb)) | |
| 139 u = a; fu = fa; | |
| 140 else | |
| 141 u = b; fu = fb; | |
| 142 endif | |
| 143 | |
| 144 d = e = u; | |
| 145 fd = fe = fu; | |
| 146 mba = mu*(b - a); | |
| 147 while (niter < maxiter && nfev < maxfev) | |
| 148 switch (itype) | |
| 149 case 1 | |
| 8507 | 150 ## The initial test. |
| 8305 | 151 if (b - a <= 2*(2 * abs (u) * eps + tolx)) |
| 152 x = u; fval = fu; | |
| 153 info = 1; | |
| 154 break; | |
| 155 endif | |
| 156 if (abs (fa) <= 1e3*abs (fb) && abs (fb) <= 1e3*abs (fa)) | |
| 8507 | 157 ## Secant step. |
| 8305 | 158 c = u - (a - b) / (fa - fb) * fu; |
| 159 else | |
| 8507 | 160 ## Bisection step. |
| 8305 | 161 c = 0.5*(a + b); |
| 162 endif | |
| 163 d = u; fd = fu; | |
| 164 itype = 5; | |
| 165 case {2, 3} | |
| 166 l = length (unique ([fa, fb, fd, fe])); | |
| 167 if (l == 4) | |
| 8507 | 168 ## Inverse cubic interpolation. |
| 8305 | 169 q11 = (d - e) * fd / (fe - fd); |
| 170 q21 = (b - d) * fb / (fd - fb); | |
| 171 q31 = (a - b) * fa / (fb - fa); | |
| 172 d21 = (b - d) * fd / (fd - fb); | |
| 173 d31 = (a - b) * fb / (fb - fa); | |
| 174 q22 = (d21 - q11) * fb / (fe - fb); | |
| 175 q32 = (d31 - q21) * fa / (fd - fa); | |
| 176 d32 = (d31 - q21) * fd / (fd - fa); | |
| 177 q33 = (d32 - q22) * fa / (fe - fa); | |
| 178 c = a + q31 + q32 + q33; | |
| 179 endif | |
| 180 if (l < 4 || sign (c - a) * sign (c - b) > 0) | |
| 8507 | 181 ## Quadratic interpolation + newton. |
| 8305 | 182 a0 = fa; |
| 183 a1 = (fb - fa)/(b - a); | |
| 184 a2 = ((fd - fb)/(d - b) - a1) / (d - a); | |
| 8507 | 185 ## Modification 1: this is simpler and does not seem to be worse. |
| 8305 | 186 c = a - a0/a1; |
| 187 if (a2 != 0) | |
| 188 c = a - a0/a1; | |
| 189 for i = 1:itype | |
| 190 pc = a0 + (a1 + a2*(c - b))*(c - a); | |
| 191 pdc = a1 + a2*(2*c - a - b); | |
| 192 if (pdc == 0) | |
| 193 c = a - a0/a1; | |
| 194 break; | |
| 195 endif | |
| 196 c -= pc/pdc; | |
| 197 endfor | |
| 198 endif | |
| 199 endif | |
| 200 itype += 1; | |
| 201 case 4 | |
| 8507 | 202 ## Double secant step. |
| 8305 | 203 c = u - 2*(b - a)/(fb - fa)*fu; |
| 8507 | 204 ## Bisect if too far. |
| 8305 | 205 if (abs (c - u) > 0.5*(b - a)) |
| 206 c = 0.5 * (b + a); | |
| 207 endif | |
| 208 itype = 5; | |
| 209 case 5 | |
| 8507 | 210 ## Bisection step. |
| 8305 | 211 c = 0.5 * (b + a); |
| 212 itype = 2; | |
| 213 endswitch | |
| 214 | |
| 8507 | 215 ## Don't let c come too close to a or b. |
| 8305 | 216 delta = 2*0.7*(2 * abs (u) * eps + tolx); |
| 217 if ((b - a) <= 2*delta) | |
| 218 c = (a + b)/2; | |
| 219 else | |
| 220 c = max (a + delta, min (b - delta, c)); | |
| 221 endif | |
| 222 | |
| 8507 | 223 ## Calculate new point. |
| 8305 | 224 x = c; |
| 225 fval = fc = fun (c); | |
| 226 niter ++; nfev ++; | |
| 227 | |
| 8507 | 228 ## Modification 2: skip inverse cubic interpolation if |
| 229 ## nonmonotonicity is detected. | |
| 8305 | 230 if (sign (fc - fa) * sign (fc - fb) >= 0) |
| 8507 | 231 ## The new point broke monotonicity. |
| 232 ## Disable inverse cubic. | |
| 8305 | 233 fe = fc; |
| 234 else | |
| 235 e = d; fe = fd; | |
| 236 endif | |
| 237 | |
| 8507 | 238 ## Bracketing. |
| 8305 | 239 if (sign (fa) * sign (fc) < 0) |
| 240 d = b; fd = fb; | |
| 241 b = c; fb = fc; | |
| 242 elseif (sign (fb) * sign (fc) < 0) | |
| 243 d = a; fd = fa; | |
| 244 a = c; fa = fc; | |
| 245 elseif (fc == 0) | |
| 246 a = b = c; fa = fb = fc; | |
| 247 info = 1; | |
| 248 break; | |
| 249 else | |
| 8507 | 250 ## This should never happen. |
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251 error ("fzero:bracket", "fzero: zero point is not bracketed"); |
| 8305 | 252 endif |
| 253 | |
| 8507 | 254 ## If there's an output function, use it now. |
| 8305 | 255 if (outfcn) |
| 10296 | 256 optv.funccount = nfev; |
| 8305 | 257 optv.fval = fval; |
| 258 optv.iteration = niter; | |
| 259 if (outfcn (x, optv, "iter")) | |
| 260 info = -1; | |
| 261 break; | |
| 262 endif | |
| 263 endif | |
| 264 | |
| 265 if (abs (fa) < abs (fb)) | |
| 266 u = a; fu = fa; | |
| 267 else | |
| 268 u = b; fu = fb; | |
| 269 endif | |
| 270 if (b - a <= 2*(2 * abs (u) * eps + tolx)) | |
| 271 info = 1; | |
| 272 break; | |
| 273 endif | |
| 274 | |
| 8507 | 275 ## Skip bisection step if successful reduction. |
| 8305 | 276 if (itype == 5 && (b - a) <= mba) |
| 277 itype = 2; | |
| 278 endif | |
| 279 if (itype == 2) | |
| 280 mba = mu * (b - a); | |
| 281 endif | |
| 282 endwhile | |
| 283 | |
| 284 output.iterations = niter; | |
| 10296 | 285 output.funcCount = nfev; |
| 8305 | 286 output.bracket = [a, b]; |
| 287 output.bracketf = [fa, fb]; | |
| 288 | |
| 289 endfunction | |
| 290 | |
| 8507 | 291 ## An assistant function that evaluates a function handle and checks for |
|
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fsolve.m, fzero.m: style fixes; use strcmpi to compare options
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parents:
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diff
changeset
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292 ## bad results. |
| 8305 | 293 function fx = guarded_eval (fun, x) |
| 294 fx = fun (x); | |
| 295 fx = fx(1); | |
| 296 if (! isreal (fx)) | |
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John W. Eaton <jwe@octave.org>
parents:
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changeset
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297 error ("fzero:notreal", "fzero: non-real value encountered"); |
| 8305 | 298 elseif (isnan (fx)) |
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fsolve.m, fzero.m: undo part of previous change
John W. Eaton <jwe@octave.org>
parents:
8467
diff
changeset
|
299 error ("fzero:isnan", "fzero: NaN value encountered"); |
| 8305 | 300 endif |
| 301 endfunction | |
| 302 | |
| 10296 | 303 %!shared opt0 |
| 304 %! opt0 = optimset ("tolx", 0); | |
| 305 %!assert(fzero(@cos, [0, 3], opt0), pi/2, 10*eps) | |
| 306 %!assert(fzero(@(x) x^(1/3) - 1e-8, [0,1], opt0), 1e-24, 1e-22*eps) |