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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Sat, 07 Mar 2009 10:41:27 -0500 |
| parents | ff61b53eb294 |
| children | 1bf0ce0930be |
| 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 ## @itemize |
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34 ## @item 1 |
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35 ## The algorithm converged to a solution. |
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36 ## @item 0 |
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37 ## Maximum number of iterations or function evaluations has been exhausted. |
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38 ## @item -1 |
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39 ## The algorithm has been terminated from user output function. |
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40 ## @item -2 |
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41 ## A general unexpected error. |
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42 ## @item -3 |
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43 ## A non-real value encountered. |
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44 ## @item -4 |
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45 ## A NaN value encountered. |
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46 ## @end itemize |
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47 ## @seealso{optimset, fsolve} |
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48 ## @end deftypefn |
| 8305 | 49 |
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50 ## This is essentially the ACM algorithm 748: Enclosing Zeros of |
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51 ## Continuous Functions due to Alefeld, Potra and Shi, ACM Transactions |
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52 ## on Mathematical Software, Vol. 21, No. 3, September 1995. Although |
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53 ## the workflow should be the same, the structure of the algorithm has |
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54 ## been transformed non-trivially; instead of the authors' approach of |
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55 ## sequentially calling building blocks subprograms we implement here a |
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56 ## FSM version using one interior point determination and one bracketing |
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57 ## per iteration, thus reducing the number of temporary variables and |
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58 ## simplifying the algorithm structure. Further, this approach reduces |
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59 ## the need for external functions and error handling. The algorithm has |
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60 ## also been slightly modified. |
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61 |
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62 ## PKG_ADD: __all_opts__ ("fzero"); |
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63 |
| 8305 | 64 function [x, fval, info, output] = fzero (fun, x0, options = struct ()) |
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65 |
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66 ## Get default options if requested. |
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67 if (nargin == 1 && ischar (fun) && strcmp (fun, 'defaults')) |
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68 x = optimset ("MaxIter", Inf, "MaxFunEvals", Inf, "TolX", 0, \ |
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69 "OutputFcn", [], "FunValCheck", "off"); |
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70 return; |
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71 endif |
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72 |
| 8305 | 73 if (nargin < 2 || nargin > 3) |
| 74 print_usage (); | |
| 75 endif | |
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76 |
| 8305 | 77 if (ischar (fun)) |
| 78 fun = str2func (fun); | |
| 79 endif | |
| 80 | |
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81 ## TODO |
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82 ## displev = optimget (options, "Display", "notify"); |
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83 funvalchk = strcmpi (optimget (options, "FunValCheck", "off"), "on"); |
| 8305 | 84 outfcn = optimget (options, "OutputFcn"); |
| 85 tolx = optimget (options, "TolX", 0); | |
| 86 maxiter = optimget (options, "MaxIter", Inf); | |
| 87 maxfev = optimget (options, "MaxFunEvals", Inf); | |
| 88 | |
| 89 persistent mu = 0.5; | |
| 90 | |
| 91 if (funvalchk) | |
| 8507 | 92 ## Replace fun with a guarded version. |
| 8305 | 93 fun = @(x) guarded_eval (fun, x); |
| 94 endif | |
| 95 | |
| 8507 | 96 ## The default exit flag if exceeded number of iterations. |
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97 info = 0; |
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98 niter = 0; |
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99 nfev = 0; |
| 8305 | 100 |
| 101 x = fval = a = fa = b = fb = NaN; | |
| 102 | |
| 8507 | 103 ## Prepare... |
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104 a = x0(1); |
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105 fa = fun (a); |
| 8305 | 106 nfev = 1; |
| 107 if (length (x0) > 1) | |
| 108 b = x0(2); | |
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109 fb = fun (b); |
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110 nfev += 1; |
| 8305 | 111 else |
| 8507 | 112 ## Try to get b. |
| 8305 | 113 if (a == 0) |
| 114 aa = 1; | |
| 115 else | |
| 116 aa = a; | |
| 117 endif | |
| 118 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] | |
| 119 fb = fun (b); nfev += 1; | |
| 120 if (sign (fa) * sign (fb) <= 0) | |
| 121 break; | |
| 122 endif | |
| 123 endfor | |
| 124 endif | |
| 125 | |
| 126 if (b < a) | |
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127 u = a; |
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128 a = b; |
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129 b = u; |
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130 |
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131 fu = fa; |
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132 fa = fb; |
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133 fb = fu; |
| 8305 | 134 endif |
| 135 | |
| 136 if (! (sign (fa) * sign (fb) <= 0)) | |
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137 error ("fzero:bracket", "fzero: not a valid initial bracketing"); |
| 8305 | 138 endif |
| 139 | |
| 140 itype = 1; | |
| 141 | |
| 142 if (abs (fa) < abs (fb)) | |
| 143 u = a; fu = fa; | |
| 144 else | |
| 145 u = b; fu = fb; | |
| 146 endif | |
| 147 | |
| 148 d = e = u; | |
| 149 fd = fe = fu; | |
| 150 mba = mu*(b - a); | |
| 151 while (niter < maxiter && nfev < maxfev) | |
| 152 switch (itype) | |
| 153 case 1 | |
| 8507 | 154 ## The initial test. |
| 8305 | 155 if (b - a <= 2*(2 * abs (u) * eps + tolx)) |
| 156 x = u; fval = fu; | |
| 157 info = 1; | |
| 158 break; | |
| 159 endif | |
| 160 if (abs (fa) <= 1e3*abs (fb) && abs (fb) <= 1e3*abs (fa)) | |
| 8507 | 161 ## Secant step. |
| 8305 | 162 c = u - (a - b) / (fa - fb) * fu; |
| 163 else | |
| 8507 | 164 ## Bisection step. |
| 8305 | 165 c = 0.5*(a + b); |
| 166 endif | |
| 167 d = u; fd = fu; | |
| 168 itype = 5; | |
| 169 case {2, 3} | |
| 170 l = length (unique ([fa, fb, fd, fe])); | |
| 171 if (l == 4) | |
| 8507 | 172 ## Inverse cubic interpolation. |
| 8305 | 173 q11 = (d - e) * fd / (fe - fd); |
| 174 q21 = (b - d) * fb / (fd - fb); | |
| 175 q31 = (a - b) * fa / (fb - fa); | |
| 176 d21 = (b - d) * fd / (fd - fb); | |
| 177 d31 = (a - b) * fb / (fb - fa); | |
| 178 q22 = (d21 - q11) * fb / (fe - fb); | |
| 179 q32 = (d31 - q21) * fa / (fd - fa); | |
| 180 d32 = (d31 - q21) * fd / (fd - fa); | |
| 181 q33 = (d32 - q22) * fa / (fe - fa); | |
| 182 c = a + q31 + q32 + q33; | |
| 183 endif | |
| 184 if (l < 4 || sign (c - a) * sign (c - b) > 0) | |
| 8507 | 185 ## Quadratic interpolation + newton. |
| 8305 | 186 a0 = fa; |
| 187 a1 = (fb - fa)/(b - a); | |
| 188 a2 = ((fd - fb)/(d - b) - a1) / (d - a); | |
| 8507 | 189 ## Modification 1: this is simpler and does not seem to be worse. |
| 8305 | 190 c = a - a0/a1; |
| 191 if (a2 != 0) | |
| 192 c = a - a0/a1; | |
| 193 for i = 1:itype | |
| 194 pc = a0 + (a1 + a2*(c - b))*(c - a); | |
| 195 pdc = a1 + a2*(2*c - a - b); | |
| 196 if (pdc == 0) | |
| 197 c = a - a0/a1; | |
| 198 break; | |
| 199 endif | |
| 200 c -= pc/pdc; | |
| 201 endfor | |
| 202 endif | |
| 203 endif | |
| 204 itype += 1; | |
| 205 case 4 | |
| 8507 | 206 ## Double secant step. |
| 8305 | 207 c = u - 2*(b - a)/(fb - fa)*fu; |
| 8507 | 208 ## Bisect if too far. |
| 8305 | 209 if (abs (c - u) > 0.5*(b - a)) |
| 210 c = 0.5 * (b + a); | |
| 211 endif | |
| 212 itype = 5; | |
| 213 case 5 | |
| 8507 | 214 ## Bisection step. |
| 8305 | 215 c = 0.5 * (b + a); |
| 216 itype = 2; | |
| 217 endswitch | |
| 218 | |
| 8507 | 219 ## Don't let c come too close to a or b. |
| 8305 | 220 delta = 2*0.7*(2 * abs (u) * eps + tolx); |
| 221 if ((b - a) <= 2*delta) | |
| 222 c = (a + b)/2; | |
| 223 else | |
| 224 c = max (a + delta, min (b - delta, c)); | |
| 225 endif | |
| 226 | |
| 8507 | 227 ## Calculate new point. |
| 8305 | 228 x = c; |
| 229 fval = fc = fun (c); | |
| 230 niter ++; nfev ++; | |
| 231 | |
| 8507 | 232 ## Modification 2: skip inverse cubic interpolation if |
| 233 ## nonmonotonicity is detected. | |
| 8305 | 234 if (sign (fc - fa) * sign (fc - fb) >= 0) |
| 8507 | 235 ## The new point broke monotonicity. |
| 236 ## Disable inverse cubic. | |
| 8305 | 237 fe = fc; |
| 238 else | |
| 239 e = d; fe = fd; | |
| 240 endif | |
| 241 | |
| 8507 | 242 ## Bracketing. |
| 8305 | 243 if (sign (fa) * sign (fc) < 0) |
| 244 d = b; fd = fb; | |
| 245 b = c; fb = fc; | |
| 246 elseif (sign (fb) * sign (fc) < 0) | |
| 247 d = a; fd = fa; | |
| 248 a = c; fa = fc; | |
| 249 elseif (fc == 0) | |
| 250 a = b = c; fa = fb = fc; | |
| 251 info = 1; | |
| 252 break; | |
| 253 else | |
| 8507 | 254 ## This should never happen. |
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255 error ("fzero:bracket", "fzero: zero point is not bracketed"); |
| 8305 | 256 endif |
| 257 | |
| 8507 | 258 ## If there's an output function, use it now. |
| 8305 | 259 if (outfcn) |
| 260 optv.funccount = niter + 2; | |
| 261 optv.fval = fval; | |
| 262 optv.iteration = niter; | |
| 263 if (outfcn (x, optv, "iter")) | |
| 264 info = -1; | |
| 265 break; | |
| 266 endif | |
| 267 endif | |
| 268 | |
| 269 if (abs (fa) < abs (fb)) | |
| 270 u = a; fu = fa; | |
| 271 else | |
| 272 u = b; fu = fb; | |
| 273 endif | |
| 274 if (b - a <= 2*(2 * abs (u) * eps + tolx)) | |
| 275 info = 1; | |
| 276 break; | |
| 277 endif | |
| 278 | |
| 8507 | 279 ## Skip bisection step if successful reduction. |
| 8305 | 280 if (itype == 5 && (b - a) <= mba) |
| 281 itype = 2; | |
| 282 endif | |
| 283 if (itype == 2) | |
| 284 mba = mu * (b - a); | |
| 285 endif | |
| 286 endwhile | |
| 287 | |
| 288 output.iterations = niter; | |
| 289 output.funcCount = niter + 2; | |
| 290 output.bracket = [a, b]; | |
| 291 output.bracketf = [fa, fb]; | |
| 292 | |
| 293 endfunction | |
| 294 | |
| 8507 | 295 ## An assistant function that evaluates a function handle and checks for |
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296 ## bad results. |
| 8305 | 297 function fx = guarded_eval (fun, x) |
| 298 fx = fun (x); | |
| 299 fx = fx(1); | |
| 300 if (! isreal (fx)) | |
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parents:
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changeset
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301 error ("fzero:notreal", "fzero: non-real value encountered"); |
| 8305 | 302 elseif (isnan (fx)) |
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John W. Eaton <jwe@octave.org>
parents:
8467
diff
changeset
|
303 error ("fzero:isnan", "fzero: NaN value encountered"); |
| 8305 | 304 endif |
| 305 endfunction | |
| 306 | |
| 307 %!assert(fzero(@cos, [0, 3]), pi/2, 10*eps) | |
| 308 %!assert(fzero(@(x) x^(1/3) - 1e-8, [0,1]), 1e-24, 1e-22*eps) |