Raymond | dee0849 | 2015-04-02 10:43:13 -0700 | [diff] [blame] | 1 | /* |
| 2 | * Licensed to the Apache Software Foundation (ASF) under one or more |
| 3 | * contributor license agreements. See the NOTICE file distributed with |
| 4 | * this work for additional information regarding copyright ownership. |
| 5 | * The ASF licenses this file to You under the Apache License, Version 2.0 |
| 6 | * (the "License"); you may not use this file except in compliance with |
| 7 | * the License. You may obtain a copy of the License at |
| 8 | * |
| 9 | * http://www.apache.org/licenses/LICENSE-2.0 |
| 10 | * |
| 11 | * Unless required by applicable law or agreed to in writing, software |
| 12 | * distributed under the License is distributed on an "AS IS" BASIS, |
| 13 | * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| 14 | * See the License for the specific language governing permissions and |
| 15 | * limitations under the License. |
| 16 | */ |
| 17 | package org.apache.commons.math.analysis.solvers; |
| 18 | |
| 19 | import org.apache.commons.math.ConvergenceException; |
| 20 | import org.apache.commons.math.FunctionEvaluationException; |
| 21 | import org.apache.commons.math.MaxIterationsExceededException; |
| 22 | import org.apache.commons.math.analysis.UnivariateRealFunction; |
| 23 | import org.apache.commons.math.util.FastMath; |
| 24 | import org.apache.commons.math.util.MathUtils; |
| 25 | |
| 26 | /** |
| 27 | * Implements the <a href="http://mathworld.wolfram.com/MullersMethod.html"> |
| 28 | * Muller's Method</a> for root finding of real univariate functions. For |
| 29 | * reference, see <b>Elementary Numerical Analysis</b>, ISBN 0070124477, |
| 30 | * chapter 3. |
| 31 | * <p> |
| 32 | * Muller's method applies to both real and complex functions, but here we |
| 33 | * restrict ourselves to real functions. Methods solve() and solve2() find |
| 34 | * real zeros, using different ways to bypass complex arithmetics.</p> |
| 35 | * |
| 36 | * @version $Revision: 1070725 $ $Date: 2011-02-15 02:31:12 +0100 (mar. 15 févr. 2011) $ |
| 37 | * @since 1.2 |
| 38 | */ |
| 39 | public class MullerSolver extends UnivariateRealSolverImpl { |
| 40 | |
| 41 | /** |
| 42 | * Construct a solver for the given function. |
| 43 | * |
| 44 | * @param f function to solve |
| 45 | * @deprecated as of 2.0 the function to solve is passed as an argument |
| 46 | * to the {@link #solve(UnivariateRealFunction, double, double)} or |
| 47 | * {@link UnivariateRealSolverImpl#solve(UnivariateRealFunction, double, double, double)} |
| 48 | * method. |
| 49 | */ |
| 50 | @Deprecated |
| 51 | public MullerSolver(UnivariateRealFunction f) { |
| 52 | super(f, 100, 1E-6); |
| 53 | } |
| 54 | |
| 55 | /** |
| 56 | * Construct a solver. |
| 57 | * @deprecated in 2.2 (to be removed in 3.0). |
| 58 | */ |
| 59 | @Deprecated |
| 60 | public MullerSolver() { |
| 61 | super(100, 1E-6); |
| 62 | } |
| 63 | |
| 64 | /** {@inheritDoc} */ |
| 65 | @Deprecated |
| 66 | public double solve(final double min, final double max) |
| 67 | throws ConvergenceException, FunctionEvaluationException { |
| 68 | return solve(f, min, max); |
| 69 | } |
| 70 | |
| 71 | /** {@inheritDoc} */ |
| 72 | @Deprecated |
| 73 | public double solve(final double min, final double max, final double initial) |
| 74 | throws ConvergenceException, FunctionEvaluationException { |
| 75 | return solve(f, min, max, initial); |
| 76 | } |
| 77 | |
| 78 | /** |
| 79 | * Find a real root in the given interval with initial value. |
| 80 | * <p> |
| 81 | * Requires bracketing condition.</p> |
| 82 | * |
| 83 | * @param f the function to solve |
| 84 | * @param min the lower bound for the interval |
| 85 | * @param max the upper bound for the interval |
| 86 | * @param initial the start value to use |
| 87 | * @param maxEval Maximum number of evaluations. |
| 88 | * @return the point at which the function value is zero |
| 89 | * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| 90 | * or the solver detects convergence problems otherwise |
| 91 | * @throws FunctionEvaluationException if an error occurs evaluating the function |
| 92 | * @throws IllegalArgumentException if any parameters are invalid |
| 93 | */ |
| 94 | @Override |
| 95 | public double solve(int maxEval, final UnivariateRealFunction f, |
| 96 | final double min, final double max, final double initial) |
| 97 | throws MaxIterationsExceededException, FunctionEvaluationException { |
| 98 | setMaximalIterationCount(maxEval); |
| 99 | return solve(f, min, max, initial); |
| 100 | } |
| 101 | |
| 102 | /** |
| 103 | * Find a real root in the given interval with initial value. |
| 104 | * <p> |
| 105 | * Requires bracketing condition.</p> |
| 106 | * |
| 107 | * @param f the function to solve |
| 108 | * @param min the lower bound for the interval |
| 109 | * @param max the upper bound for the interval |
| 110 | * @param initial the start value to use |
| 111 | * @return the point at which the function value is zero |
| 112 | * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| 113 | * or the solver detects convergence problems otherwise |
| 114 | * @throws FunctionEvaluationException if an error occurs evaluating the function |
| 115 | * @throws IllegalArgumentException if any parameters are invalid |
| 116 | * @deprecated in 2.2 (to be removed in 3.0). |
| 117 | */ |
| 118 | @Deprecated |
| 119 | public double solve(final UnivariateRealFunction f, |
| 120 | final double min, final double max, final double initial) |
| 121 | throws MaxIterationsExceededException, FunctionEvaluationException { |
| 122 | |
| 123 | // check for zeros before verifying bracketing |
| 124 | if (f.value(min) == 0.0) { return min; } |
| 125 | if (f.value(max) == 0.0) { return max; } |
| 126 | if (f.value(initial) == 0.0) { return initial; } |
| 127 | |
| 128 | verifyBracketing(min, max, f); |
| 129 | verifySequence(min, initial, max); |
| 130 | if (isBracketing(min, initial, f)) { |
| 131 | return solve(f, min, initial); |
| 132 | } else { |
| 133 | return solve(f, initial, max); |
| 134 | } |
| 135 | } |
| 136 | |
| 137 | /** |
| 138 | * Find a real root in the given interval. |
| 139 | * <p> |
| 140 | * Original Muller's method would have function evaluation at complex point. |
| 141 | * Since our f(x) is real, we have to find ways to avoid that. Bracketing |
| 142 | * condition is one way to go: by requiring bracketing in every iteration, |
| 143 | * the newly computed approximation is guaranteed to be real.</p> |
| 144 | * <p> |
| 145 | * Normally Muller's method converges quadratically in the vicinity of a |
| 146 | * zero, however it may be very slow in regions far away from zeros. For |
| 147 | * example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use |
| 148 | * bisection as a safety backup if it performs very poorly.</p> |
| 149 | * <p> |
| 150 | * The formulas here use divided differences directly.</p> |
| 151 | * |
| 152 | * @param f the function to solve |
| 153 | * @param min the lower bound for the interval |
| 154 | * @param max the upper bound for the interval |
| 155 | * @param maxEval Maximum number of evaluations. |
| 156 | * @return the point at which the function value is zero |
| 157 | * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| 158 | * or the solver detects convergence problems otherwise |
| 159 | * @throws FunctionEvaluationException if an error occurs evaluating the function |
| 160 | * @throws IllegalArgumentException if any parameters are invalid |
| 161 | */ |
| 162 | @Override |
| 163 | public double solve(int maxEval, final UnivariateRealFunction f, |
| 164 | final double min, final double max) |
| 165 | throws MaxIterationsExceededException, FunctionEvaluationException { |
| 166 | setMaximalIterationCount(maxEval); |
| 167 | return solve(f, min, max); |
| 168 | } |
| 169 | |
| 170 | /** |
| 171 | * Find a real root in the given interval. |
| 172 | * <p> |
| 173 | * Original Muller's method would have function evaluation at complex point. |
| 174 | * Since our f(x) is real, we have to find ways to avoid that. Bracketing |
| 175 | * condition is one way to go: by requiring bracketing in every iteration, |
| 176 | * the newly computed approximation is guaranteed to be real.</p> |
| 177 | * <p> |
| 178 | * Normally Muller's method converges quadratically in the vicinity of a |
| 179 | * zero, however it may be very slow in regions far away from zeros. For |
| 180 | * example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use |
| 181 | * bisection as a safety backup if it performs very poorly.</p> |
| 182 | * <p> |
| 183 | * The formulas here use divided differences directly.</p> |
| 184 | * |
| 185 | * @param f the function to solve |
| 186 | * @param min the lower bound for the interval |
| 187 | * @param max the upper bound for the interval |
| 188 | * @return the point at which the function value is zero |
| 189 | * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| 190 | * or the solver detects convergence problems otherwise |
| 191 | * @throws FunctionEvaluationException if an error occurs evaluating the function |
| 192 | * @throws IllegalArgumentException if any parameters are invalid |
| 193 | * @deprecated in 2.2 (to be removed in 3.0). |
| 194 | */ |
| 195 | @Deprecated |
| 196 | public double solve(final UnivariateRealFunction f, |
| 197 | final double min, final double max) |
| 198 | throws MaxIterationsExceededException, FunctionEvaluationException { |
| 199 | |
| 200 | // [x0, x2] is the bracketing interval in each iteration |
| 201 | // x1 is the last approximation and an interpolation point in (x0, x2) |
| 202 | // x is the new root approximation and new x1 for next round |
| 203 | // d01, d12, d012 are divided differences |
| 204 | |
| 205 | double x0 = min; |
| 206 | double y0 = f.value(x0); |
| 207 | double x2 = max; |
| 208 | double y2 = f.value(x2); |
| 209 | double x1 = 0.5 * (x0 + x2); |
| 210 | double y1 = f.value(x1); |
| 211 | |
| 212 | // check for zeros before verifying bracketing |
| 213 | if (y0 == 0.0) { |
| 214 | return min; |
| 215 | } |
| 216 | if (y2 == 0.0) { |
| 217 | return max; |
| 218 | } |
| 219 | verifyBracketing(min, max, f); |
| 220 | |
| 221 | double oldx = Double.POSITIVE_INFINITY; |
| 222 | for (int i = 1; i <= maximalIterationCount; ++i) { |
| 223 | // Muller's method employs quadratic interpolation through |
| 224 | // x0, x1, x2 and x is the zero of the interpolating parabola. |
| 225 | // Due to bracketing condition, this parabola must have two |
| 226 | // real roots and we choose one in [x0, x2] to be x. |
| 227 | final double d01 = (y1 - y0) / (x1 - x0); |
| 228 | final double d12 = (y2 - y1) / (x2 - x1); |
| 229 | final double d012 = (d12 - d01) / (x2 - x0); |
| 230 | final double c1 = d01 + (x1 - x0) * d012; |
| 231 | final double delta = c1 * c1 - 4 * y1 * d012; |
| 232 | final double xplus = x1 + (-2.0 * y1) / (c1 + FastMath.sqrt(delta)); |
| 233 | final double xminus = x1 + (-2.0 * y1) / (c1 - FastMath.sqrt(delta)); |
| 234 | // xplus and xminus are two roots of parabola and at least |
| 235 | // one of them should lie in (x0, x2) |
| 236 | final double x = isSequence(x0, xplus, x2) ? xplus : xminus; |
| 237 | final double y = f.value(x); |
| 238 | |
| 239 | // check for convergence |
| 240 | final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy); |
| 241 | if (FastMath.abs(x - oldx) <= tolerance) { |
| 242 | setResult(x, i); |
| 243 | return result; |
| 244 | } |
| 245 | if (FastMath.abs(y) <= functionValueAccuracy) { |
| 246 | setResult(x, i); |
| 247 | return result; |
| 248 | } |
| 249 | |
| 250 | // Bisect if convergence is too slow. Bisection would waste |
| 251 | // our calculation of x, hopefully it won't happen often. |
| 252 | // the real number equality test x == x1 is intentional and |
| 253 | // completes the proximity tests above it |
| 254 | boolean bisect = (x < x1 && (x1 - x0) > 0.95 * (x2 - x0)) || |
| 255 | (x > x1 && (x2 - x1) > 0.95 * (x2 - x0)) || |
| 256 | (x == x1); |
| 257 | // prepare the new bracketing interval for next iteration |
| 258 | if (!bisect) { |
| 259 | x0 = x < x1 ? x0 : x1; |
| 260 | y0 = x < x1 ? y0 : y1; |
| 261 | x2 = x > x1 ? x2 : x1; |
| 262 | y2 = x > x1 ? y2 : y1; |
| 263 | x1 = x; y1 = y; |
| 264 | oldx = x; |
| 265 | } else { |
| 266 | double xm = 0.5 * (x0 + x2); |
| 267 | double ym = f.value(xm); |
| 268 | if (MathUtils.sign(y0) + MathUtils.sign(ym) == 0.0) { |
| 269 | x2 = xm; y2 = ym; |
| 270 | } else { |
| 271 | x0 = xm; y0 = ym; |
| 272 | } |
| 273 | x1 = 0.5 * (x0 + x2); |
| 274 | y1 = f.value(x1); |
| 275 | oldx = Double.POSITIVE_INFINITY; |
| 276 | } |
| 277 | } |
| 278 | throw new MaxIterationsExceededException(maximalIterationCount); |
| 279 | } |
| 280 | |
| 281 | /** |
| 282 | * Find a real root in the given interval. |
| 283 | * <p> |
| 284 | * solve2() differs from solve() in the way it avoids complex operations. |
| 285 | * Except for the initial [min, max], solve2() does not require bracketing |
| 286 | * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex |
| 287 | * number arises in the computation, we simply use its modulus as real |
| 288 | * approximation.</p> |
| 289 | * <p> |
| 290 | * Because the interval may not be bracketing, bisection alternative is |
| 291 | * not applicable here. However in practice our treatment usually works |
| 292 | * well, especially near real zeros where the imaginary part of complex |
| 293 | * approximation is often negligible.</p> |
| 294 | * <p> |
| 295 | * The formulas here do not use divided differences directly.</p> |
| 296 | * |
| 297 | * @param min the lower bound for the interval |
| 298 | * @param max the upper bound for the interval |
| 299 | * @return the point at which the function value is zero |
| 300 | * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| 301 | * or the solver detects convergence problems otherwise |
| 302 | * @throws FunctionEvaluationException if an error occurs evaluating the function |
| 303 | * @throws IllegalArgumentException if any parameters are invalid |
| 304 | * @deprecated replaced by {@link #solve2(UnivariateRealFunction, double, double)} |
| 305 | * since 2.0 |
| 306 | */ |
| 307 | @Deprecated |
| 308 | public double solve2(final double min, final double max) |
| 309 | throws MaxIterationsExceededException, FunctionEvaluationException { |
| 310 | return solve2(f, min, max); |
| 311 | } |
| 312 | |
| 313 | /** |
| 314 | * Find a real root in the given interval. |
| 315 | * <p> |
| 316 | * solve2() differs from solve() in the way it avoids complex operations. |
| 317 | * Except for the initial [min, max], solve2() does not require bracketing |
| 318 | * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex |
| 319 | * number arises in the computation, we simply use its modulus as real |
| 320 | * approximation.</p> |
| 321 | * <p> |
| 322 | * Because the interval may not be bracketing, bisection alternative is |
| 323 | * not applicable here. However in practice our treatment usually works |
| 324 | * well, especially near real zeros where the imaginary part of complex |
| 325 | * approximation is often negligible.</p> |
| 326 | * <p> |
| 327 | * The formulas here do not use divided differences directly.</p> |
| 328 | * |
| 329 | * @param f the function to solve |
| 330 | * @param min the lower bound for the interval |
| 331 | * @param max the upper bound for the interval |
| 332 | * @return the point at which the function value is zero |
| 333 | * @throws MaxIterationsExceededException if the maximum iteration count is exceeded |
| 334 | * or the solver detects convergence problems otherwise |
| 335 | * @throws FunctionEvaluationException if an error occurs evaluating the function |
| 336 | * @throws IllegalArgumentException if any parameters are invalid |
| 337 | * @deprecated in 2.2 (to be removed in 3.0). |
| 338 | */ |
| 339 | @Deprecated |
| 340 | public double solve2(final UnivariateRealFunction f, |
| 341 | final double min, final double max) |
| 342 | throws MaxIterationsExceededException, FunctionEvaluationException { |
| 343 | |
| 344 | // x2 is the last root approximation |
| 345 | // x is the new approximation and new x2 for next round |
| 346 | // x0 < x1 < x2 does not hold here |
| 347 | |
| 348 | double x0 = min; |
| 349 | double y0 = f.value(x0); |
| 350 | double x1 = max; |
| 351 | double y1 = f.value(x1); |
| 352 | double x2 = 0.5 * (x0 + x1); |
| 353 | double y2 = f.value(x2); |
| 354 | |
| 355 | // check for zeros before verifying bracketing |
| 356 | if (y0 == 0.0) { return min; } |
| 357 | if (y1 == 0.0) { return max; } |
| 358 | verifyBracketing(min, max, f); |
| 359 | |
| 360 | double oldx = Double.POSITIVE_INFINITY; |
| 361 | for (int i = 1; i <= maximalIterationCount; ++i) { |
| 362 | // quadratic interpolation through x0, x1, x2 |
| 363 | final double q = (x2 - x1) / (x1 - x0); |
| 364 | final double a = q * (y2 - (1 + q) * y1 + q * y0); |
| 365 | final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0; |
| 366 | final double c = (1 + q) * y2; |
| 367 | final double delta = b * b - 4 * a * c; |
| 368 | double x; |
| 369 | final double denominator; |
| 370 | if (delta >= 0.0) { |
| 371 | // choose a denominator larger in magnitude |
| 372 | double dplus = b + FastMath.sqrt(delta); |
| 373 | double dminus = b - FastMath.sqrt(delta); |
| 374 | denominator = FastMath.abs(dplus) > FastMath.abs(dminus) ? dplus : dminus; |
| 375 | } else { |
| 376 | // take the modulus of (B +/- FastMath.sqrt(delta)) |
| 377 | denominator = FastMath.sqrt(b * b - delta); |
| 378 | } |
| 379 | if (denominator != 0) { |
| 380 | x = x2 - 2.0 * c * (x2 - x1) / denominator; |
| 381 | // perturb x if it exactly coincides with x1 or x2 |
| 382 | // the equality tests here are intentional |
| 383 | while (x == x1 || x == x2) { |
| 384 | x += absoluteAccuracy; |
| 385 | } |
| 386 | } else { |
| 387 | // extremely rare case, get a random number to skip it |
| 388 | x = min + FastMath.random() * (max - min); |
| 389 | oldx = Double.POSITIVE_INFINITY; |
| 390 | } |
| 391 | final double y = f.value(x); |
| 392 | |
| 393 | // check for convergence |
| 394 | final double tolerance = FastMath.max(relativeAccuracy * FastMath.abs(x), absoluteAccuracy); |
| 395 | if (FastMath.abs(x - oldx) <= tolerance) { |
| 396 | setResult(x, i); |
| 397 | return result; |
| 398 | } |
| 399 | if (FastMath.abs(y) <= functionValueAccuracy) { |
| 400 | setResult(x, i); |
| 401 | return result; |
| 402 | } |
| 403 | |
| 404 | // prepare the next iteration |
| 405 | x0 = x1; |
| 406 | y0 = y1; |
| 407 | x1 = x2; |
| 408 | y1 = y2; |
| 409 | x2 = x; |
| 410 | y2 = y; |
| 411 | oldx = x; |
| 412 | } |
| 413 | throw new MaxIterationsExceededException(maximalIterationCount); |
| 414 | } |
| 415 | } |