001 /* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 package org.apache.commons.math.analysis.solvers; 018 019 import org.apache.commons.math.ConvergenceException; 020 import org.apache.commons.math.FunctionEvaluationException; 021 import org.apache.commons.math.MaxIterationsExceededException; 022 import org.apache.commons.math.analysis.UnivariateRealFunction; 023 import org.apache.commons.math.util.MathUtils; 024 025 /** 026 * Implements the <a href="http://mathworld.wolfram.com/MullersMethod.html"> 027 * Muller's Method</a> for root finding of real univariate functions. For 028 * reference, see <b>Elementary Numerical Analysis</b>, ISBN 0070124477, 029 * chapter 3. 030 * <p> 031 * Muller's method applies to both real and complex functions, but here we 032 * restrict ourselves to real functions. Methods solve() and solve2() find 033 * real zeros, using different ways to bypass complex arithmetics.</p> 034 * 035 * @version $Revision: 825919 $ $Date: 2009-10-16 10:51:55 -0400 (Fri, 16 Oct 2009) $ 036 * @since 1.2 037 */ 038 public class MullerSolver extends UnivariateRealSolverImpl { 039 040 /** 041 * Construct a solver for the given function. 042 * 043 * @param f function to solve 044 * @deprecated as of 2.0 the function to solve is passed as an argument 045 * to the {@link #solve(UnivariateRealFunction, double, double)} or 046 * {@link UnivariateRealSolverImpl#solve(UnivariateRealFunction, double, double, double)} 047 * method. 048 */ 049 @Deprecated 050 public MullerSolver(UnivariateRealFunction f) { 051 super(f, 100, 1E-6); 052 } 053 054 /** 055 * Construct a solver. 056 */ 057 public MullerSolver() { 058 super(100, 1E-6); 059 } 060 061 /** {@inheritDoc} */ 062 @Deprecated 063 public double solve(final double min, final double max) 064 throws ConvergenceException, FunctionEvaluationException { 065 return solve(f, min, max); 066 } 067 068 /** {@inheritDoc} */ 069 @Deprecated 070 public double solve(final double min, final double max, final double initial) 071 throws ConvergenceException, FunctionEvaluationException { 072 return solve(f, min, max, initial); 073 } 074 075 /** 076 * Find a real root in the given interval with initial value. 077 * <p> 078 * Requires bracketing condition.</p> 079 * 080 * @param f the function to solve 081 * @param min the lower bound for the interval 082 * @param max the upper bound for the interval 083 * @param initial the start value to use 084 * @return the point at which the function value is zero 085 * @throws MaxIterationsExceededException if the maximum iteration count is exceeded 086 * or the solver detects convergence problems otherwise 087 * @throws FunctionEvaluationException if an error occurs evaluating the 088 * function 089 * @throws IllegalArgumentException if any parameters are invalid 090 */ 091 public double solve(final UnivariateRealFunction f, 092 final double min, final double max, final double initial) 093 throws MaxIterationsExceededException, FunctionEvaluationException { 094 095 // check for zeros before verifying bracketing 096 if (f.value(min) == 0.0) { return min; } 097 if (f.value(max) == 0.0) { return max; } 098 if (f.value(initial) == 0.0) { return initial; } 099 100 verifyBracketing(min, max, f); 101 verifySequence(min, initial, max); 102 if (isBracketing(min, initial, f)) { 103 return solve(f, min, initial); 104 } else { 105 return solve(f, initial, max); 106 } 107 } 108 109 /** 110 * Find a real root in the given interval. 111 * <p> 112 * Original Muller's method would have function evaluation at complex point. 113 * Since our f(x) is real, we have to find ways to avoid that. Bracketing 114 * condition is one way to go: by requiring bracketing in every iteration, 115 * the newly computed approximation is guaranteed to be real.</p> 116 * <p> 117 * Normally Muller's method converges quadratically in the vicinity of a 118 * zero, however it may be very slow in regions far away from zeros. For 119 * example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use 120 * bisection as a safety backup if it performs very poorly.</p> 121 * <p> 122 * The formulas here use divided differences directly.</p> 123 * 124 * @param f the function to solve 125 * @param min the lower bound for the interval 126 * @param max the upper bound for the interval 127 * @return the point at which the function value is zero 128 * @throws MaxIterationsExceededException if the maximum iteration count is exceeded 129 * or the solver detects convergence problems otherwise 130 * @throws FunctionEvaluationException if an error occurs evaluating the 131 * function 132 * @throws IllegalArgumentException if any parameters are invalid 133 */ 134 public double solve(final UnivariateRealFunction f, 135 final double min, final double max) 136 throws MaxIterationsExceededException, FunctionEvaluationException { 137 138 // [x0, x2] is the bracketing interval in each iteration 139 // x1 is the last approximation and an interpolation point in (x0, x2) 140 // x is the new root approximation and new x1 for next round 141 // d01, d12, d012 are divided differences 142 143 double x0 = min; 144 double y0 = f.value(x0); 145 double x2 = max; 146 double y2 = f.value(x2); 147 double x1 = 0.5 * (x0 + x2); 148 double y1 = f.value(x1); 149 150 // check for zeros before verifying bracketing 151 if (y0 == 0.0) { 152 return min; 153 } 154 if (y2 == 0.0) { 155 return max; 156 } 157 verifyBracketing(min, max, f); 158 159 double oldx = Double.POSITIVE_INFINITY; 160 for (int i = 1; i <= maximalIterationCount; ++i) { 161 // Muller's method employs quadratic interpolation through 162 // x0, x1, x2 and x is the zero of the interpolating parabola. 163 // Due to bracketing condition, this parabola must have two 164 // real roots and we choose one in [x0, x2] to be x. 165 final double d01 = (y1 - y0) / (x1 - x0); 166 final double d12 = (y2 - y1) / (x2 - x1); 167 final double d012 = (d12 - d01) / (x2 - x0); 168 final double c1 = d01 + (x1 - x0) * d012; 169 final double delta = c1 * c1 - 4 * y1 * d012; 170 final double xplus = x1 + (-2.0 * y1) / (c1 + Math.sqrt(delta)); 171 final double xminus = x1 + (-2.0 * y1) / (c1 - Math.sqrt(delta)); 172 // xplus and xminus are two roots of parabola and at least 173 // one of them should lie in (x0, x2) 174 final double x = isSequence(x0, xplus, x2) ? xplus : xminus; 175 final double y = f.value(x); 176 177 // check for convergence 178 final double tolerance = Math.max(relativeAccuracy * Math.abs(x), absoluteAccuracy); 179 if (Math.abs(x - oldx) <= tolerance) { 180 setResult(x, i); 181 return result; 182 } 183 if (Math.abs(y) <= functionValueAccuracy) { 184 setResult(x, i); 185 return result; 186 } 187 188 // Bisect if convergence is too slow. Bisection would waste 189 // our calculation of x, hopefully it won't happen often. 190 // the real number equality test x == x1 is intentional and 191 // completes the proximity tests above it 192 boolean bisect = (x < x1 && (x1 - x0) > 0.95 * (x2 - x0)) || 193 (x > x1 && (x2 - x1) > 0.95 * (x2 - x0)) || 194 (x == x1); 195 // prepare the new bracketing interval for next iteration 196 if (!bisect) { 197 x0 = x < x1 ? x0 : x1; 198 y0 = x < x1 ? y0 : y1; 199 x2 = x > x1 ? x2 : x1; 200 y2 = x > x1 ? y2 : y1; 201 x1 = x; y1 = y; 202 oldx = x; 203 } else { 204 double xm = 0.5 * (x0 + x2); 205 double ym = f.value(xm); 206 if (MathUtils.sign(y0) + MathUtils.sign(ym) == 0.0) { 207 x2 = xm; y2 = ym; 208 } else { 209 x0 = xm; y0 = ym; 210 } 211 x1 = 0.5 * (x0 + x2); 212 y1 = f.value(x1); 213 oldx = Double.POSITIVE_INFINITY; 214 } 215 } 216 throw new MaxIterationsExceededException(maximalIterationCount); 217 } 218 219 /** 220 * Find a real root in the given interval. 221 * <p> 222 * solve2() differs from solve() in the way it avoids complex operations. 223 * Except for the initial [min, max], solve2() does not require bracketing 224 * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex 225 * number arises in the computation, we simply use its modulus as real 226 * approximation.</p> 227 * <p> 228 * Because the interval may not be bracketing, bisection alternative is 229 * not applicable here. However in practice our treatment usually works 230 * well, especially near real zeros where the imaginary part of complex 231 * approximation is often negligible.</p> 232 * <p> 233 * The formulas here do not use divided differences directly.</p> 234 * 235 * @param min the lower bound for the interval 236 * @param max the upper bound for the interval 237 * @return the point at which the function value is zero 238 * @throws MaxIterationsExceededException if the maximum iteration count is exceeded 239 * or the solver detects convergence problems otherwise 240 * @throws FunctionEvaluationException if an error occurs evaluating the 241 * function 242 * @throws IllegalArgumentException if any parameters are invalid 243 * @deprecated replaced by {@link #solve2(UnivariateRealFunction, double, double)} 244 * since 2.0 245 */ 246 @Deprecated 247 public double solve2(final double min, final double max) 248 throws MaxIterationsExceededException, FunctionEvaluationException { 249 return solve2(f, min, max); 250 } 251 252 /** 253 * Find a real root in the given interval. 254 * <p> 255 * solve2() differs from solve() in the way it avoids complex operations. 256 * Except for the initial [min, max], solve2() does not require bracketing 257 * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex 258 * number arises in the computation, we simply use its modulus as real 259 * approximation.</p> 260 * <p> 261 * Because the interval may not be bracketing, bisection alternative is 262 * not applicable here. However in practice our treatment usually works 263 * well, especially near real zeros where the imaginary part of complex 264 * approximation is often negligible.</p> 265 * <p> 266 * The formulas here do not use divided differences directly.</p> 267 * 268 * @param f the function to solve 269 * @param min the lower bound for the interval 270 * @param max the upper bound for the interval 271 * @return the point at which the function value is zero 272 * @throws MaxIterationsExceededException if the maximum iteration count is exceeded 273 * or the solver detects convergence problems otherwise 274 * @throws FunctionEvaluationException if an error occurs evaluating the 275 * function 276 * @throws IllegalArgumentException if any parameters are invalid 277 */ 278 public double solve2(final UnivariateRealFunction f, 279 final double min, final double max) 280 throws MaxIterationsExceededException, FunctionEvaluationException { 281 282 // x2 is the last root approximation 283 // x is the new approximation and new x2 for next round 284 // x0 < x1 < x2 does not hold here 285 286 double x0 = min; 287 double y0 = f.value(x0); 288 double x1 = max; 289 double y1 = f.value(x1); 290 double x2 = 0.5 * (x0 + x1); 291 double y2 = f.value(x2); 292 293 // check for zeros before verifying bracketing 294 if (y0 == 0.0) { return min; } 295 if (y1 == 0.0) { return max; } 296 verifyBracketing(min, max, f); 297 298 double oldx = Double.POSITIVE_INFINITY; 299 for (int i = 1; i <= maximalIterationCount; ++i) { 300 // quadratic interpolation through x0, x1, x2 301 final double q = (x2 - x1) / (x1 - x0); 302 final double a = q * (y2 - (1 + q) * y1 + q * y0); 303 final double b = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0; 304 final double c = (1 + q) * y2; 305 final double delta = b * b - 4 * a * c; 306 double x; 307 final double denominator; 308 if (delta >= 0.0) { 309 // choose a denominator larger in magnitude 310 double dplus = b + Math.sqrt(delta); 311 double dminus = b - Math.sqrt(delta); 312 denominator = Math.abs(dplus) > Math.abs(dminus) ? dplus : dminus; 313 } else { 314 // take the modulus of (B +/- Math.sqrt(delta)) 315 denominator = Math.sqrt(b * b - delta); 316 } 317 if (denominator != 0) { 318 x = x2 - 2.0 * c * (x2 - x1) / denominator; 319 // perturb x if it exactly coincides with x1 or x2 320 // the equality tests here are intentional 321 while (x == x1 || x == x2) { 322 x += absoluteAccuracy; 323 } 324 } else { 325 // extremely rare case, get a random number to skip it 326 x = min + Math.random() * (max - min); 327 oldx = Double.POSITIVE_INFINITY; 328 } 329 final double y = f.value(x); 330 331 // check for convergence 332 final double tolerance = Math.max(relativeAccuracy * Math.abs(x), absoluteAccuracy); 333 if (Math.abs(x - oldx) <= tolerance) { 334 setResult(x, i); 335 return result; 336 } 337 if (Math.abs(y) <= functionValueAccuracy) { 338 setResult(x, i); 339 return result; 340 } 341 342 // prepare the next iteration 343 x0 = x1; 344 y0 = y1; 345 x1 = x2; 346 y1 = y2; 347 x2 = x; 348 y2 = y; 349 oldx = x; 350 } 351 throw new MaxIterationsExceededException(maximalIterationCount); 352 } 353 }