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 018package org.apache.commons.math3.ode.nonstiff; 019 020import java.util.Arrays; 021import java.util.HashMap; 022import java.util.Map; 023 024import org.apache.commons.math3.fraction.BigFraction; 025import org.apache.commons.math3.linear.Array2DRowFieldMatrix; 026import org.apache.commons.math3.linear.Array2DRowRealMatrix; 027import org.apache.commons.math3.linear.ArrayFieldVector; 028import org.apache.commons.math3.linear.FieldDecompositionSolver; 029import org.apache.commons.math3.linear.FieldLUDecomposition; 030import org.apache.commons.math3.linear.FieldMatrix; 031import org.apache.commons.math3.linear.MatrixUtils; 032import org.apache.commons.math3.linear.QRDecomposition; 033import org.apache.commons.math3.linear.RealMatrix; 034 035/** Transformer to Nordsieck vectors for Adams integrators. 036 * <p>This class is used by {@link AdamsBashforthIntegrator Adams-Bashforth} and 037 * {@link AdamsMoultonIntegrator Adams-Moulton} integrators to convert between 038 * classical representation with several previous first derivatives and Nordsieck 039 * representation with higher order scaled derivatives.</p> 040 * 041 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 042 * <pre> 043 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 044 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 045 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 046 * ... 047 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative 048 * </pre></p> 049 * 050 * <p>With the previous definition, the classical representation of multistep methods 051 * uses first derivatives only, i.e. it handles y<sub>n</sub>, s<sub>1</sub>(n) and 052 * q<sub>n</sub> where q<sub>n</sub> is defined as: 053 * <pre> 054 * q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup> 055 * </pre> 056 * (we omit the k index in the notation for clarity).</p> 057 * 058 * <p>Another possible representation uses the Nordsieck vector with 059 * higher degrees scaled derivatives all taken at the same step, i.e it handles y<sub>n</sub>, 060 * s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as: 061 * <pre> 062 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 063 * </pre> 064 * (here again we omit the k index in the notation for clarity) 065 * </p> 066 * 067 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 068 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 069 * for degree k polynomials. 070 * <pre> 071 * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n) 072 * </pre> 073 * The previous formula can be used with several values for i to compute the transform between 074 * classical representation and Nordsieck vector at step end. The transform between r<sub>n</sub> 075 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 076 * <pre> 077 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 078 * </pre> 079 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 080 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being 081 * the column number starting from 1: 082 * <pre> 083 * [ -2 3 -4 5 ... ] 084 * [ -4 12 -32 80 ... ] 085 * P = [ -6 27 -108 405 ... ] 086 * [ -8 48 -256 1280 ... ] 087 * [ ... ] 088 * </pre></p> 089 * 090 * <p>Changing -i into +i in the formula above can be used to compute a similar transform between 091 * classical representation and Nordsieck vector at step start. The resulting matrix is simply 092 * the absolute value of matrix P.</p> 093 * 094 * <p>For {@link AdamsBashforthIntegrator Adams-Bashforth} method, the Nordsieck vector 095 * at step n+1 is computed from the Nordsieck vector at step n as follows: 096 * <ul> 097 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 098 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 099 * <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li> 100 * </ul> 101 * where A is a rows shifting matrix (the lower left part is an identity matrix): 102 * <pre> 103 * [ 0 0 ... 0 0 | 0 ] 104 * [ ---------------+---] 105 * [ 1 0 ... 0 0 | 0 ] 106 * A = [ 0 1 ... 0 0 | 0 ] 107 * [ ... | 0 ] 108 * [ 0 0 ... 1 0 | 0 ] 109 * [ 0 0 ... 0 1 | 0 ] 110 * </pre></p> 111 * 112 * <p>For {@link AdamsMoultonIntegrator Adams-Moulton} method, the predicted Nordsieck vector 113 * at step n+1 is computed from the Nordsieck vector at step n as follows: 114 * <ul> 115 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 116 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> 117 * <li>R<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li> 118 * </ul> 119 * From this predicted vector, the corrected vector is computed as follows: 120 * <ul> 121 * <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub></li> 122 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 123 * <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li> 124 * </ul> 125 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the 126 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> 127 * represent the corrected states.</p> 128 * 129 * <p>We observe that both methods use similar update formulas. In both cases a P<sup>-1</sup>u 130 * vector and a P<sup>-1</sup> A P matrix are used that do not depend on the state, 131 * they only depend on k. This class handles these transformations.</p> 132 * 133 * @since 2.0 134 */ 135public class AdamsNordsieckTransformer { 136 137 /** Cache for already computed coefficients. */ 138 private static final Map<Integer, AdamsNordsieckTransformer> CACHE = 139 new HashMap<Integer, AdamsNordsieckTransformer>(); 140 141 /** Update matrix for the higher order derivatives h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ... */ 142 private final Array2DRowRealMatrix update; 143 144 /** Update coefficients of the higher order derivatives wrt y'. */ 145 private final double[] c1; 146 147 /** Simple constructor. 148 * @param n number of steps of the multistep method 149 * (excluding the one being computed) 150 */ 151 private AdamsNordsieckTransformer(final int n) { 152 153 final int rows = n - 1; 154 155 // compute exact coefficients 156 FieldMatrix<BigFraction> bigP = buildP(rows); 157 FieldDecompositionSolver<BigFraction> pSolver = 158 new FieldLUDecomposition<BigFraction>(bigP).getSolver(); 159 160 BigFraction[] u = new BigFraction[rows]; 161 Arrays.fill(u, BigFraction.ONE); 162 BigFraction[] bigC1 = pSolver.solve(new ArrayFieldVector<BigFraction>(u, false)).toArray(); 163 164 // update coefficients are computed by combining transform from 165 // Nordsieck to multistep, then shifting rows to represent step advance 166 // then applying inverse transform 167 BigFraction[][] shiftedP = bigP.getData(); 168 for (int i = shiftedP.length - 1; i > 0; --i) { 169 // shift rows 170 shiftedP[i] = shiftedP[i - 1]; 171 } 172 shiftedP[0] = new BigFraction[rows]; 173 Arrays.fill(shiftedP[0], BigFraction.ZERO); 174 FieldMatrix<BigFraction> bigMSupdate = 175 pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false)); 176 177 // convert coefficients to double 178 update = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate); 179 c1 = new double[rows]; 180 for (int i = 0; i < rows; ++i) { 181 c1[i] = bigC1[i].doubleValue(); 182 } 183 184 } 185 186 /** Get the Nordsieck transformer for a given number of steps. 187 * @param nSteps number of steps of the multistep method 188 * (excluding the one being computed) 189 * @return Nordsieck transformer for the specified number of steps 190 */ 191 public static AdamsNordsieckTransformer getInstance(final int nSteps) { 192 synchronized(CACHE) { 193 AdamsNordsieckTransformer t = CACHE.get(nSteps); 194 if (t == null) { 195 t = new AdamsNordsieckTransformer(nSteps); 196 CACHE.put(nSteps, t); 197 } 198 return t; 199 } 200 } 201 202 /** Get the number of steps of the method 203 * (excluding the one being computed). 204 * @return number of steps of the method 205 * (excluding the one being computed) 206 * @deprecated as of 3.6, this method is not used anymore 207 */ 208 @Deprecated 209 public int getNSteps() { 210 return c1.length; 211 } 212 213 /** Build the P matrix. 214 * <p>The P matrix general terms are shifted (j+1) (-i)<sup>j</sup> terms 215 * with i being the row number starting from 1 and j being the column 216 * number starting from 1: 217 * <pre> 218 * [ -2 3 -4 5 ... ] 219 * [ -4 12 -32 80 ... ] 220 * P = [ -6 27 -108 405 ... ] 221 * [ -8 48 -256 1280 ... ] 222 * [ ... ] 223 * </pre></p> 224 * @param rows number of rows of the matrix 225 * @return P matrix 226 */ 227 private FieldMatrix<BigFraction> buildP(final int rows) { 228 229 final BigFraction[][] pData = new BigFraction[rows][rows]; 230 231 for (int i = 1; i <= pData.length; ++i) { 232 // build the P matrix elements from Taylor series formulas 233 final BigFraction[] pI = pData[i - 1]; 234 final int factor = -i; 235 int aj = factor; 236 for (int j = 1; j <= pI.length; ++j) { 237 pI[j - 1] = new BigFraction(aj * (j + 1)); 238 aj *= factor; 239 } 240 } 241 242 return new Array2DRowFieldMatrix<BigFraction>(pData, false); 243 244 } 245 246 /** Initialize the high order scaled derivatives at step start. 247 * @param h step size to use for scaling 248 * @param t first steps times 249 * @param y first steps states 250 * @param yDot first steps derivatives 251 * @return Nordieck vector at start of first step (h<sup>2</sup>/2 y''<sub>n</sub>, 252 * h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>) 253 */ 254 255 public Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t, 256 final double[][] y, 257 final double[][] yDot) { 258 259 // using Taylor series with di = ti - t0, we get: 260 // y(ti) - y(t0) - di y'(t0) = di^2 / h^2 s2 + ... + di^k / h^k sk + O(h^k) 261 // y'(ti) - y'(t0) = 2 di / h^2 s2 + ... + k di^(k-1) / h^k sk + O(h^(k-1)) 262 // we write these relations for i = 1 to i= 1+n/2 as a set of n + 2 linear 263 // equations depending on the Nordsieck vector [s2 ... sk rk], so s2 to sk correspond 264 // to the appropriately truncated Taylor expansion, and rk is the Taylor remainder. 265 // The goal is to have s2 to sk as accurate as possible considering the fact the sum is 266 // truncated and we don't want the error terms to be included in s2 ... sk, so we need 267 // to solve also for the remainder 268 final double[][] a = new double[c1.length + 1][c1.length + 1]; 269 final double[][] b = new double[c1.length + 1][y[0].length]; 270 final double[] y0 = y[0]; 271 final double[] yDot0 = yDot[0]; 272 for (int i = 1; i < y.length; ++i) { 273 274 final double di = t[i] - t[0]; 275 final double ratio = di / h; 276 double dikM1Ohk = 1 / h; 277 278 // linear coefficients of equations 279 // y(ti) - y(t0) - di y'(t0) and y'(ti) - y'(t0) 280 final double[] aI = a[2 * i - 2]; 281 final double[] aDotI = (2 * i - 1) < a.length ? a[2 * i - 1] : null; 282 for (int j = 0; j < aI.length; ++j) { 283 dikM1Ohk *= ratio; 284 aI[j] = di * dikM1Ohk; 285 if (aDotI != null) { 286 aDotI[j] = (j + 2) * dikM1Ohk; 287 } 288 } 289 290 // expected value of the previous equations 291 final double[] yI = y[i]; 292 final double[] yDotI = yDot[i]; 293 final double[] bI = b[2 * i - 2]; 294 final double[] bDotI = (2 * i - 1) < b.length ? b[2 * i - 1] : null; 295 for (int j = 0; j < yI.length; ++j) { 296 bI[j] = yI[j] - y0[j] - di * yDot0[j]; 297 if (bDotI != null) { 298 bDotI[j] = yDotI[j] - yDot0[j]; 299 } 300 } 301 302 } 303 304 // solve the linear system to get the best estimate of the Nordsieck vector [s2 ... sk], 305 // with the additional terms s(k+1) and c grabbing the parts after the truncated Taylor expansion 306 final QRDecomposition decomposition = new QRDecomposition(new Array2DRowRealMatrix(a, false)); 307 final RealMatrix x = decomposition.getSolver().solve(new Array2DRowRealMatrix(b, false)); 308 309 // extract just the Nordsieck vector [s2 ... sk] 310 final Array2DRowRealMatrix truncatedX = new Array2DRowRealMatrix(x.getRowDimension() - 1, x.getColumnDimension()); 311 for (int i = 0; i < truncatedX.getRowDimension(); ++i) { 312 for (int j = 0; j < truncatedX.getColumnDimension(); ++j) { 313 truncatedX.setEntry(i, j, x.getEntry(i, j)); 314 } 315 } 316 return truncatedX; 317 318 } 319 320 /** Update the high order scaled derivatives for Adams integrators (phase 1). 321 * <p>The complete update of high order derivatives has a form similar to: 322 * <pre> 323 * r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub> 324 * </pre> 325 * this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.</p> 326 * @param highOrder high order scaled derivatives 327 * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k)) 328 * @return updated high order derivatives 329 * @see #updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix) 330 */ 331 public Array2DRowRealMatrix updateHighOrderDerivativesPhase1(final Array2DRowRealMatrix highOrder) { 332 return update.multiply(highOrder); 333 } 334 335 /** Update the high order scaled derivatives Adams integrators (phase 2). 336 * <p>The complete update of high order derivatives has a form similar to: 337 * <pre> 338 * r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub> 339 * </pre> 340 * this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.</p> 341 * <p>Phase 1 of the update must already have been performed.</p> 342 * @param start first order scaled derivatives at step start 343 * @param end first order scaled derivatives at step end 344 * @param highOrder high order scaled derivatives, will be modified 345 * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k)) 346 * @see #updateHighOrderDerivativesPhase1(Array2DRowRealMatrix) 347 */ 348 public void updateHighOrderDerivativesPhase2(final double[] start, 349 final double[] end, 350 final Array2DRowRealMatrix highOrder) { 351 final double[][] data = highOrder.getDataRef(); 352 for (int i = 0; i < data.length; ++i) { 353 final double[] dataI = data[i]; 354 final double c1I = c1[i]; 355 for (int j = 0; j < dataI.length; ++j) { 356 dataI[j] += c1I * (start[j] - end[j]); 357 } 358 } 359 } 360 361}