| /* |
| * Licensed to the Apache Software Foundation (ASF) under one or more |
| * contributor license agreements. See the NOTICE file distributed with |
| * this work for additional information regarding copyright ownership. |
| * The ASF licenses this file to You under the Apache License, Version 2.0 |
| * (the "License"); you may not use this file except in compliance with |
| * the License. You may obtain a copy of the License at |
| * |
| * http://www.apache.org/licenses/LICENSE-2.0 |
| * |
| * Unless required by applicable law or agreed to in writing, software |
| * distributed under the License is distributed on an "AS IS" BASIS, |
| * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| * See the License for the specific language governing permissions and |
| * limitations under the License. |
| */ |
| |
| package org.apache.commons.math.ode.nonstiff; |
| |
| import java.util.Arrays; |
| import java.util.HashMap; |
| import java.util.Map; |
| |
| import org.apache.commons.math.fraction.BigFraction; |
| import org.apache.commons.math.linear.Array2DRowFieldMatrix; |
| import org.apache.commons.math.linear.Array2DRowRealMatrix; |
| import org.apache.commons.math.linear.DefaultFieldMatrixChangingVisitor; |
| import org.apache.commons.math.linear.FieldDecompositionSolver; |
| import org.apache.commons.math.linear.FieldLUDecompositionImpl; |
| import org.apache.commons.math.linear.FieldMatrix; |
| import org.apache.commons.math.linear.MatrixUtils; |
| |
| /** Transformer to Nordsieck vectors for Adams integrators. |
| * <p>This class i used by {@link AdamsBashforthIntegrator Adams-Bashforth} and |
| * {@link AdamsMoultonIntegrator Adams-Moulton} integrators to convert between |
| * classical representation with several previous first derivatives and Nordsieck |
| * representation with higher order scaled derivatives.</p> |
| * |
| * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: |
| * <pre> |
| * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative |
| * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative |
| * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative |
| * ... |
| * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative |
| * </pre></p> |
| * |
| * <p>With the previous definition, the classical representation of multistep methods |
| * uses first derivatives only, i.e. it handles y<sub>n</sub>, s<sub>1</sub>(n) and |
| * q<sub>n</sub> where q<sub>n</sub> is defined as: |
| * <pre> |
| * 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> |
| * </pre> |
| * (we omit the k index in the notation for clarity).</p> |
| * |
| * <p>Another possible representation uses the Nordsieck vector with |
| * higher degrees scaled derivatives all taken at the same step, i.e it handles y<sub>n</sub>, |
| * s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as: |
| * <pre> |
| * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> |
| * </pre> |
| * (here again we omit the k index in the notation for clarity) |
| * </p> |
| * |
| * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be |
| * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact |
| * for degree k polynomials. |
| * <pre> |
| * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(n) |
| * </pre> |
| * The previous formula can be used with several values for i to compute the transform between |
| * classical representation and Nordsieck vector at step end. The transform between r<sub>n</sub> |
| * and q<sub>n</sub> resulting from the Taylor series formulas above is: |
| * <pre> |
| * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> |
| * </pre> |
| * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built |
| * with the j (-i)<sup>j-1</sup> terms: |
| * <pre> |
| * [ -2 3 -4 5 ... ] |
| * [ -4 12 -32 80 ... ] |
| * P = [ -6 27 -108 405 ... ] |
| * [ -8 48 -256 1280 ... ] |
| * [ ... ] |
| * </pre></p> |
| * |
| * <p>Changing -i into +i in the formula above can be used to compute a similar transform between |
| * classical representation and Nordsieck vector at step start. The resulting matrix is simply |
| * the absolute value of matrix P.</p> |
| * |
| * <p>For {@link AdamsBashforthIntegrator Adams-Bashforth} method, the Nordsieck vector |
| * at step n+1 is computed from the Nordsieck vector at step n as follows: |
| * <ul> |
| * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> |
| * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> |
| * <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> |
| * </ul> |
| * where A is a rows shifting matrix (the lower left part is an identity matrix): |
| * <pre> |
| * [ 0 0 ... 0 0 | 0 ] |
| * [ ---------------+---] |
| * [ 1 0 ... 0 0 | 0 ] |
| * A = [ 0 1 ... 0 0 | 0 ] |
| * [ ... | 0 ] |
| * [ 0 0 ... 1 0 | 0 ] |
| * [ 0 0 ... 0 1 | 0 ] |
| * </pre></p> |
| * |
| * <p>For {@link AdamsMoultonIntegrator Adams-Moulton} method, the predicted Nordsieck vector |
| * at step n+1 is computed from the Nordsieck vector at step n as follows: |
| * <ul> |
| * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> |
| * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> |
| * <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> |
| * </ul> |
| * From this predicted vector, the corrected vector is computed as follows: |
| * <ul> |
| * <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> |
| * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> |
| * <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> |
| * </ul> |
| * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the |
| * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> |
| * represent the corrected states.</p> |
| * |
| * <p>We observe that both methods use similar update formulas. In both cases a P<sup>-1</sup>u |
| * vector and a P<sup>-1</sup> A P matrix are used that do not depend on the state, |
| * they only depend on k. This class handles these transformations.</p> |
| * |
| * @version $Revision: 810196 $ $Date: 2009-09-01 21:47:46 +0200 (mar. 01 sept. 2009) $ |
| * @since 2.0 |
| */ |
| public class AdamsNordsieckTransformer { |
| |
| /** Cache for already computed coefficients. */ |
| private static final Map<Integer, AdamsNordsieckTransformer> CACHE = |
| new HashMap<Integer, AdamsNordsieckTransformer>(); |
| |
| /** Initialization matrix for the higher order derivatives wrt y'', y''' ... */ |
| private final Array2DRowRealMatrix initialization; |
| |
| /** Update matrix for the higher order derivatives h<sup>2</sup>/2y'', h<sup>3</sup>/6 y''' ... */ |
| private final Array2DRowRealMatrix update; |
| |
| /** Update coefficients of the higher order derivatives wrt y'. */ |
| private final double[] c1; |
| |
| /** Simple constructor. |
| * @param nSteps number of steps of the multistep method |
| * (excluding the one being computed) |
| */ |
| private AdamsNordsieckTransformer(final int nSteps) { |
| |
| // compute exact coefficients |
| FieldMatrix<BigFraction> bigP = buildP(nSteps); |
| FieldDecompositionSolver<BigFraction> pSolver = |
| new FieldLUDecompositionImpl<BigFraction>(bigP).getSolver(); |
| |
| BigFraction[] u = new BigFraction[nSteps]; |
| Arrays.fill(u, BigFraction.ONE); |
| BigFraction[] bigC1 = pSolver.solve(u); |
| |
| // update coefficients are computed by combining transform from |
| // Nordsieck to multistep, then shifting rows to represent step advance |
| // then applying inverse transform |
| BigFraction[][] shiftedP = bigP.getData(); |
| for (int i = shiftedP.length - 1; i > 0; --i) { |
| // shift rows |
| shiftedP[i] = shiftedP[i - 1]; |
| } |
| shiftedP[0] = new BigFraction[nSteps]; |
| Arrays.fill(shiftedP[0], BigFraction.ZERO); |
| FieldMatrix<BigFraction> bigMSupdate = |
| pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false)); |
| |
| // initialization coefficients, computed from a R matrix = abs(P) |
| bigP.walkInOptimizedOrder(new DefaultFieldMatrixChangingVisitor<BigFraction>(BigFraction.ZERO) { |
| /** {@inheritDoc} */ |
| @Override |
| public BigFraction visit(int row, int column, BigFraction value) { |
| return ((column & 0x1) == 0x1) ? value : value.negate(); |
| } |
| }); |
| FieldMatrix<BigFraction> bigRInverse = |
| new FieldLUDecompositionImpl<BigFraction>(bigP).getSolver().getInverse(); |
| |
| // convert coefficients to double |
| initialization = MatrixUtils.bigFractionMatrixToRealMatrix(bigRInverse); |
| update = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate); |
| c1 = new double[nSteps]; |
| for (int i = 0; i < nSteps; ++i) { |
| c1[i] = bigC1[i].doubleValue(); |
| } |
| |
| } |
| |
| /** Get the Nordsieck transformer for a given number of steps. |
| * @param nSteps number of steps of the multistep method |
| * (excluding the one being computed) |
| * @return Nordsieck transformer for the specified number of steps |
| */ |
| public static AdamsNordsieckTransformer getInstance(final int nSteps) { |
| synchronized(CACHE) { |
| AdamsNordsieckTransformer t = CACHE.get(nSteps); |
| if (t == null) { |
| t = new AdamsNordsieckTransformer(nSteps); |
| CACHE.put(nSteps, t); |
| } |
| return t; |
| } |
| } |
| |
| /** Get the number of steps of the method |
| * (excluding the one being computed). |
| * @return number of steps of the method |
| * (excluding the one being computed) |
| */ |
| public int getNSteps() { |
| return c1.length; |
| } |
| |
| /** Build the P matrix. |
| * <p>The P matrix general terms are shifted j (-i)<sup>j-1</sup> terms: |
| * <pre> |
| * [ -2 3 -4 5 ... ] |
| * [ -4 12 -32 80 ... ] |
| * P = [ -6 27 -108 405 ... ] |
| * [ -8 48 -256 1280 ... ] |
| * [ ... ] |
| * </pre></p> |
| * @param nSteps number of steps of the multistep method |
| * (excluding the one being computed) |
| * @return P matrix |
| */ |
| private FieldMatrix<BigFraction> buildP(final int nSteps) { |
| |
| final BigFraction[][] pData = new BigFraction[nSteps][nSteps]; |
| |
| for (int i = 0; i < pData.length; ++i) { |
| // build the P matrix elements from Taylor series formulas |
| final BigFraction[] pI = pData[i]; |
| final int factor = -(i + 1); |
| int aj = factor; |
| for (int j = 0; j < pI.length; ++j) { |
| pI[j] = new BigFraction(aj * (j + 2)); |
| aj *= factor; |
| } |
| } |
| |
| return new Array2DRowFieldMatrix<BigFraction>(pData, false); |
| |
| } |
| |
| /** Initialize the high order scaled derivatives at step start. |
| * @param first first scaled derivative at step start |
| * @param multistep scaled derivatives after step start (hy'1, ..., hy'k-1) |
| * will be modified |
| * @return high order derivatives at step start |
| */ |
| public Array2DRowRealMatrix initializeHighOrderDerivatives(final double[] first, |
| final double[][] multistep) { |
| for (int i = 0; i < multistep.length; ++i) { |
| final double[] msI = multistep[i]; |
| for (int j = 0; j < first.length; ++j) { |
| msI[j] -= first[j]; |
| } |
| } |
| return initialization.multiply(new Array2DRowRealMatrix(multistep, false)); |
| } |
| |
| /** Update the high order scaled derivatives for Adams integrators (phase 1). |
| * <p>The complete update of high order derivatives has a form similar to: |
| * <pre> |
| * 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> |
| * </pre> |
| * this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.</p> |
| * @param highOrder high order scaled derivatives |
| * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k)) |
| * @return updated high order derivatives |
| * @see #updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix) |
| */ |
| public Array2DRowRealMatrix updateHighOrderDerivativesPhase1(final Array2DRowRealMatrix highOrder) { |
| return update.multiply(highOrder); |
| } |
| |
| /** Update the high order scaled derivatives Adams integrators (phase 2). |
| * <p>The complete update of high order derivatives has a form similar to: |
| * <pre> |
| * 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> |
| * </pre> |
| * this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.</p> |
| * <p>Phase 1 of the update must already have been performed.</p> |
| * @param start first order scaled derivatives at step start |
| * @param end first order scaled derivatives at step end |
| * @param highOrder high order scaled derivatives, will be modified |
| * (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k)) |
| * @see #updateHighOrderDerivativesPhase1(Array2DRowRealMatrix) |
| */ |
| public void updateHighOrderDerivativesPhase2(final double[] start, |
| final double[] end, |
| final Array2DRowRealMatrix highOrder) { |
| final double[][] data = highOrder.getDataRef(); |
| for (int i = 0; i < data.length; ++i) { |
| final double[] dataI = data[i]; |
| final double c1I = c1[i]; |
| for (int j = 0; j < dataI.length; ++j) { |
| dataI[j] += c1I * (start[j] - end[j]); |
| } |
| } |
| } |
| |
| } |
