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AdamsNordsieckTransformer
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CACHE: Map<Integer, AdamsNordsieckTransformer>
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update: Array2DRowRealMatrix
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c1: double[]
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AdamsNordsieckTransformer(int): void
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getInstance(int): AdamsNordsieckTransformer
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getNSteps(): int
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buildP(int): FieldMatrix<BigFraction>
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initializeHighOrderDerivatives(double, double[], double[][], double[][]): Array2DRowRealMatrix
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updateHighOrderDerivativesPhase1(Array2DRowRealMatrix): Array2DRowRealMatrix
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updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix): void
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/*
* 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.math3.ode.nonstiff;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;
import org.apache.commons.math3.fraction.BigFraction;
import org.apache.commons.math3.linear.Array2DRowFieldMatrix;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.ArrayFieldVector;
import org.apache.commons.math3.linear.FieldDecompositionSolver;
import org.apache.commons.math3.linear.FieldLUDecomposition;
import org.apache.commons.math3.linear.FieldMatrix;
import org.apache.commons.math3.linear.MatrixUtils;
import org.apache.commons.math3.linear.QRDecomposition;
import org.apache.commons.math3.linear.RealMatrix;
Transformer to Nordsieck vectors for Adams integrators.
This class is used by Adams-Bashforth and Adams-Moulton integrators to convert between classical representation with several previous first derivatives and Nordsieck representation with higher order scaled derivatives.
We define scaled derivatives si(n) at step n as:
s1(n) = h y'n for first derivative
s2(n) = h2/2 y''n for second derivative
s3(n) = h3/6 y'''n for third derivative
...
sk(n) = hk/k! y(k)n for kth derivative
With the previous definition, the classical representation of multistep methods
uses first derivatives only, i.e. it handles yn, s1(n) and
qn where qn is defined as:
qn = [ s1(n-1) s1(n-2) ... s1(n-(k-1)) ]T
(we omit the k index in the notation for clarity).
Another possible representation uses the Nordsieck vector with
higher degrees scaled derivatives all taken at the same step, i.e it handles yn,
s1(n) and rn) where rn is defined as:
rn = [ s2(n), s3(n) ... sk(n) ]T
(here again we omit the k index in the notation for clarity)
Taylor series formulas show that for any index offset i, s1(n-i) can be
computed from s1(n), s2(n) ... sk(n), the formula being exact
for degree k polynomials.
s1(n-i) = s1(n) + ∑j>0 (j+1) (-i)j sj+1(n)
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 rn
and qn resulting from the Taylor series formulas above is:
qn = s1(n) u + P rn
where u is the [ 1 1 ... 1 ]T vector and P is the (k-1)×(k-1) matrix built
with the (j+1) (-i)j terms with i being the row number starting from 1 and j being
the column number starting from 1:
[ -2 3 -4 5 ... ]
[ -4 12 -32 80 ... ]
P = [ -6 27 -108 405 ... ]
[ -8 48 -256 1280 ... ]
[ ... ]
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.
For Adams-Bashforth method, the Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
- yn+1 = yn + s1(n) + uT rn
- s1(n+1) = h f(tn+1, yn+1)
- rn+1 = (s1(n) - s1(n+1)) P-1 u + P-1 A P rn
where A is a rows shifting matrix (the lower left part is an identity matrix):
[ 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 ]
For Adams-Moulton method, the predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
- Yn+1 = yn + s1(n) + uT rn
- S1(n+1) = h f(tn+1, Yn+1)
- Rn+1 = (s1(n) - s1(n+1)) P-1 u + P-1 A P rn
From this predicted vector, the corrected vector is computed as follows:
- yn+1 = yn + S1(n+1) + [ -1 +1 -1 +1 ... ±1 ] rn+1
- s1(n+1) = h f(tn+1, yn+1)
- rn+1 = Rn+1 + (s1(n+1) - S1(n+1)) P-1 u
where the upper case Yn+1, S1(n+1) and Rn+1 represent the
predicted states whereas the lower case yn+1, sn+1 and rn+1
represent the corrected states.
We observe that both methods use similar update formulas. In both cases a P-1u
vector and a P-1 A P matrix are used that do not depend on the state,
they only depend on k. This class handles these transformations.
Since: 2.0
/** Transformer to Nordsieck vectors for Adams integrators.
* <p>This class is 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<sup>(k)</sup><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>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</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+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
* the column number starting from 1:
* <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>
*
* @since 2.0
*/
public class AdamsNordsieckTransformer {
Cache for already computed coefficients. /** Cache for already computed coefficients. */
private static final Map<Integer, AdamsNordsieckTransformer> CACHE =
new HashMap<Integer, AdamsNordsieckTransformer>();
Update matrix for the higher order derivatives h2/2 y'', h3/6 y''' ... /** Update matrix for the higher order derivatives h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ... */
private final Array2DRowRealMatrix update;
Update coefficients of the higher order derivatives wrt y'. /** Update coefficients of the higher order derivatives wrt y'. */
private final double[] c1;
Simple constructor.
Params: - n – number of steps of the multistep method
(excluding the one being computed)
/** Simple constructor.
* @param n number of steps of the multistep method
* (excluding the one being computed)
*/
private AdamsNordsieckTransformer(final int n) {
final int rows = n - 1;
// compute exact coefficients
FieldMatrix<BigFraction> bigP = buildP(rows);
FieldDecompositionSolver<BigFraction> pSolver =
new FieldLUDecomposition<BigFraction>(bigP).getSolver();
BigFraction[] u = new BigFraction[rows];
Arrays.fill(u, BigFraction.ONE);
BigFraction[] bigC1 = pSolver.solve(new ArrayFieldVector<BigFraction>(u, false)).toArray();
// 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[rows];
Arrays.fill(shiftedP[0], BigFraction.ZERO);
FieldMatrix<BigFraction> bigMSupdate =
pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false));
// convert coefficients to double
update = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate);
c1 = new double[rows];
for (int i = 0; i < rows; ++i) {
c1[i] = bigC1[i].doubleValue();
}
}
Get the Nordsieck transformer for a given number of steps.
Params: - nSteps – number of steps of the multistep method
(excluding the one being computed)
Returns: Nordsieck transformer for the specified number of steps
/** 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).
Returns: number of steps of the method
(excluding the one being computed) Deprecated: as of 3.6, this method is not used anymore
/** 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)
* @deprecated as of 3.6, this method is not used anymore
*/
@Deprecated
public int getNSteps() {
return c1.length;
}
Build the P matrix.
The P matrix general terms are shifted (j+1) (-i)j terms
with i being the row number starting from 1 and j being the column
number starting from 1:
[ -2 3 -4 5 ... ]
[ -4 12 -32 80 ... ]
P = [ -6 27 -108 405 ... ]
[ -8 48 -256 1280 ... ]
[ ... ]
Params: - rows – number of rows of the matrix
Returns: P matrix
/** Build the P matrix.
* <p>The P matrix general terms are shifted (j+1) (-i)<sup>j</sup> terms
* with i being the row number starting from 1 and j being the column
* number starting from 1:
* <pre>
* [ -2 3 -4 5 ... ]
* [ -4 12 -32 80 ... ]
* P = [ -6 27 -108 405 ... ]
* [ -8 48 -256 1280 ... ]
* [ ... ]
* </pre></p>
* @param rows number of rows of the matrix
* @return P matrix
*/
private FieldMatrix<BigFraction> buildP(final int rows) {
final BigFraction[][] pData = new BigFraction[rows][rows];
for (int i = 1; i <= pData.length; ++i) {
// build the P matrix elements from Taylor series formulas
final BigFraction[] pI = pData[i - 1];
final int factor = -i;
int aj = factor;
for (int j = 1; j <= pI.length; ++j) {
pI[j - 1] = new BigFraction(aj * (j + 1));
aj *= factor;
}
}
return new Array2DRowFieldMatrix<BigFraction>(pData, false);
}
Initialize the high order scaled derivatives at step start.
Params: - h – step size to use for scaling
- t – first steps times
- y – first steps states
- yDot – first steps derivatives
Returns: Nordieck vector at start of first step (h2/2 y''n,
h3/6 y'''n ... hk/k! y(k)n)
/** Initialize the high order scaled derivatives at step start.
* @param h step size to use for scaling
* @param t first steps times
* @param y first steps states
* @param yDot first steps derivatives
* @return Nordieck vector at start of first step (h<sup>2</sup>/2 y''<sub>n</sub>,
* h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
*/
public Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
final double[][] y,
final double[][] yDot) {
// using Taylor series with di = ti - t0, we get:
// y(ti) - y(t0) - di y'(t0) = di^2 / h^2 s2 + ... + di^k / h^k sk + O(h^k)
// y'(ti) - y'(t0) = 2 di / h^2 s2 + ... + k di^(k-1) / h^k sk + O(h^(k-1))
// we write these relations for i = 1 to i= 1+n/2 as a set of n + 2 linear
// equations depending on the Nordsieck vector [s2 ... sk rk], so s2 to sk correspond
// to the appropriately truncated Taylor expansion, and rk is the Taylor remainder.
// The goal is to have s2 to sk as accurate as possible considering the fact the sum is
// truncated and we don't want the error terms to be included in s2 ... sk, so we need
// to solve also for the remainder
final double[][] a = new double[c1.length + 1][c1.length + 1];
final double[][] b = new double[c1.length + 1][y[0].length];
final double[] y0 = y[0];
final double[] yDot0 = yDot[0];
for (int i = 1; i < y.length; ++i) {
final double di = t[i] - t[0];
final double ratio = di / h;
double dikM1Ohk = 1 / h;
// linear coefficients of equations
// y(ti) - y(t0) - di y'(t0) and y'(ti) - y'(t0)
final double[] aI = a[2 * i - 2];
final double[] aDotI = (2 * i - 1) < a.length ? a[2 * i - 1] : null;
for (int j = 0; j < aI.length; ++j) {
dikM1Ohk *= ratio;
aI[j] = di * dikM1Ohk;
if (aDotI != null) {
aDotI[j] = (j + 2) * dikM1Ohk;
}
}
// expected value of the previous equations
final double[] yI = y[i];
final double[] yDotI = yDot[i];
final double[] bI = b[2 * i - 2];
final double[] bDotI = (2 * i - 1) < b.length ? b[2 * i - 1] : null;
for (int j = 0; j < yI.length; ++j) {
bI[j] = yI[j] - y0[j] - di * yDot0[j];
if (bDotI != null) {
bDotI[j] = yDotI[j] - yDot0[j];
}
}
}
// solve the linear system to get the best estimate of the Nordsieck vector [s2 ... sk],
// with the additional terms s(k+1) and c grabbing the parts after the truncated Taylor expansion
final QRDecomposition decomposition = new QRDecomposition(new Array2DRowRealMatrix(a, false));
final RealMatrix x = decomposition.getSolver().solve(new Array2DRowRealMatrix(b, false));
// extract just the Nordsieck vector [s2 ... sk]
final Array2DRowRealMatrix truncatedX = new Array2DRowRealMatrix(x.getRowDimension() - 1, x.getColumnDimension());
for (int i = 0; i < truncatedX.getRowDimension(); ++i) {
for (int j = 0; j < truncatedX.getColumnDimension(); ++j) {
truncatedX.setEntry(i, j, x.getEntry(i, j));
}
}
return truncatedX;
}
Update the high order scaled derivatives for Adams integrators (phase 1).
The complete update of high order derivatives has a form similar to:
rn+1 = (s1(n) - s1(n+1)) P-1 u + P-1 A P rn
this method computes the P-1 A P rn part.
Params: - highOrder – high order scaled derivatives
(h2/2 y'', ... hk/k! y(k))
See Also: Returns: updated high order derivatives
/** 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).
The complete update of high order derivatives has a form similar to:
rn+1 = (s1(n) - s1(n+1)) P-1 u + P-1 A P rn
this method computes the (s1(n) - s1(n+1)) P-1 u part.
Phase 1 of the update must already have been performed.
Params: - start – first order scaled derivatives at step start
- end – first order scaled derivatives at step end
- highOrder – high order scaled derivatives, will be modified
(h2/2 y'', ... hk/k! y(k))
See Also:
/** 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]);
}
}
}
}