AdamsNordsieckFieldTransformer.java
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package org.apache.commons.math4.legacy.ode.nonstiff;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;
import org.apache.commons.math4.legacy.core.Field;
import org.apache.commons.math4.legacy.core.RealFieldElement;
import org.apache.commons.math4.legacy.linear.Array2DRowFieldMatrix;
import org.apache.commons.math4.legacy.linear.ArrayFieldVector;
import org.apache.commons.math4.legacy.linear.FieldDecompositionSolver;
import org.apache.commons.math4.legacy.linear.FieldLUDecomposition;
import org.apache.commons.math4.legacy.linear.FieldMatrix;
import org.apache.commons.math4.legacy.core.MathArrays;
/** 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:
* <div style="white-space: pre"><code>
* 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
* </code></div>
*
* <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:
* <div style="white-space: pre"><code>
* 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>
* </code></div>
* (we omit the k index in the notation for clarity).
*
* <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:
* <div style="white-space: pre"><code>
* r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
* </code></div>
* (here again we omit the k index in the notation for clarity)
*
* <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.
* <div style="white-space: pre"><code>
* 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)
* </code></div>
* 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:
* <div style="white-space: pre"><code>
* q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
* </code></div>
* 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>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>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>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>
*
* @param <T> the type of the field elements
* @since 3.6
*/
public final class AdamsNordsieckFieldTransformer<T extends RealFieldElement<T>> {
/** Cache for already computed coefficients. */
private static final Map<Integer,
Map<Field<? extends RealFieldElement<?>>,
AdamsNordsieckFieldTransformer<? extends RealFieldElement<?>>>> CACHE =
new HashMap<>();
/** Field to which the time and state vector elements belong. */
private final Field<T> field;
/** Update matrix for the higher order derivatives h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ... */
private final Array2DRowFieldMatrix<T> update;
/** Update coefficients of the higher order derivatives wrt y'. */
private final T[] c1;
/** Simple constructor.
* @param field field to which the time and state vector elements belong
* @param n number of steps of the multistep method
* (excluding the one being computed)
*/
private AdamsNordsieckFieldTransformer(final Field<T> field, final int n) {
this.field = field;
final int rows = n - 1;
// compute coefficients
FieldMatrix<T> bigP = buildP(rows);
FieldDecompositionSolver<T> pSolver =
new FieldLUDecomposition<>(bigP).getSolver();
T[] u = MathArrays.buildArray(field, rows);
Arrays.fill(u, field.getOne());
c1 = pSolver.solve(new ArrayFieldVector<>(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
T[][] shiftedP = bigP.getData();
// shift rows
if (shiftedP.length - 1 > 0){
System.arraycopy(shiftedP, 0, shiftedP, 1, shiftedP.length - 1);
}
shiftedP[0] = MathArrays.buildArray(field, rows);
Arrays.fill(shiftedP[0], field.getZero());
update = new Array2DRowFieldMatrix<>(pSolver.solve(new Array2DRowFieldMatrix<>(shiftedP, false)).getData());
}
/** Get the Nordsieck transformer for a given field and number of steps.
* @param field field to which the time and state vector elements belong
* @param nSteps number of steps of the multistep method
* (excluding the one being computed)
* @return Nordsieck transformer for the specified field and number of steps
* @param <T> the type of the field elements
*/
public static <T extends RealFieldElement<T>> AdamsNordsieckFieldTransformer<T>
getInstance(final Field<T> field, final int nSteps) {
synchronized(CACHE) {
Map<Field<? extends RealFieldElement<?>>,
AdamsNordsieckFieldTransformer<? extends RealFieldElement<?>>> map = CACHE.get(nSteps);
if (map == null) {
map = new HashMap<>();
CACHE.put(nSteps, map);
}
@SuppressWarnings("unchecked")
AdamsNordsieckFieldTransformer<T> t = (AdamsNordsieckFieldTransformer<T>) map.get(field);
if (t == null) {
t = new AdamsNordsieckFieldTransformer<>(field, nSteps);
map.put(field, t);
}
return t;
}
}
/** 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>
* @param rows number of rows of the matrix
* @return P matrix
*/
private FieldMatrix<T> buildP(final int rows) {
final T[][] pData = MathArrays.buildArray(field, rows, rows);
for (int i = 1; i <= pData.length; ++i) {
// build the P matrix elements from Taylor series formulas
final T[] pI = pData[i - 1];
final int factor = -i;
T aj = field.getZero().add(factor);
for (int j = 1; j <= pI.length; ++j) {
pI[j - 1] = aj.multiply(j + 1);
aj = aj.multiply(factor);
}
}
return new Array2DRowFieldMatrix<>(pData, false);
}
/** 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 Array2DRowFieldMatrix<T> initializeHighOrderDerivatives(final T h, final T[] t,
final T[][] y,
final T[][] 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 T[][] a = MathArrays.buildArray(field, c1.length + 1, c1.length + 1);
final T[][] b = MathArrays.buildArray(field, c1.length + 1, y[0].length);
final T[] y0 = y[0];
final T[] yDot0 = yDot[0];
for (int i = 1; i < y.length; ++i) {
final T di = t[i].subtract(t[0]);
final T ratio = di.divide(h);
T dikM1Ohk = h.reciprocal();
// linear coefficients of equations
// y(ti) - y(t0) - di y'(t0) and y'(ti) - y'(t0)
final T[] aI = a[2 * i - 2];
final T[] aDotI = (2 * i - 1) < a.length ? a[2 * i - 1] : null;
for (int j = 0; j < aI.length; ++j) {
dikM1Ohk = dikM1Ohk.multiply(ratio);
aI[j] = di.multiply(dikM1Ohk);
if (aDotI != null) {
aDotI[j] = dikM1Ohk.multiply(j + 2);
}
}
// expected value of the previous equations
final T[] yI = y[i];
final T[] yDotI = yDot[i];
final T[] bI = b[2 * i - 2];
final T[] bDotI = (2 * i - 1) < b.length ? b[2 * i - 1] : null;
for (int j = 0; j < yI.length; ++j) {
bI[j] = yI[j].subtract(y0[j]).subtract(di.multiply(yDot0[j]));
if (bDotI != null) {
bDotI[j] = yDotI[j].subtract(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 FieldLUDecomposition<T> decomposition = new FieldLUDecomposition<>(new Array2DRowFieldMatrix<>(a, false));
final FieldMatrix<T> x = decomposition.getSolver().solve(new Array2DRowFieldMatrix<>(b, false));
// extract just the Nordsieck vector [s2 ... sk]
final Array2DRowFieldMatrix<T> truncatedX =
new Array2DRowFieldMatrix<>(field, 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).
* <p>The complete update of high order derivatives has a form similar to:
* <div style="white-space: pre"><code>
* 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>
* </code></div>
* this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.
* @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(RealFieldElement[], RealFieldElement[], Array2DRowFieldMatrix)
*/
public Array2DRowFieldMatrix<T> updateHighOrderDerivativesPhase1(final Array2DRowFieldMatrix<T> 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:
* <div style="white-space: pre"><code>
* 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>
* </code></div>
* this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.
* <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(Array2DRowFieldMatrix)
*/
public void updateHighOrderDerivativesPhase2(final T[] start,
final T[] end,
final Array2DRowFieldMatrix<T> highOrder) {
final T[][] data = highOrder.getDataRef();
for (int i = 0; i < data.length; ++i) {
final T[] dataI = data[i];
final T c1I = c1[i];
for (int j = 0; j < dataI.length; ++j) {
dataI[j] = dataI[j].add(c1I.multiply(start[j].subtract(end[j])));
}
}
}
}