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* 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
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*
* 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,
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* See the License for the specific language governing permissions and
* limitations under the License.
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package org.apache.commons.math3.ode.nonstiff;
import java.util.Arrays;
import org.apache.commons.math3.Field;
import org.apache.commons.math3.RealFieldElement;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.linear.Array2DRowFieldMatrix;
import org.apache.commons.math3.linear.FieldMatrixPreservingVisitor;
import org.apache.commons.math3.ode.FieldExpandableODE;
import org.apache.commons.math3.ode.FieldODEState;
import org.apache.commons.math3.ode.FieldODEStateAndDerivative;
import org.apache.commons.math3.util.MathArrays;
import org.apache.commons.math3.util.MathUtils;
/**
* This class implements implicit Adams-Moulton integrators for Ordinary
* Differential Equations.
*
* Adams-Moulton methods (in fact due to Adams alone) are implicit
* multistep ODE solvers. This implementation is a variation of the classical
* one: it uses adaptive stepsize to implement error control, whereas
* classical implementations are fixed step size. The value of state vector
* at step n+1 is a simple combination of the value at step n and of the
* derivatives at steps n+1, n, n-1 ... Since y'n+1 is needed to
* compute yn+1, another method must be used to compute a first
* estimate of yn+1, then compute y'n+1, then compute
* a final estimate of yn+1 using the following formulas. Depending
* on the number k of previous steps one wants to use for computing the next
* value, different formulas are available for the final estimate:
*
* - k = 1: yn+1 = yn + h y'n+1
* - k = 2: yn+1 = yn + h (y'n+1+y'n)/2
* - k = 3: yn+1 = yn + h (5y'n+1+8y'n-y'n-1)/12
* - k = 4: yn+1 = yn + h (9y'n+1+19y'n-5y'n-1+y'n-2)/24
* - ...
*
*
* A k-steps Adams-Moulton method is of order k+1.
*
* Implementation details
*
* 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
*
*
* The definitions above use the classical representation with several previous first
* derivatives. Lets define
*
* qn = [ s1(n-1) s1(n-2) ... s1(n-(k-1)) ]T
*
* (we omit the k index in the notation for clarity). With these definitions,
* Adams-Moulton methods can be written:
*
* - k = 1: yn+1 = yn + s1(n+1)
* - k = 2: yn+1 = yn + 1/2 s1(n+1) + [ 1/2 ] qn+1
* - k = 3: yn+1 = yn + 5/12 s1(n+1) + [ 8/12 -1/12 ] qn+1
* - k = 4: yn+1 = yn + 9/24 s1(n+1) + [ 19/24 -5/24 1/24 ] qn+1
* - ...
*
*
* Instead of using the classical representation with first derivatives only (yn,
* s1(n+1) and qn+1), our implementation uses the Nordsieck vector with
* higher degrees scaled derivatives all taken at the same step (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. 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 ... ]
* [ ... ]
*
*
* Using the Nordsieck vector has several advantages:
*
* - it greatly simplifies step interpolation as the interpolator mainly applies
* Taylor series formulas,
* - it simplifies step changes that occur when discrete events that truncate
* the step are triggered,
* - it allows to extend the methods in order to support adaptive stepsize.
*
*
* 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
*
* 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 ]
*
* 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.
*
* The P-1u vector and the P-1 A P matrix do not depend on the state,
* they only depend on k and therefore are precomputed once for all.
*
* @param the type of the field elements
* @since 3.6
*/
public class AdamsMoultonFieldIntegrator> extends AdamsFieldIntegrator {
/** Integrator method name. */
private static final String METHOD_NAME = "Adams-Moulton";
/**
* Build an Adams-Moulton integrator with the given order and error control parameters.
* @param field field to which the time and state vector elements belong
* @param nSteps number of steps of the method excluding the one being computed
* @param minStep minimal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param maxStep maximal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param scalAbsoluteTolerance allowed absolute error
* @param scalRelativeTolerance allowed relative error
* @exception NumberIsTooSmallException if order is 1 or less
*/
public AdamsMoultonFieldIntegrator(final Field field, final int nSteps,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance)
throws NumberIsTooSmallException {
super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
scalAbsoluteTolerance, scalRelativeTolerance);
}
/**
* Build an Adams-Moulton integrator with the given order and error control parameters.
* @param field field to which the time and state vector elements belong
* @param nSteps number of steps of the method excluding the one being computed
* @param minStep minimal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param maxStep maximal step (sign is irrelevant, regardless of
* integration direction, forward or backward), the last step can
* be smaller than this
* @param vecAbsoluteTolerance allowed absolute error
* @param vecRelativeTolerance allowed relative error
* @exception IllegalArgumentException if order is 1 or less
*/
public AdamsMoultonFieldIntegrator(final Field field, final int nSteps,
final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance)
throws IllegalArgumentException {
super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
vecAbsoluteTolerance, vecRelativeTolerance);
}
/** {@inheritDoc} */
@Override
public FieldODEStateAndDerivative integrate(final FieldExpandableODE equations,
final FieldODEState initialState,
final T finalTime)
throws NumberIsTooSmallException, DimensionMismatchException,
MaxCountExceededException, NoBracketingException {
sanityChecks(initialState, finalTime);
final T t0 = initialState.getTime();
final T[] y = equations.getMapper().mapState(initialState);
setStepStart(initIntegration(equations, t0, y, finalTime));
final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0;
// compute the initial Nordsieck vector using the configured starter integrator
start(equations, getStepStart(), finalTime);
// reuse the step that was chosen by the starter integrator
FieldODEStateAndDerivative stepStart = getStepStart();
FieldODEStateAndDerivative stepEnd =
AdamsFieldStepInterpolator.taylor(stepStart,
stepStart.getTime().add(getStepSize()),
getStepSize(), scaled, nordsieck);
// main integration loop
setIsLastStep(false);
do {
T[] predictedY = null;
final T[] predictedScaled = MathArrays.buildArray(getField(), y.length);
Array2DRowFieldMatrix predictedNordsieck = null;
T error = getField().getZero().add(10);
while (error.subtract(1.0).getReal() >= 0.0) {
// predict a first estimate of the state at step end (P in the PECE sequence)
predictedY = stepEnd.getState();
// evaluate a first estimate of the derivative (first E in the PECE sequence)
final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY);
// update Nordsieck vector
for (int j = 0; j < predictedScaled.length; ++j) {
predictedScaled[j] = getStepSize().multiply(yDot[j]);
}
predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);
// apply correction (C in the PECE sequence)
error = predictedNordsieck.walkInOptimizedOrder(new Corrector(y, predictedScaled, predictedY));
if (error.subtract(1.0).getReal() >= 0.0) {
// reject the step and attempt to reduce error by stepsize control
final T factor = computeStepGrowShrinkFactor(error);
rescale(filterStep(getStepSize().multiply(factor), forward, false));
stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(),
getStepStart().getTime().add(getStepSize()),
getStepSize(),
scaled,
nordsieck);
}
}
// evaluate a final estimate of the derivative (second E in the PECE sequence)
final T[] correctedYDot = computeDerivatives(stepEnd.getTime(), predictedY);
// update Nordsieck vector
final T[] correctedScaled = MathArrays.buildArray(getField(), y.length);
for (int j = 0; j < correctedScaled.length; ++j) {
correctedScaled[j] = getStepSize().multiply(correctedYDot[j]);
}
updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, predictedNordsieck);
// discrete events handling
stepEnd = new FieldODEStateAndDerivative(stepEnd.getTime(), predictedY, correctedYDot);
setStepStart(acceptStep(new AdamsFieldStepInterpolator(getStepSize(), stepEnd,
correctedScaled, predictedNordsieck, forward,
getStepStart(), stepEnd,
equations.getMapper()),
finalTime));
scaled = correctedScaled;
nordsieck = predictedNordsieck;
if (!isLastStep()) {
System.arraycopy(predictedY, 0, y, 0, y.length);
if (resetOccurred()) {
// some events handler has triggered changes that
// invalidate the derivatives, we need to restart from scratch
start(equations, getStepStart(), finalTime);
}
// stepsize control for next step
final T factor = computeStepGrowShrinkFactor(error);
final T scaledH = getStepSize().multiply(factor);
final T nextT = getStepStart().getTime().add(scaledH);
final boolean nextIsLast = forward ?
nextT.subtract(finalTime).getReal() >= 0 :
nextT.subtract(finalTime).getReal() <= 0;
T hNew = filterStep(scaledH, forward, nextIsLast);
final T filteredNextT = getStepStart().getTime().add(hNew);
final boolean filteredNextIsLast = forward ?
filteredNextT.subtract(finalTime).getReal() >= 0 :
filteredNextT.subtract(finalTime).getReal() <= 0;
if (filteredNextIsLast) {
hNew = finalTime.subtract(getStepStart().getTime());
}
rescale(hNew);
stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), getStepStart().getTime().add(getStepSize()),
getStepSize(), scaled, nordsieck);
}
} while (!isLastStep());
final FieldODEStateAndDerivative finalState = getStepStart();
setStepStart(null);
setStepSize(null);
return finalState;
}
/** Corrector for current state in Adams-Moulton method.
*
* This visitor implements the Taylor series formula:
*
* Yn+1 = yn + s1(n+1) + [ -1 +1 -1 +1 ... ±1 ] rn+1
*
*
*/
private class Corrector implements FieldMatrixPreservingVisitor {
/** Previous state. */
private final T[] previous;
/** Current scaled first derivative. */
private final T[] scaled;
/** Current state before correction. */
private final T[] before;
/** Current state after correction. */
private final T[] after;
/** Simple constructor.
* @param previous previous state
* @param scaled current scaled first derivative
* @param state state to correct (will be overwritten after visit)
*/
Corrector(final T[] previous, final T[] scaled, final T[] state) {
this.previous = previous;
this.scaled = scaled;
this.after = state;
this.before = state.clone();
}
/** {@inheritDoc} */
public void start(int rows, int columns,
int startRow, int endRow, int startColumn, int endColumn) {
Arrays.fill(after, getField().getZero());
}
/** {@inheritDoc} */
public void visit(int row, int column, T value) {
if ((row & 0x1) == 0) {
after[column] = after[column].subtract(value);
} else {
after[column] = after[column].add(value);
}
}
/**
* End visiting the Nordsieck vector.
* The correction is used to control stepsize. So its amplitude is
* considered to be an error, which must be normalized according to
* error control settings. If the normalized value is greater than 1,
* the correction was too large and the step must be rejected.
* @return the normalized correction, if greater than 1, the step
* must be rejected
*/
public T end() {
T error = getField().getZero();
for (int i = 0; i < after.length; ++i) {
after[i] = after[i].add(previous[i].add(scaled[i]));
if (i < mainSetDimension) {
final T yScale = MathUtils.max(previous[i].abs(), after[i].abs());
final T tol = (vecAbsoluteTolerance == null) ?
yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) :
yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]);
final T ratio = after[i].subtract(before[i]).divide(tol); // (corrected-predicted)/tol
error = error.add(ratio.multiply(ratio));
}
}
return error.divide(mainSetDimension).sqrt();
}
}
}