Field-based implementation of Adams-Moulton ODE integrator.
This commit is contained in:
Luc Maisonobe
parent
2a690ee895
commit
82cf2774a2
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@ -316,6 +316,33 @@ public abstract class MultistepFieldIntegrator<T extends RealFieldElement<T>>
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return nSteps;
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}
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/** Rescale the instance.
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* <p>Since the scaled and Nordsieck arrays are shared with the caller,
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* this method has the side effect of rescaling this arrays in the caller too.</p>
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* @param newStepSize new step size to use in the scaled and Nordsieck arrays
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*/
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protected void rescale(final T newStepSize) {
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final T ratio = newStepSize.divide(getStepSize());
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for (int i = 0; i < scaled.length; ++i) {
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scaled[i] = scaled[i].multiply(ratio);
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}
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final T[][] nData = nordsieck.getDataRef();
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T power = ratio;
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for (int i = 0; i < nData.length; ++i) {
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power = power.multiply(ratio);
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final T[] nDataI = nData[i];
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for (int j = 0; j < nDataI.length; ++j) {
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nDataI[j] = nDataI[j].multiply(power);
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}
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}
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setStepSize(newStepSize);
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}
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/** Compute step grow/shrink factor according to normalized error.
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* @param error normalized error of the current step
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* @return grow/shrink factor for next step
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@ -255,9 +255,11 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
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start(equations, getStepStart(), finalTime);
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// reuse the step that was chosen by the starter integrator
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AdamsFieldStepInterpolator<T> interpolator =
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new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
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forward, equations.getMapper());
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FieldODEStateAndDerivative<T> stepStart = getStepStart();
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FieldODEStateAndDerivative<T> stepEnd =
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AdamsFieldStepInterpolator.taylor(stepStart,
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stepStart.getTime().add(getStepSize()),
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getStepSize(), scaled, nordsieck);
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// main integration loop
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setIsLastStep(false);
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@ -270,7 +272,6 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
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while (error.subtract(1.0).getReal() >= 0.0) {
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// predict a first estimate of the state at step end
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final FieldODEStateAndDerivative<T> stepEnd = interpolator.getCurrentState();
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predictedY = stepEnd.getState();
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// evaluate the derivative
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@ -290,26 +291,32 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
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// reject the step and attempt to reduce error by stepsize control
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final T factor = computeStepGrowShrinkFactor(error);
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rescale(filterStep(getStepSize().multiply(factor), forward, false));
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interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
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forward, equations.getMapper());
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stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(),
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getStepStart().getTime().add(getStepSize()),
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getStepSize(),
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scaled,
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nordsieck);
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}
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}
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// discrete events handling
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System.arraycopy(predictedY, 0, y, 0, y.length);
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setStepStart(acceptStep(interpolator, finalTime));
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setStepStart(acceptStep(new AdamsFieldStepInterpolator<T>(getStepSize(), stepEnd,
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predictedScaled, predictedNordsieck, forward,
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getStepStart(), stepEnd,
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equations.getMapper()),
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finalTime));
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scaled = predictedScaled;
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nordsieck = predictedNordsieck;
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if (!isLastStep()) {
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System.arraycopy(predictedY, 0, y, 0, y.length);
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if (resetOccurred()) {
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// some events handler has triggered changes that
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// invalidate the derivatives, we need to restart from scratch
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start(equations, getStepStart(), finalTime);
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interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
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forward, equations.getMapper());
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}
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// stepsize control for next step
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@ -330,8 +337,8 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
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}
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rescale(hNew);
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interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
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forward, equations.getMapper());
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stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), getStepStart().getTime().add(getStepSize()),
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getStepSize(), scaled, nordsieck);
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}
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@ -344,31 +351,4 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
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}
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/** Rescale the instance.
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* <p>Since the scaled and Nordsieck arrays are shared with the caller,
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* this method has the side effect of rescaling this arrays in the caller too.</p>
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* @param newStepSize new step size to use in the scaled and Nordsieck arrays
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*/
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public void rescale(final T newStepSize) {
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final T ratio = newStepSize.divide(getStepSize());
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for (int i = 0; i < scaled.length; ++i) {
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scaled[i] = scaled[i].multiply(ratio);
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}
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final T[][] nData = nordsieck.getDataRef();
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T power = ratio;
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for (int i = 0; i < nData.length; ++i) {
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power = power.multiply(ratio);
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final T[] nDataI = nData[i];
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for (int j = 0; j < nDataI.length; ++j) {
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nDataI[j] = nDataI[j].multiply(power);
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}
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}
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setStepSize(newStepSize);
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}
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}
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@ -43,6 +43,14 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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/** Step size used in the first scaled derivative and Nordsieck vector. */
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private T scalingH;
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/** Reference state.
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* <p>Sometimes, the reference state is the same as globalPreviousState,
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* sometimes it is the same as globalCurrentState, so we use a separate
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* field to avoid any confusion.
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* </p>
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*/
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private final FieldODEStateAndDerivative<T> reference;
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/** First scaled derivative. */
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private final T[] scaled;
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@ -51,22 +59,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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/** Simple constructor.
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* @param stepSize step size used in the scaled and Nordsieck arrays
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* @param referenceState reference state from which Taylor expansion are estimated
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* @param scaled first scaled derivative
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* @param nordsieck Nordsieck vector
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* @param isForward integration direction indicator
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* @param equationsMapper mapper for ODE equations primary and secondary components
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*/
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AdamsFieldStepInterpolator(final T stepSize, final FieldODEStateAndDerivative<T> referenceState,
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final T[] scaled, final Array2DRowFieldMatrix<T> nordsieck,
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final boolean isForward, final FieldEquationsMapper<T> equationsMapper) {
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this(stepSize, scaled, nordsieck, isForward,
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referenceState, taylor(referenceState, referenceState.getTime().add(stepSize), stepSize, scaled, nordsieck),
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equationsMapper);
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}
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/** Simple constructor.
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* @param stepSize step size used in the scaled and Nordsieck arrays
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* @param reference reference state from which Taylor expansion are estimated
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* @param scaled first scaled derivative
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* @param nordsieck Nordsieck vector
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* @param isForward integration direction indicator
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@ -74,19 +67,20 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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* @param globalCurrentState end of the global step
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* @param equationsMapper mapper for ODE equations primary and secondary components
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*/
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private AdamsFieldStepInterpolator(final T stepSize, final T[] scaled,
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final Array2DRowFieldMatrix<T> nordsieck,
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final boolean isForward,
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final FieldODEStateAndDerivative<T> globalPreviousState,
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final FieldODEStateAndDerivative<T> globalCurrentState,
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final FieldEquationsMapper<T> equationsMapper) {
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this(stepSize, scaled, nordsieck,
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AdamsFieldStepInterpolator(final T stepSize, final FieldODEStateAndDerivative<T> reference,
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final T[] scaled, final Array2DRowFieldMatrix<T> nordsieck,
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final boolean isForward,
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final FieldODEStateAndDerivative<T> globalPreviousState,
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final FieldODEStateAndDerivative<T> globalCurrentState,
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final FieldEquationsMapper<T> equationsMapper) {
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this(stepSize, reference, scaled, nordsieck,
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isForward, globalPreviousState, globalCurrentState,
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globalPreviousState, globalCurrentState, equationsMapper);
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}
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/** Simple constructor.
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* @param stepSize step size used in the scaled and Nordsieck arrays
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* @param reference reference state from which Taylor expansion are estimated
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* @param scaled first scaled derivative
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* @param nordsieck Nordsieck vector
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* @param isForward integration direction indicator
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@ -96,8 +90,8 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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* @param softCurrentState end of the restricted step
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* @param equationsMapper mapper for ODE equations primary and secondary components
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*/
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private AdamsFieldStepInterpolator(final T stepSize, final T[] scaled,
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final Array2DRowFieldMatrix<T> nordsieck,
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private AdamsFieldStepInterpolator(final T stepSize, final FieldODEStateAndDerivative<T> reference,
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final T[] scaled, final Array2DRowFieldMatrix<T> nordsieck,
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final boolean isForward,
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final FieldODEStateAndDerivative<T> globalPreviousState,
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final FieldODEStateAndDerivative<T> globalCurrentState,
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@ -107,6 +101,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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super(isForward, globalPreviousState, globalCurrentState,
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softPreviousState, softCurrentState, equationsMapper);
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this.scalingH = stepSize;
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this.reference = reference;
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this.scaled = scaled.clone();
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this.nordsieck = new Array2DRowFieldMatrix<T>(nordsieck.getData(), false);
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}
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@ -126,7 +121,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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FieldODEStateAndDerivative<T> newSoftPreviousState,
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FieldODEStateAndDerivative<T> newSoftCurrentState,
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FieldEquationsMapper<T> newMapper) {
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return new AdamsFieldStepInterpolator<T>(scalingH, scaled, nordsieck,
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return new AdamsFieldStepInterpolator<T>(scalingH, reference, scaled, nordsieck,
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newForward,
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newGlobalPreviousState, newGlobalCurrentState,
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newSoftPreviousState, newSoftCurrentState,
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@ -139,11 +134,11 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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protected FieldODEStateAndDerivative<T> computeInterpolatedStateAndDerivatives(final FieldEquationsMapper<T> equationsMapper,
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final T time, final T theta,
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final T thetaH, final T oneMinusThetaH) {
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return taylor(getPreviousState(), time, scalingH, scaled, nordsieck);
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return taylor(reference, time, scalingH, scaled, nordsieck);
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}
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/** Estimate state by applying Taylor formula.
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* @param referenceState reference state
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* @param reference reference state
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* @param time time at which state must be estimated
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* @param stepSize step size used in the scaled and Nordsieck arrays
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* @param scaled first scaled derivative
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@ -151,12 +146,12 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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* @return estimated state
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* @param <S> the type of the field elements
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*/
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private static <S extends RealFieldElement<S>> FieldODEStateAndDerivative<S> taylor(final FieldODEStateAndDerivative<S> referenceState,
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final S time, final S stepSize,
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final S[] scaled,
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final Array2DRowFieldMatrix<S> nordsieck) {
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public static <S extends RealFieldElement<S>> FieldODEStateAndDerivative<S> taylor(final FieldODEStateAndDerivative<S> reference,
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final S time, final S stepSize,
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final S[] scaled,
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final Array2DRowFieldMatrix<S> nordsieck) {
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final S x = time.subtract(referenceState.getTime());
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final S x = time.subtract(reference.getTime());
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final S normalizedAbscissa = x.divide(stepSize);
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S[] stateVariation = MathArrays.buildArray(time.getField(), scaled.length);
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@ -178,7 +173,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
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}
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}
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S[] estimatedState = referenceState.getState();
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S[] estimatedState = reference.getState();
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for (int j = 0; j < stateVariation.length; ++j) {
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stateVariation[j] = stateVariation[j].add(scaled[j].multiply(normalizedAbscissa));
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estimatedState[j] = estimatedState[j].add(stateVariation[j]);
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@ -0,0 +1,416 @@
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package org.apache.commons.math4.ode.nonstiff;
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import java.util.Arrays;
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import org.apache.commons.math4.Field;
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import org.apache.commons.math4.RealFieldElement;
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import org.apache.commons.math4.exception.DimensionMismatchException;
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import org.apache.commons.math4.exception.MaxCountExceededException;
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import org.apache.commons.math4.exception.NoBracketingException;
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import org.apache.commons.math4.exception.NumberIsTooSmallException;
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import org.apache.commons.math4.linear.Array2DRowFieldMatrix;
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import org.apache.commons.math4.linear.FieldMatrixPreservingVisitor;
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import org.apache.commons.math4.ode.FieldExpandableODE;
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import org.apache.commons.math4.ode.FieldODEState;
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import org.apache.commons.math4.ode.FieldODEStateAndDerivative;
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import org.apache.commons.math4.util.MathArrays;
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import org.apache.commons.math4.util.MathUtils;
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/**
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* This class implements implicit Adams-Moulton integrators for Ordinary
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* Differential Equations.
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*
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* <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
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* multistep ODE solvers. This implementation is a variation of the classical
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* one: it uses adaptive stepsize to implement error control, whereas
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* classical implementations are fixed step size. The value of state vector
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* at step n+1 is a simple combination of the value at step n and of the
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* derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
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* compute y<sub>n+1</sub>, another method must be used to compute a first
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* estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
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* a final estimate of y<sub>n+1</sub> using the following formulas. Depending
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* on the number k of previous steps one wants to use for computing the next
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* value, different formulas are available for the final estimate:</p>
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* <ul>
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* <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
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* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
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* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li>
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* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li>
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* <li>...</li>
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* </ul>
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*
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* <p>A k-steps Adams-Moulton method is of order k+1.</p>
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*
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* <h3>Implementation details</h3>
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*
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* <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
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* <pre>
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* s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
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* s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
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* s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
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* ...
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* s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
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* </pre></p>
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*
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* <p>The definitions above use the classical representation with several previous first
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* derivatives. Lets define
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* <pre>
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* 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>
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* </pre>
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* (we omit the k index in the notation for clarity). With these definitions,
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* Adams-Moulton methods can be written:
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* <ul>
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* <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
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* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li>
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* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li>
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* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</sub></li>
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* <li>...</li>
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* </ul></p>
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*
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* <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
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* s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
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* higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n)
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* and r<sub>n</sub>) where r<sub>n</sub> is defined as:
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* <pre>
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* r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
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* </pre>
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* (here again we omit the k index in the notation for clarity)
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* </p>
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*
|
||||
* <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. 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>Using the Nordsieck vector has several advantages:
|
||||
* <ul>
|
||||
* <li>it greatly simplifies step interpolation as the interpolator mainly applies
|
||||
* Taylor series formulas,</li>
|
||||
* <li>it simplifies step changes that occur when discrete events that truncate
|
||||
* the step are triggered,</li>
|
||||
* <li>it allows to extend the methods in order to support adaptive stepsize.</li>
|
||||
* </ul></p>
|
||||
*
|
||||
* <p>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>
|
||||
* 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>
|
||||
* 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>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
|
||||
* they only depend on k and therefore are precomputed once for all.</p>
|
||||
*
|
||||
* @param <T> the type of the field elements
|
||||
* @since 3.6
|
||||
*/
|
||||
public class AdamsMoultonFieldIntegrator<T extends RealFieldElement<T>> extends AdamsFieldIntegrator<T> {
|
||||
|
||||
/** 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<T> 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<T> 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<T> integrate(final FieldExpandableODE<T> equations,
|
||||
final FieldODEState<T> 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<T> stepStart = getStepStart();
|
||||
FieldODEStateAndDerivative<T> 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<T> 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<T>(stepEnd.getTime(), predictedY, correctedYDot);
|
||||
setStepStart(acceptStep(new AdamsFieldStepInterpolator<T>(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<T> finalState = getStepStart();
|
||||
setStepStart(null);
|
||||
setStepSize(null);
|
||||
return finalState;
|
||||
|
||||
}
|
||||
|
||||
/** Corrector for current state in Adams-Moulton method.
|
||||
* <p>
|
||||
* This visitor implements the Taylor series formula:
|
||||
* <pre>
|
||||
* 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>
|
||||
* </pre>
|
||||
* </p>
|
||||
*/
|
||||
private class Corrector implements FieldMatrixPreservingVisitor<T> {
|
||||
|
||||
/** 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.
|
||||
* <p>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.</p>
|
||||
* @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();
|
||||
|
||||
}
|
||||
}
|
||||
|
||||
}
|
||||
|
|
@ -74,10 +74,10 @@ public abstract class AbstractAdamsFieldIntegratorTest {
|
|||
public abstract void testIncreasingTolerance();
|
||||
|
||||
protected <T extends RealFieldElement<T>> void doTestIncreasingTolerance(final Field<T> field,
|
||||
int ratioMin, int ratioMax) {
|
||||
double ratioMin, double ratioMax) {
|
||||
|
||||
int previousCalls = Integer.MAX_VALUE;
|
||||
for (int i = -12; i < -5; ++i) {
|
||||
for (int i = -12; i < -2; ++i) {
|
||||
TestFieldProblem1<T> pb = new TestFieldProblem1<T>(field);
|
||||
double minStep = 0;
|
||||
double maxStep = pb.getFinalTime().subtract(pb.getInitialState().getTime()).getReal();
|
||||
|
|
@ -106,7 +106,7 @@ public abstract class AbstractAdamsFieldIntegratorTest {
|
|||
@Test(expected = MaxCountExceededException.class)
|
||||
public abstract void exceedMaxEvaluations();
|
||||
|
||||
protected <T extends RealFieldElement<T>> void doExceedMaxEvaluations(final Field<T> field) {
|
||||
protected <T extends RealFieldElement<T>> void doExceedMaxEvaluations(final Field<T> field, final int max) {
|
||||
|
||||
TestFieldProblem1<T> pb = new TestFieldProblem1<T>(field);
|
||||
double range = pb.getFinalTime().subtract(pb.getInitialState().getTime()).getReal();
|
||||
|
|
@ -114,7 +114,7 @@ public abstract class AbstractAdamsFieldIntegratorTest {
|
|||
FirstOrderFieldIntegrator<T> integ = createIntegrator(field, 2, 0, range, 1.0e-12, 1.0e-12);
|
||||
TestFieldProblemHandler<T> handler = new TestFieldProblemHandler<T>(pb, integ);
|
||||
integ.addStepHandler(handler);
|
||||
integ.setMaxEvaluations(650);
|
||||
integ.setMaxEvaluations(max);
|
||||
integ.integrate(new FieldExpandableODE<T>(pb), pb.getInitialState(), pb.getFinalTime());
|
||||
|
||||
}
|
||||
|
|
@ -132,7 +132,6 @@ public abstract class AbstractAdamsFieldIntegratorTest {
|
|||
double range = pb.getFinalTime().subtract(pb.getInitialState().getTime()).getReal();
|
||||
|
||||
AdamsFieldIntegrator<T> integ = createIntegrator(field, 4, 0, range, 1.0e-12, 1.0e-12);
|
||||
integ.setStarterIntegrator(new PerfectStarter<T>(pb, (integ.getNSteps() + 5) / 2));
|
||||
TestFieldProblemHandler<T> handler = new TestFieldProblemHandler<T>(pb, integ);
|
||||
integ.addStepHandler(handler);
|
||||
integ.integrate(new FieldExpandableODE<T>(pb), pb.getInitialState(), pb.getFinalTime());
|
||||
|
|
|
|||
|
|
@ -49,15 +49,15 @@ public class AdamsBashforthFieldIntegratorTest extends AbstractAdamsFieldIntegra
|
|||
|
||||
@Test
|
||||
public void testIncreasingTolerance() {
|
||||
// the 7 and 121 factors are only valid for this test
|
||||
// the 2.6 and 122 factors are only valid for this test
|
||||
// and has been obtained from trial and error
|
||||
// there are no general relationship between local and global errors
|
||||
doTestIncreasingTolerance(Decimal64Field.getInstance(), 7, 121);
|
||||
doTestIncreasingTolerance(Decimal64Field.getInstance(), 2.6, 122);
|
||||
}
|
||||
|
||||
@Test(expected = MaxCountExceededException.class)
|
||||
public void exceedMaxEvaluations() {
|
||||
doExceedMaxEvaluations(Decimal64Field.getInstance());
|
||||
doExceedMaxEvaluations(Decimal64Field.getInstance(), 650);
|
||||
}
|
||||
|
||||
@Test
|
||||
|
|
|
|||
|
|
@ -77,7 +77,7 @@ public class AdamsBashforthIntegratorTest {
|
|||
public void testIncreasingTolerance() throws DimensionMismatchException, NumberIsTooSmallException, MaxCountExceededException, NoBracketingException {
|
||||
|
||||
int previousCalls = Integer.MAX_VALUE;
|
||||
for (int i = -12; i < -5; ++i) {
|
||||
for (int i = -12; i < -2; ++i) {
|
||||
TestProblem1 pb = new TestProblem1();
|
||||
double minStep = 0;
|
||||
double maxStep = pb.getFinalTime() - pb.getInitialTime();
|
||||
|
|
@ -93,10 +93,10 @@ public class AdamsBashforthIntegratorTest {
|
|||
pb.getInitialTime(), pb.getInitialState(),
|
||||
pb.getFinalTime(), new double[pb.getDimension()]);
|
||||
|
||||
// the 8 and 122 factors are only valid for this test
|
||||
// the 2.6 and 122 factors are only valid for this test
|
||||
// and has been obtained from trial and error
|
||||
// there are no general relationship between local and global errors
|
||||
Assert.assertTrue(handler.getMaximalValueError() > ( 8 * scalAbsoluteTolerance));
|
||||
Assert.assertTrue(handler.getMaximalValueError() > (2.6 * scalAbsoluteTolerance));
|
||||
Assert.assertTrue(handler.getMaximalValueError() < (122 * scalAbsoluteTolerance));
|
||||
|
||||
int calls = pb.getCalls();
|
||||
|
|
|
|||
|
|
@ -0,0 +1,78 @@
|
|||
/*
|
||||
* 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.math4.ode.nonstiff;
|
||||
|
||||
|
||||
import org.apache.commons.math4.Field;
|
||||
import org.apache.commons.math4.RealFieldElement;
|
||||
import org.apache.commons.math4.exception.MathIllegalStateException;
|
||||
import org.apache.commons.math4.exception.MaxCountExceededException;
|
||||
import org.apache.commons.math4.exception.NumberIsTooSmallException;
|
||||
import org.apache.commons.math4.util.Decimal64Field;
|
||||
import org.junit.Test;
|
||||
|
||||
public class AdamsMoultonFieldIntegratorTest extends AbstractAdamsFieldIntegratorTest {
|
||||
|
||||
protected <T extends RealFieldElement<T>> AdamsFieldIntegrator<T>
|
||||
createIntegrator(Field<T> field, final int nSteps, final double minStep, final double maxStep,
|
||||
final double scalAbsoluteTolerance, final double scalRelativeTolerance) {
|
||||
return new AdamsMoultonFieldIntegrator<T>(field, nSteps, minStep, maxStep,
|
||||
scalAbsoluteTolerance, scalRelativeTolerance);
|
||||
}
|
||||
|
||||
protected <T extends RealFieldElement<T>> AdamsFieldIntegrator<T>
|
||||
createIntegrator(Field<T> field, final int nSteps, final double minStep, final double maxStep,
|
||||
final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance) {
|
||||
return new AdamsMoultonFieldIntegrator<T>(field, nSteps, minStep, maxStep,
|
||||
vecAbsoluteTolerance, vecRelativeTolerance);
|
||||
}
|
||||
|
||||
@Test(expected=NumberIsTooSmallException.class)
|
||||
public void testMinStep() {
|
||||
doDimensionCheck(Decimal64Field.getInstance());
|
||||
}
|
||||
|
||||
@Test
|
||||
public void testIncreasingTolerance() {
|
||||
// the 0.45 and 8.69 factors are only valid for this test
|
||||
// and has been obtained from trial and error
|
||||
// there are no general relationship between local and global errors
|
||||
doTestIncreasingTolerance(Decimal64Field.getInstance(), 0.45, 8.69);
|
||||
}
|
||||
|
||||
@Test(expected = MaxCountExceededException.class)
|
||||
public void exceedMaxEvaluations() {
|
||||
doExceedMaxEvaluations(Decimal64Field.getInstance(), 650);
|
||||
}
|
||||
|
||||
@Test
|
||||
public void backward() {
|
||||
doBackward(Decimal64Field.getInstance(), 3.0e-9, 3.0e-9, 1.0e-16, "Adams-Moulton");
|
||||
}
|
||||
|
||||
@Test
|
||||
public void polynomial() {
|
||||
doPolynomial(Decimal64Field.getInstance(), 5, 2.2e-05, 1.1e-11);
|
||||
}
|
||||
|
||||
@Test(expected=MathIllegalStateException.class)
|
||||
public void testStartFailure() {
|
||||
doTestStartFailure(Decimal64Field.getInstance());
|
||||
}
|
||||
|
||||
}
|
||||
Loading…
Reference in New Issue