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Posted to commits@commons.apache.org by lu...@apache.org on 2016/01/06 14:51:01 UTC
[45/50] [abbrv] [math] Field-based implementation of Adams-Moulton
ODE integrator.
Field-based implementation of Adams-Moulton ODE integrator.
Project: http://git-wip-us.apache.org/repos/asf/commons-math/repo
Commit: http://git-wip-us.apache.org/repos/asf/commons-math/commit/82cf2774
Tree: http://git-wip-us.apache.org/repos/asf/commons-math/tree/82cf2774
Diff: http://git-wip-us.apache.org/repos/asf/commons-math/diff/82cf2774
Branch: refs/heads/master
Commit: 82cf2774a215ae46477e4b35decf77321e20ab34
Parents: 2a690ee
Author: Luc Maisonobe <lu...@apache.org>
Authored: Wed Jan 6 14:19:07 2016 +0100
Committer: Luc Maisonobe <lu...@apache.org>
Committed: Wed Jan 6 14:19:07 2016 +0100
----------------------------------------------------------------------
.../math4/ode/MultistepFieldIntegrator.java | 27 ++
.../nonstiff/AdamsBashforthFieldIntegrator.java | 58 +--
.../nonstiff/AdamsFieldStepInterpolator.java | 63 ++-
.../nonstiff/AdamsMoultonFieldIntegrator.java | 416 +++++++++++++++++++
.../AbstractAdamsFieldIntegratorTest.java | 9 +-
.../AdamsBashforthFieldIntegratorTest.java | 6 +-
.../nonstiff/AdamsBashforthIntegratorTest.java | 6 +-
.../AdamsMoultonFieldIntegratorTest.java | 78 ++++
8 files changed, 579 insertions(+), 84 deletions(-)
----------------------------------------------------------------------
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/main/java/org/apache/commons/math4/ode/MultistepFieldIntegrator.java
----------------------------------------------------------------------
diff --git a/src/main/java/org/apache/commons/math4/ode/MultistepFieldIntegrator.java b/src/main/java/org/apache/commons/math4/ode/MultistepFieldIntegrator.java
index feec974..d1ad3c8 100644
--- a/src/main/java/org/apache/commons/math4/ode/MultistepFieldIntegrator.java
+++ b/src/main/java/org/apache/commons/math4/ode/MultistepFieldIntegrator.java
@@ -316,6 +316,33 @@ public abstract class MultistepFieldIntegrator<T extends RealFieldElement<T>>
return nSteps;
}
+ /** Rescale the instance.
+ * <p>Since the scaled and Nordsieck arrays are shared with the caller,
+ * this method has the side effect of rescaling this arrays in the caller too.</p>
+ * @param newStepSize new step size to use in the scaled and Nordsieck arrays
+ */
+ protected void rescale(final T newStepSize) {
+
+ final T ratio = newStepSize.divide(getStepSize());
+ for (int i = 0; i < scaled.length; ++i) {
+ scaled[i] = scaled[i].multiply(ratio);
+ }
+
+ final T[][] nData = nordsieck.getDataRef();
+ T power = ratio;
+ for (int i = 0; i < nData.length; ++i) {
+ power = power.multiply(ratio);
+ final T[] nDataI = nData[i];
+ for (int j = 0; j < nDataI.length; ++j) {
+ nDataI[j] = nDataI[j].multiply(power);
+ }
+ }
+
+ setStepSize(newStepSize);
+
+ }
+
+
/** Compute step grow/shrink factor according to normalized error.
* @param error normalized error of the current step
* @return grow/shrink factor for next step
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java
----------------------------------------------------------------------
diff --git a/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java
index db6bf4f..977573e 100644
--- a/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java
+++ b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java
@@ -255,9 +255,11 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
start(equations, getStepStart(), finalTime);
// reuse the step that was chosen by the starter integrator
- AdamsFieldStepInterpolator<T> interpolator =
- new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
- forward, equations.getMapper());
+ FieldODEStateAndDerivative<T> stepStart = getStepStart();
+ FieldODEStateAndDerivative<T> stepEnd =
+ AdamsFieldStepInterpolator.taylor(stepStart,
+ stepStart.getTime().add(getStepSize()),
+ getStepSize(), scaled, nordsieck);
// main integration loop
setIsLastStep(false);
@@ -270,7 +272,6 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
while (error.subtract(1.0).getReal() >= 0.0) {
// predict a first estimate of the state at step end
- final FieldODEStateAndDerivative<T> stepEnd = interpolator.getCurrentState();
predictedY = stepEnd.getState();
// evaluate the derivative
@@ -290,26 +291,32 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
// reject the step and attempt to reduce error by stepsize control
final T factor = computeStepGrowShrinkFactor(error);
rescale(filterStep(getStepSize().multiply(factor), forward, false));
- interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
- forward, equations.getMapper());
+ stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(),
+ getStepStart().getTime().add(getStepSize()),
+ getStepSize(),
+ scaled,
+ nordsieck);
}
}
// discrete events handling
- System.arraycopy(predictedY, 0, y, 0, y.length);
- setStepStart(acceptStep(interpolator, finalTime));
+ setStepStart(acceptStep(new AdamsFieldStepInterpolator<T>(getStepSize(), stepEnd,
+ predictedScaled, predictedNordsieck, forward,
+ getStepStart(), stepEnd,
+ equations.getMapper()),
+ finalTime));
scaled = predictedScaled;
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);
- interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
- forward, equations.getMapper());
}
// stepsize control for next step
@@ -330,8 +337,8 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
}
rescale(hNew);
- interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(), scaled, nordsieck,
- forward, equations.getMapper());
+ stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), getStepStart().getTime().add(getStepSize()),
+ getStepSize(), scaled, nordsieck);
}
@@ -344,31 +351,4 @@ public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extend
}
- /** Rescale the instance.
- * <p>Since the scaled and Nordsieck arrays are shared with the caller,
- * this method has the side effect of rescaling this arrays in the caller too.</p>
- * @param newStepSize new step size to use in the scaled and Nordsieck arrays
- */
- public void rescale(final T newStepSize) {
-
- final T ratio = newStepSize.divide(getStepSize());
- for (int i = 0; i < scaled.length; ++i) {
- scaled[i] = scaled[i].multiply(ratio);
- }
-
- final T[][] nData = nordsieck.getDataRef();
- T power = ratio;
- for (int i = 0; i < nData.length; ++i) {
- power = power.multiply(ratio);
- final T[] nDataI = nData[i];
- for (int j = 0; j < nDataI.length; ++j) {
- nDataI[j] = nDataI[j].multiply(power);
- }
- }
-
- setStepSize(newStepSize);
-
- }
-
-
}
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsFieldStepInterpolator.java
----------------------------------------------------------------------
diff --git a/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsFieldStepInterpolator.java b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsFieldStepInterpolator.java
index 78c3c8e..b4b5357 100644
--- a/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsFieldStepInterpolator.java
+++ b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsFieldStepInterpolator.java
@@ -43,6 +43,14 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
/** Step size used in the first scaled derivative and Nordsieck vector. */
private T scalingH;
+ /** Reference state.
+ * <p>Sometimes, the reference state is the same as globalPreviousState,
+ * sometimes it is the same as globalCurrentState, so we use a separate
+ * field to avoid any confusion.
+ * </p>
+ */
+ private final FieldODEStateAndDerivative<T> reference;
+
/** First scaled derivative. */
private final T[] scaled;
@@ -51,22 +59,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
/** Simple constructor.
* @param stepSize step size used in the scaled and Nordsieck arrays
- * @param referenceState reference state from which Taylor expansion are estimated
- * @param scaled first scaled derivative
- * @param nordsieck Nordsieck vector
- * @param isForward integration direction indicator
- * @param equationsMapper mapper for ODE equations primary and secondary components
- */
- AdamsFieldStepInterpolator(final T stepSize, final FieldODEStateAndDerivative<T> referenceState,
- final T[] scaled, final Array2DRowFieldMatrix<T> nordsieck,
- final boolean isForward, final FieldEquationsMapper<T> equationsMapper) {
- this(stepSize, scaled, nordsieck, isForward,
- referenceState, taylor(referenceState, referenceState.getTime().add(stepSize), stepSize, scaled, nordsieck),
- equationsMapper);
- }
-
- /** Simple constructor.
- * @param stepSize step size used in the scaled and Nordsieck arrays
+ * @param reference reference state from which Taylor expansion are estimated
* @param scaled first scaled derivative
* @param nordsieck Nordsieck vector
* @param isForward integration direction indicator
@@ -74,19 +67,20 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
* @param globalCurrentState end of the global step
* @param equationsMapper mapper for ODE equations primary and secondary components
*/
- private AdamsFieldStepInterpolator(final T stepSize, final T[] scaled,
- final Array2DRowFieldMatrix<T> nordsieck,
- final boolean isForward,
- final FieldODEStateAndDerivative<T> globalPreviousState,
- final FieldODEStateAndDerivative<T> globalCurrentState,
- final FieldEquationsMapper<T> equationsMapper) {
- this(stepSize, scaled, nordsieck,
+ AdamsFieldStepInterpolator(final T stepSize, final FieldODEStateAndDerivative<T> reference,
+ final T[] scaled, final Array2DRowFieldMatrix<T> nordsieck,
+ final boolean isForward,
+ final FieldODEStateAndDerivative<T> globalPreviousState,
+ final FieldODEStateAndDerivative<T> globalCurrentState,
+ final FieldEquationsMapper<T> equationsMapper) {
+ this(stepSize, reference, scaled, nordsieck,
isForward, globalPreviousState, globalCurrentState,
globalPreviousState, globalCurrentState, equationsMapper);
}
/** Simple constructor.
* @param stepSize step size used in the scaled and Nordsieck arrays
+ * @param reference reference state from which Taylor expansion are estimated
* @param scaled first scaled derivative
* @param nordsieck Nordsieck vector
* @param isForward integration direction indicator
@@ -96,8 +90,8 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
* @param softCurrentState end of the restricted step
* @param equationsMapper mapper for ODE equations primary and secondary components
*/
- private AdamsFieldStepInterpolator(final T stepSize, final T[] scaled,
- final Array2DRowFieldMatrix<T> nordsieck,
+ private AdamsFieldStepInterpolator(final T stepSize, final FieldODEStateAndDerivative<T> reference,
+ final T[] scaled, final Array2DRowFieldMatrix<T> nordsieck,
final boolean isForward,
final FieldODEStateAndDerivative<T> globalPreviousState,
final FieldODEStateAndDerivative<T> globalCurrentState,
@@ -107,6 +101,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
super(isForward, globalPreviousState, globalCurrentState,
softPreviousState, softCurrentState, equationsMapper);
this.scalingH = stepSize;
+ this.reference = reference;
this.scaled = scaled.clone();
this.nordsieck = new Array2DRowFieldMatrix<T>(nordsieck.getData(), false);
}
@@ -126,7 +121,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
FieldODEStateAndDerivative<T> newSoftPreviousState,
FieldODEStateAndDerivative<T> newSoftCurrentState,
FieldEquationsMapper<T> newMapper) {
- return new AdamsFieldStepInterpolator<T>(scalingH, scaled, nordsieck,
+ return new AdamsFieldStepInterpolator<T>(scalingH, reference, scaled, nordsieck,
newForward,
newGlobalPreviousState, newGlobalCurrentState,
newSoftPreviousState, newSoftCurrentState,
@@ -139,11 +134,11 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
protected FieldODEStateAndDerivative<T> computeInterpolatedStateAndDerivatives(final FieldEquationsMapper<T> equationsMapper,
final T time, final T theta,
final T thetaH, final T oneMinusThetaH) {
- return taylor(getPreviousState(), time, scalingH, scaled, nordsieck);
+ return taylor(reference, time, scalingH, scaled, nordsieck);
}
/** Estimate state by applying Taylor formula.
- * @param referenceState reference state
+ * @param reference reference state
* @param time time at which state must be estimated
* @param stepSize step size used in the scaled and Nordsieck arrays
* @param scaled first scaled derivative
@@ -151,12 +146,12 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
* @return estimated state
* @param <S> the type of the field elements
*/
- private static <S extends RealFieldElement<S>> FieldODEStateAndDerivative<S> taylor(final FieldODEStateAndDerivative<S> referenceState,
- final S time, final S stepSize,
- final S[] scaled,
- final Array2DRowFieldMatrix<S> nordsieck) {
+ public static <S extends RealFieldElement<S>> FieldODEStateAndDerivative<S> taylor(final FieldODEStateAndDerivative<S> reference,
+ final S time, final S stepSize,
+ final S[] scaled,
+ final Array2DRowFieldMatrix<S> nordsieck) {
- final S x = time.subtract(referenceState.getTime());
+ final S x = time.subtract(reference.getTime());
final S normalizedAbscissa = x.divide(stepSize);
S[] stateVariation = MathArrays.buildArray(time.getField(), scaled.length);
@@ -178,7 +173,7 @@ class AdamsFieldStepInterpolator<T extends RealFieldElement<T>> extends Abstract
}
}
- S[] estimatedState = referenceState.getState();
+ S[] estimatedState = reference.getState();
for (int j = 0; j < stateVariation.length; ++j) {
stateVariation[j] = stateVariation[j].add(scaled[j].multiply(normalizedAbscissa));
estimatedState[j] = estimatedState[j].add(stateVariation[j]);
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegrator.java
----------------------------------------------------------------------
diff --git a/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegrator.java b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegrator.java
new file mode 100644
index 0000000..b09942d
--- /dev/null
+++ b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegrator.java
@@ -0,0 +1,416 @@
+/*
+ * 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 java.util.Arrays;
+
+import org.apache.commons.math4.Field;
+import org.apache.commons.math4.RealFieldElement;
+import org.apache.commons.math4.exception.DimensionMismatchException;
+import org.apache.commons.math4.exception.MaxCountExceededException;
+import org.apache.commons.math4.exception.NoBracketingException;
+import org.apache.commons.math4.exception.NumberIsTooSmallException;
+import org.apache.commons.math4.linear.Array2DRowFieldMatrix;
+import org.apache.commons.math4.linear.FieldMatrixPreservingVisitor;
+import org.apache.commons.math4.ode.FieldExpandableODE;
+import org.apache.commons.math4.ode.FieldODEState;
+import org.apache.commons.math4.ode.FieldODEStateAndDerivative;
+import org.apache.commons.math4.util.MathArrays;
+import org.apache.commons.math4.util.MathUtils;
+
+
+/**
+ * This class implements implicit Adams-Moulton integrators for Ordinary
+ * Differential Equations.
+ *
+ * <p>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'<sub>n+1</sub> is needed to
+ * compute y<sub>n+1</sub>, another method must be used to compute a first
+ * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
+ * a final estimate of y<sub>n+1</sub> 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:</p>
+ * <ul>
+ * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
+ * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
+ * <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>
+ * <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>
+ * <li>...</li>
+ * </ul>
+ *
+ * <p>A k-steps Adams-Moulton method is of order k+1.</p>
+ *
+ * <h3>Implementation details</h3>
+ *
+ * <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>The definitions above use the classical representation with several previous first
+ * derivatives. Lets define
+ * <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). With these definitions,
+ * Adams-Moulton methods can be written:
+ * <ul>
+ * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
+ * <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>
+ * <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>
+ * <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>
+ * <li>...</li>
+ * </ul></p>
+ *
+ * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>,
+ * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with
+ * higher degrees scaled derivatives all taken at the same step (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. 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();
+
+ }
+ }
+
+}
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/test/java/org/apache/commons/math4/ode/nonstiff/AbstractAdamsFieldIntegratorTest.java
----------------------------------------------------------------------
diff --git a/src/test/java/org/apache/commons/math4/ode/nonstiff/AbstractAdamsFieldIntegratorTest.java b/src/test/java/org/apache/commons/math4/ode/nonstiff/AbstractAdamsFieldIntegratorTest.java
index 0d22831..606c9e8 100644
--- a/src/test/java/org/apache/commons/math4/ode/nonstiff/AbstractAdamsFieldIntegratorTest.java
+++ b/src/test/java/org/apache/commons/math4/ode/nonstiff/AbstractAdamsFieldIntegratorTest.java
@@ -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());
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java
----------------------------------------------------------------------
diff --git a/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java
index 408e646..e9a046c 100644
--- a/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java
+++ b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.java
@@ -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
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthIntegratorTest.java
----------------------------------------------------------------------
diff --git a/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthIntegratorTest.java b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthIntegratorTest.java
index 85c7e43..f655238 100644
--- a/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthIntegratorTest.java
+++ b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthIntegratorTest.java
@@ -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();
http://git-wip-us.apache.org/repos/asf/commons-math/blob/82cf2774/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegratorTest.java
----------------------------------------------------------------------
diff --git a/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegratorTest.java b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegratorTest.java
new file mode 100644
index 0000000..c44124a
--- /dev/null
+++ b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsMoultonFieldIntegratorTest.java
@@ -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());
+ }
+
+}