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From l..@apache.org
Subject [43/50] [abbrv] [math] Field-based Adams-Bashforth integrator.
Date Wed, 06 Jan 2016 13:50:59 GMT
Field-based Adams-Bashforth 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/305934df
Tree: http://git-wip-us.apache.org/repos/asf/commons-math/tree/305934df
Diff: http://git-wip-us.apache.org/repos/asf/commons-math/diff/305934df

Branch: refs/heads/master
Commit: 305934dfbd2b37deb50cf93732e442c51bb1603b
Parents: dd9dc94
Author: Luc Maisonobe <luc@apache.org>
Authored: Wed Jan 6 14:18:39 2016 +0100
Committer: Luc Maisonobe <luc@apache.org>
Committed: Wed Jan 6 14:18:39 2016 +0100

----------------------------------------------------------------------
 .../nonstiff/AdamsBashforthFieldIntegrator.java | 374 +++++++++++++++++++
 .../AdamsBashforthFieldIntegratorTest.java      |  78 ++++
 2 files changed, 452 insertions(+)
----------------------------------------------------------------------


http://git-wip-us.apache.org/repos/asf/commons-math/blob/305934df/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
new file mode 100644
index 0000000..db6bf4f
--- /dev/null
+++ b/src/main/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegrator.java
@@ -0,0 +1,374 @@
+/*
+ * 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.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.FieldMatrix;
+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;
+
+
+/**
+ * This class implements explicit Adams-Bashforth integrators for Ordinary
+ * Differential Equations.
+ *
+ * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
+ * 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, n-1, n-2 ... Depending on the number k of previous
+ * steps one wants to use for computing the next value, different formulas
+ * are available:</p>
+ * <ul>
+ *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
+ *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
+ *   <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (23y'<sub>n</sub>-16y'<sub>n-1</sub>+5y'<sub>n-2</sub>)/12</li>
+ *   <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (55y'<sub>n</sub>-59y'<sub>n-1</sub>+37y'<sub>n-2</sub>-9y'<sub>n-3</sub>)/24</li>
+ *   <li>...</li>
+ * </ul>
+ *
+ * <p>A k-steps Adams-Bashforth method is of order k.</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-Bashforth methods can be written:
+ * <ul>
+ *   <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
+ *   <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 3/2 s<sub>1</sub>(n)
+ [ -1/2 ] q<sub>n</sub></li>
+ *   <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 23/12 s<sub>1</sub>(n)
+ [ -16/12 5/12 ] q<sub>n</sub></li>
+ *   <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 55/24 s<sub>1</sub>(n)
+ [ -59/24 37/24 -9/24 ] q<sub>n</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) and q<sub>n</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) + &sum;<sub>j&gt;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)&times;(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 Nordsieck vector at step n+1 is computed from the Nordsieck vector at step
n as follows:
+ * <ul>
+ *   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)
+ u<sup>T</sup> r<sub>n</sub></li>
+ *   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
+ *   <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1))
P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li>
+ * </ul>
+ * where A is a rows shifting matrix (the lower left part is an identity matrix):
+ * <pre>
+ *        [ 0 0   ...  0 0 | 0 ]
+ *        [ ---------------+---]
+ *        [ 1 0   ...  0 0 | 0 ]
+ *    A = [ 0 1   ...  0 0 | 0 ]
+ *        [       ...      | 0 ]
+ *        [ 0 0   ...  1 0 | 0 ]
+ *        [ 0 0   ...  0 1 | 0 ]
+ * </pre></p>
+ *
+ * <p>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 AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extends
AdamsFieldIntegrator<T> {
+
+    /** Integrator method name. */
+    private static final String METHOD_NAME = "Adams-Bashforth";
+
+    /**
+     * Build an Adams-Bashforth integrator with the given order and step 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 AdamsBashforthFieldIntegrator(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, minStep, maxStep,
+              scalAbsoluteTolerance, scalRelativeTolerance);
+    }
+
+    /**
+     * Build an Adams-Bashforth integrator with the given order and step 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 AdamsBashforthFieldIntegrator(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, minStep, maxStep,
+              vecAbsoluteTolerance, vecRelativeTolerance);
+    }
+
+    /** Estimate error.
+     * <p>
+     * Error is estimated by interpolating back to previous state using
+     * the state Taylor expansion and comparing to real previous state.
+     * </p>
+     * @param previousState state vector at step start
+     * @param predictedState predicted state vector at step end
+     * @param predictedScaled predicted value of the scaled derivatives at step end
+     * @param predictedNordsieck predicted value of the Nordsieck vector at step end
+     * @return estimated normalized local discretization error
+     */
+    private T errorEstimation(final T[] previousState,
+                              final T[] predictedState,
+                              final T[] predictedScaled,
+                              final FieldMatrix<T> predictedNordsieck) {
+
+        T error = getField().getZero();
+        for (int i = 0; i < mainSetDimension; ++i) {
+            final T yScale = predictedState[i].abs();
+            final T tol = (vecAbsoluteTolerance == null) ?
+                          yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance)
:
+                          yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]);
+
+            // apply Taylor formula from high order to low order,
+            // for the sake of numerical accuracy
+            T variation = getField().getZero();
+            int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1;
+            for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) {
+                variation = variation.add(predictedNordsieck.getEntry(k, i).multiply(sign));
+                sign      = -sign;
+            }
+            variation = variation.subtract(predictedScaled[i]);
+
+            final T ratio  = predictedState[i].subtract(previousState[i]).add(variation).divide(tol);
+            error = error.add(ratio.multiply(ratio));
+
+        }
+
+        return error.divide(mainSetDimension).sqrt();
+
+    }
+
+    /** {@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
+        AdamsFieldStepInterpolator<T> interpolator =
+                        new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(),
scaled, nordsieck,
+                                                          forward, equations.getMapper());
+
+        // 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
+                final FieldODEStateAndDerivative<T> stepEnd = interpolator.getCurrentState();
+                predictedY = stepEnd.getState();
+
+                // evaluate the derivative
+                final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY);
+
+                // predict Nordsieck vector at step end
+                for (int j = 0; j < predictedScaled.length; ++j) {
+                    predictedScaled[j] = getStepSize().multiply(yDot[j]);
+                }
+                predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
+                updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);
+
+                // evaluate error
+                error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck);
+
+                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));
+                    interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(),
getStepStart(), scaled, nordsieck,
+                                                                     forward, equations.getMapper());
+
+                }
+            }
+
+            // discrete events handling
+            System.arraycopy(predictedY, 0, y, 0, y.length);
+            setStepStart(acceptStep(interpolator, finalTime));
+            scaled    = predictedScaled;
+            nordsieck = predictedNordsieck;
+
+            if (!isLastStep()) {
+
+                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
+                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);
+                interpolator = new AdamsFieldStepInterpolator<T>(getStepSize(), getStepStart(),
scaled, nordsieck,
+                                                                 forward, equations.getMapper());
+
+            }
+
+        } while (!isLastStep());
+
+        final FieldODEStateAndDerivative<T> finalState = getStepStart();
+        setStepStart(null);
+        setStepSize(null);
+        return finalState;
+
+    }
+
+    /** 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/305934df/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
new file mode 100644
index 0000000..408e646
--- /dev/null
+++ b/src/test/java/org/apache/commons/math4/ode/nonstiff/AdamsBashforthFieldIntegratorTest.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 AdamsBashforthFieldIntegratorTest 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 AdamsBashforthFieldIntegrator<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 AdamsBashforthFieldIntegrator<T>(field, nSteps, minStep, maxStep,
+                        vecAbsoluteTolerance, vecRelativeTolerance);
+    }
+
+    @Test(expected=NumberIsTooSmallException.class)
+    public void testMinStep() {
+        doDimensionCheck(Decimal64Field.getInstance());
+    }
+
+    @Test
+    public void testIncreasingTolerance() {
+        // the 7 and 121 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);
+    }
+
+    @Test(expected = MaxCountExceededException.class)
+    public void exceedMaxEvaluations() {
+        doExceedMaxEvaluations(Decimal64Field.getInstance());
+    }
+
+    @Test
+    public void backward() {
+        doBackward(Decimal64Field.getInstance(), 4.3e-8, 4.3e-8, 1.0e-16, "Adams-Bashforth");
+    }
+
+    @Test
+    public void polynomial() {
+        doPolynomial(Decimal64Field.getInstance(), 5, 0.004, 6.0e-10);
+    }
+
+    @Test(expected=MathIllegalStateException.class)
+    public void testStartFailure() {
+        doTestStartFailure(Decimal64Field.getInstance());
+    }
+
+}


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