Field-based Adams-Bashforth integrator.
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Luc Maisonobe
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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 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.FieldMatrix;
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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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/**
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* This class implements explicit Adams-Bashforth integrators for Ordinary
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* Differential Equations.
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*
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* <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit
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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, n-1, n-2 ... Depending on the number k of previous
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* steps one wants to use for computing the next value, different formulas
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* are available:</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</sub></li>
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* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
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* <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>
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* <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>
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* <li>...</li>
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* </ul>
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*
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* <p>A k-steps Adams-Bashforth method is of order k.</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-Bashforth 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)</li>
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* <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>
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* <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>
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* <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>
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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) and q<sub>n</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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*
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* <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
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* computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
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* for degree k polynomials.
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* <pre>
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* 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)
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* </pre>
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* The previous formula can be used with several values for i to compute the transform between
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* classical representation and Nordsieck vector. The transform between r<sub>n</sub>
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* and q<sub>n</sub> resulting from the Taylor series formulas above is:
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* <pre>
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* q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
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* </pre>
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* where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
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* with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
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* the column number starting from 1:
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* <pre>
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* [ -2 3 -4 5 ... ]
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* [ -4 12 -32 80 ... ]
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* P = [ -6 27 -108 405 ... ]
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* [ -8 48 -256 1280 ... ]
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* [ ... ]
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* </pre></p>
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*
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* <p>Using the Nordsieck vector has several advantages:
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* <ul>
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* <li>it greatly simplifies step interpolation as the interpolator mainly applies
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* Taylor series formulas,</li>
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* <li>it simplifies step changes that occur when discrete events that truncate
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* the step are triggered,</li>
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* <li>it allows to extend the methods in order to support adaptive stepsize.</li>
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* </ul></p>
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*
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* <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:
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* <ul>
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* <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
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* <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
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* <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>
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* </ul>
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* where A is a rows shifting matrix (the lower left part is an identity matrix):
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* <pre>
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* [ 0 0 ... 0 0 | 0 ]
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* [ ---------------+---]
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* [ 1 0 ... 0 0 | 0 ]
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* A = [ 0 1 ... 0 0 | 0 ]
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* [ ... | 0 ]
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* [ 0 0 ... 1 0 | 0 ]
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* [ 0 0 ... 0 1 | 0 ]
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* </pre></p>
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*
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* <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state,
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* they only depend on k and therefore are precomputed once for all.</p>
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*
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* @param <T> the type of the field elements
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* @since 3.6
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*/
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public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extends AdamsFieldIntegrator<T> {
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/** Integrator method name. */
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private static final String METHOD_NAME = "Adams-Bashforth";
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/**
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* Build an Adams-Bashforth integrator with the given order and step control parameters.
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* @param field field to which the time and state vector elements belong
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* @param nSteps number of steps of the method excluding the one being computed
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* @param minStep minimal step (sign is irrelevant, regardless of
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* integration direction, forward or backward), the last step can
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* be smaller than this
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* @param maxStep maximal step (sign is irrelevant, regardless of
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* integration direction, forward or backward), the last step can
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* be smaller than this
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* @param scalAbsoluteTolerance allowed absolute error
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* @param scalRelativeTolerance allowed relative error
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* @exception NumberIsTooSmallException if order is 1 or less
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*/
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public AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps,
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final double minStep, final double maxStep,
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final double scalAbsoluteTolerance,
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final double scalRelativeTolerance)
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throws NumberIsTooSmallException {
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super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep,
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scalAbsoluteTolerance, scalRelativeTolerance);
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}
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/**
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* Build an Adams-Bashforth integrator with the given order and step control parameters.
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* @param field field to which the time and state vector elements belong
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* @param nSteps number of steps of the method excluding the one being computed
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* @param minStep minimal step (sign is irrelevant, regardless of
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* integration direction, forward or backward), the last step can
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* be smaller than this
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* @param maxStep maximal step (sign is irrelevant, regardless of
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* integration direction, forward or backward), the last step can
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* be smaller than this
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* @param vecAbsoluteTolerance allowed absolute error
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* @param vecRelativeTolerance allowed relative error
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* @exception IllegalArgumentException if order is 1 or less
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*/
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public AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps,
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final double minStep, final double maxStep,
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final double[] vecAbsoluteTolerance,
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final double[] vecRelativeTolerance)
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throws IllegalArgumentException {
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super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep,
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vecAbsoluteTolerance, vecRelativeTolerance);
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}
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/** Estimate error.
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* <p>
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* Error is estimated by interpolating back to previous state using
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* the state Taylor expansion and comparing to real previous state.
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* </p>
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* @param previousState state vector at step start
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* @param predictedState predicted state vector at step end
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* @param predictedScaled predicted value of the scaled derivatives at step end
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* @param predictedNordsieck predicted value of the Nordsieck vector at step end
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* @return estimated normalized local discretization error
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*/
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private T errorEstimation(final T[] previousState,
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final T[] predictedState,
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final T[] predictedScaled,
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final FieldMatrix<T> predictedNordsieck) {
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T error = getField().getZero();
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for (int i = 0; i < mainSetDimension; ++i) {
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final T yScale = predictedState[i].abs();
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final T tol = (vecAbsoluteTolerance == null) ?
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yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) :
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yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]);
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// apply Taylor formula from high order to low order,
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// for the sake of numerical accuracy
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T variation = getField().getZero();
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int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1;
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for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) {
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variation = variation.add(predictedNordsieck.getEntry(k, i).multiply(sign));
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sign = -sign;
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}
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variation = variation.subtract(predictedScaled[i]);
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final T ratio = predictedState[i].subtract(previousState[i]).add(variation).divide(tol);
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error = error.add(ratio.multiply(ratio));
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}
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return error.divide(mainSetDimension).sqrt();
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}
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/** {@inheritDoc} */
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@Override
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public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations,
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final FieldODEState<T> initialState,
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final T finalTime)
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throws NumberIsTooSmallException, DimensionMismatchException,
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MaxCountExceededException, NoBracketingException {
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sanityChecks(initialState, finalTime);
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final T t0 = initialState.getTime();
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final T[] y = equations.getMapper().mapState(initialState);
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setStepStart(initIntegration(equations, t0, y, finalTime));
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final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0;
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// compute the initial Nordsieck vector using the configured starter integrator
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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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// main integration loop
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setIsLastStep(false);
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do {
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T[] predictedY = null;
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final T[] predictedScaled = MathArrays.buildArray(getField(), y.length);
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Array2DRowFieldMatrix<T> predictedNordsieck = null;
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T error = getField().getZero().add(10);
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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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final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY);
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// predict Nordsieck vector at step end
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for (int j = 0; j < predictedScaled.length; ++j) {
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predictedScaled[j] = getStepSize().multiply(yDot[j]);
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}
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predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
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updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);
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// evaluate error
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error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck);
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if (error.subtract(1.0).getReal() >= 0.0) {
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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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}
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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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scaled = predictedScaled;
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nordsieck = predictedNordsieck;
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if (!isLastStep()) {
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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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final T factor = computeStepGrowShrinkFactor(error);
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final T scaledH = getStepSize().multiply(factor);
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final T nextT = getStepStart().getTime().add(scaledH);
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final boolean nextIsLast = forward ?
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nextT.subtract(finalTime).getReal() >= 0 :
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nextT.subtract(finalTime).getReal() <= 0;
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T hNew = filterStep(scaledH, forward, nextIsLast);
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final T filteredNextT = getStepStart().getTime().add(hNew);
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final boolean filteredNextIsLast = forward ?
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filteredNextT.subtract(finalTime).getReal() >= 0 :
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filteredNextT.subtract(finalTime).getReal() <= 0;
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if (filteredNextIsLast) {
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hNew = finalTime.subtract(getStepStart().getTime());
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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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}
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} while (!isLastStep());
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final FieldODEStateAndDerivative<T> finalState = getStepStart();
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setStepStart(null);
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setStepSize(null);
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return finalState;
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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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@ -0,0 +1,78 @@
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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,
|
||||
* 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());
|
||||
}
|
||||
|
||||
}
|
||||
Loading…
Reference in New Issue