RjFieldDuplication.java
/*
* Licensed to the Hipparchus project under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The Hipparchus project 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
*
* https://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.hipparchus.special.elliptic.carlson;
import org.hipparchus.CalculusFieldElement;
import org.hipparchus.complex.Complex;
import org.hipparchus.complex.FieldComplex;
import org.hipparchus.util.FastMath;
/** Duplication algorithm for Carlson R<sub>J</sub> elliptic integral.
* @param <T> type of the field elements (really {@link Complex} or {@link FieldComplex})
* @since 2.0
*/
class RjFieldDuplication<T extends CalculusFieldElement<T>> extends FieldDuplication<T> {
/** Delta product. */
private T delta;
/** sₘ iteration parameter. */
private T sM;
/** Simple constructor.
* @param x first symmetric variable of the integral
* @param y second symmetric variable of the integral
* @param z third symmetric variable of the integral
* @param p fourth <em>not</em> symmetric variable of the integral
* @param delta precomputed value of (p-x)(p-y)(p-z)
*/
RjFieldDuplication(final T x, final T y, final T z, final T p, final T delta) {
super(x, y, z, p);
this.delta = delta;
}
/** {@inheritDoc} */
@Override
protected void initialMeanPoint(final T[] va) {
va[4] = va[0].add(va[1]).add(va[2]).add(va[3].multiply(2)).divide(5.0);
}
/** {@inheritDoc} */
@Override
protected T convergenceCriterion(final T r, final T max) {
return max.divide(FastMath.sqrt(FastMath.sqrt(FastMath.sqrt(r.multiply(0.25)))));
}
/** {@inheritDoc} */
@Override
protected void update(final int m, final T[] vaM, final T[] sqrtM, final double fourM) {
final T dM = sqrtM[3].add(sqrtM[0]).
multiply(sqrtM[3].add(sqrtM[1])).
multiply(sqrtM[3].add(sqrtM[2]));
if (m == 0) {
sM = dM.multiply(0.5);
} else {
// equation A.3 in Carlson[2000]
final T rM = sM.multiply(delta.divide(sM.multiply(sM).multiply(fourM)).add(1.0).sqrt().add(1.0));
sM = dM.multiply(rM).subtract(delta.divide(fourM * fourM)).
divide(dM.add(rM.divide(fourM)).multiply(2));
}
// equation 2.19 in Carlson[1995]
final T lambdaA = sqrtM[0].multiply(sqrtM[1]);
final T lambdaB = sqrtM[0].multiply(sqrtM[2]);
final T lambdaC = sqrtM[1].multiply(sqrtM[2]);
// equations 2.19 and 2.20 in Carlson[1995]
vaM[0] = vaM[0].linearCombination(0.25, vaM[0], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // xₘ
vaM[1] = vaM[1].linearCombination(0.25, vaM[1], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // yₘ
vaM[2] = vaM[2].linearCombination(0.25, vaM[2], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // zₘ
vaM[3] = vaM[3].linearCombination(0.25, vaM[3], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // pₘ
vaM[4] = vaM[4].linearCombination(0.25, vaM[4], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // aₘ
}
/** {@inheritDoc} */
@Override
protected T evaluate(final T[] va0, final T aM, final double fourM) {
// compute symmetric differences
final T inv = aM.multiply(fourM).reciprocal();
final T bigX = va0[4].subtract(va0[0]).multiply(inv);
final T bigY = va0[4].subtract(va0[1]).multiply(inv);
final T bigZ = va0[4].subtract(va0[2]).multiply(inv);
final T bigP = bigX.add(bigY).add(bigZ).multiply(-0.5);
final T bigP2 = bigP.multiply(bigP);
// compute elementary symmetric functions (we already know e1 = 0 by construction)
final T xyz = bigX.multiply(bigY).multiply(bigZ);
final T e2 = bigX.multiply(bigY.add(bigZ)).add(bigY.multiply(bigZ)).
subtract(bigP.multiply(bigP).multiply(3));
final T e3 = xyz.add(bigP.multiply(2).multiply(e2.add(bigP2.multiply(2))));
final T e4 = xyz.multiply(2).add(bigP.multiply(e2.add(bigP2.multiply(3)))).multiply(bigP);
final T e5 = xyz.multiply(bigP2);
final T e2e2 = e2.multiply(e2);
final T e2e3 = e2.multiply(e3);
final T e2e4 = e2.multiply(e4);
final T e2e5 = e2.multiply(e5);
final T e3e3 = e3.multiply(e3);
final T e3e4 = e3.multiply(e4);
final T e2e2e2 = e2e2.multiply(e2);
final T e2e2e3 = e2e2.multiply(e3);
// evaluate integral using equation 19.36.1 in DLMF
// (which add more terms than equation 2.7 in Carlson[1995])
final T poly = e3e4.add(e2e5).multiply(RdRealDuplication.E3_E4_P_E2_E5).
add(e2e2e3.multiply(RdRealDuplication.E2_E2_E3)).
add(e2e4.multiply(RdRealDuplication.E2_E4)).
add(e3e3.multiply(RdRealDuplication.E3_E3)).
add(e2e2e2.multiply(RdRealDuplication.E2_E2_E2)).
add(e5.multiply(RdRealDuplication.E5)).
add(e2e3.multiply(RdRealDuplication.E2_E3)).
add(e4.multiply(RdRealDuplication.E4)).
add(e2e2.multiply(RdRealDuplication.E2_E2)).
add(e3.multiply(RdRealDuplication.E3)).
add(e2.multiply(RdRealDuplication.E2)).
add(RdRealDuplication.CONSTANT).
divide(RdRealDuplication.DENOMINATOR);
final T polyTerm = poly.divide(aM.multiply(FastMath.sqrt(aM)).multiply(fourM));
// compute a single R_C term
final T rcTerm = new RcFieldDuplication<>(poly.getField().getOne(), delta.divide(sM.multiply(sM).multiply(fourM)).add(1)).
integral().
multiply(3).divide(sM);
return polyTerm.add(rcTerm);
}
}