RdRealDuplication.java
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* https://www.apache.org/licenses/LICENSE-2.0
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package org.hipparchus.special.elliptic.carlson;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.MathArrays;
/** Duplication algorithm for Carlson R<sub>D</sub> elliptic integral.
* @since 2.0
*/
class RdRealDuplication extends RealDuplication {
/** Constant term in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double CONSTANT = 4084080;
/** Coefficient of E₂ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E2 = -875160;
/** Coefficient of E₃ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E3 = 680680;
/** Coefficient of E₂² in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E2_E2 = 417690;
/** Coefficient of E₄ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E4 = -556920;
/** Coefficient of E₂E₃ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E2_E3 = -706860;
/** Coefficient of E₅ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E5 = 471240;
/** Coefficient of E₂³ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E2_E2_E2 = -255255;
/** Coefficient of E₃² in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E3_E3 = 306306;
/** Coefficient of E₂E₄ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E2_E4 = 612612;
/** Coefficient of E₂²E₃ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E2_E2_E3 = 675675;
/** Coefficient of E₃E₄+E₂E₅ in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double E3_E4_P_E2_E5 = -540540;
/** Denominator in R<sub>J</sub> and R<sub>D</sub> polynomials. */
static final double DENOMINATOR = 4084080;
/** Partial sum. */
private double sum;
/** 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
*/
RdRealDuplication(final double x, final double y, final double z) {
super(x, y, z);
sum = 0;
}
/** {@inheritDoc} */
@Override
protected void initialMeanPoint(final double[] va) {
va[3] = (va[0] + va[1] + va[2] * 3.0) / 5.0;
}
/** {@inheritDoc} */
@Override
protected double convergenceCriterion(final double r, final double max) {
return max / (FastMath.sqrt(FastMath.sqrt(FastMath.sqrt(r * 0.25))));
}
/** {@inheritDoc} */
@Override
protected void update(final int m, final double[] vaM, final double[] sqrtM, final double fourM) {
// equation 2.29 in Carlson[1995]
final double lambdaA = sqrtM[0] * sqrtM[1];
final double lambdaB = sqrtM[0] * sqrtM[2];
final double lambdaC = sqrtM[1] * sqrtM[2];
// running sum in equation 2.34 in Carlson[1995]
final double lambda = lambdaA + lambdaB + lambdaC;
sum += 1.0 / ((vaM[2] + lambda) * sqrtM[2] * fourM);
// equations 2.29 and 2.30 in Carlson[1995]
vaM[0] = MathArrays.linearCombination(0.25, vaM[0], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // xₘ
vaM[1] = MathArrays.linearCombination(0.25, vaM[1], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // yₘ
vaM[2] = MathArrays.linearCombination(0.25, vaM[2], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // zₘ
vaM[3] = MathArrays.linearCombination(0.25, vaM[3], 0.25, lambdaA, 0.25, lambdaB, 0.25, lambdaC); // aₘ
}
/** {@inheritDoc} */
@Override
protected double evaluate(final double[] va0, final double aM, final double fourM) {
// compute symmetric differences
final double inv = 1.0 / (aM * fourM);
final double bigX = (va0[3] - va0[0]) * inv;
final double bigY = (va0[3] - va0[1]) * inv;
final double bigZ = (bigX + bigY) / -3;
final double bigXY = bigX * bigY;
final double bigZ2 = bigZ * bigZ;
// compute elementary symmetric functions (we already know e1 = 0 by construction)
final double e2 = bigXY - bigZ2 * 6;
final double e3 = (bigXY * 3 - bigZ2 * 8) * bigZ;
final double e4 = (bigXY - bigZ2) * 3 * bigZ2;
final double e5 = bigXY * bigZ2 * bigZ;
final double e2e2 = e2 * e2;
final double e2e3 = e2 * e3;
final double e2e4 = e2 * e4;
final double e2e5 = e2 * e5;
final double e3e3 = e3 * e3;
final double e3e4 = e3 * e4;
final double e2e2e2 = e2e2 * e2;
final double e2e2e3 = e2e2 * e3;
// evaluate integral using equation 19.36.1 in DLMF
// (which add more terms than equation 2.7 in Carlson[1995])
final double poly = ((e3e4 + e2e5) * E3_E4_P_E2_E5 +
e2e2e3 * E2_E2_E3 +
e2e4 * E2_E4 +
e3e3 * E3_E3 +
e2e2e2 * E2_E2_E2 +
e5 * E5 +
e2e3 * E2_E3 +
e4 * E4 +
e2e2 * E2_E2 +
e3 * E3 +
e2 * E2 +
CONSTANT) /
DENOMINATOR;
final double polyTerm = poly / (aM * FastMath.sqrt(aM) * fourM);
return polyTerm + sum * 3;
}
}