RfRealDuplication.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>F</sub> elliptic integral.
* @since 2.0
*/
class RfRealDuplication extends RealDuplication {
/** Max number of iterations in the AGM scale. */
static final int AGM_MAX = 32;
/** Constant term in R<sub>F</sub> polynomial. */
static final double CONSTANT = 240240;
/** Coefficient of E₂ in R<sub>F</sub> polynomial. */
static final double E2 = -24024;
/** Coefficient of E₃ in R<sub>F</sub> polynomial. */
static final double E3 = 17160;
/** Coefficient of E₂² in R<sub>F</sub> polynomial. */
static final double E2_E2 = 10010;
/** Coefficient of E₂E₃ in R<sub>F</sub> polynomial. */
static final double E2_E3 = -16380;
/** Coefficient of E₃² in R<sub>F</sub> polynomial. */
static final double E3_E3 = 6930;
/** Coefficient of E₂³ in R<sub>F</sub> polynomial. */
static final double E2_E2_E2 = -5775;
/** Denominator in R<sub>F</sub> polynomial. */
static final double DENOMINATOR = 240240;
/** 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
*/
RfRealDuplication(final double x, final double y, final double z) {
super(x, y, z);
}
/** {@inheritDoc} */
@Override
protected void initialMeanPoint(final double[] va) {
va[3] = (va[0] + va[1] + va[2]) / 3.0;
}
/** {@inheritDoc} */
@Override
protected double convergenceCriterion(final double r, final double max) {
return max / FastMath.sqrt(FastMath.sqrt(FastMath.sqrt(r * 3.0)));
}
/** {@inheritDoc} */
@Override
protected void update(final int m, final double[] vaM, final double[] sqrtM, final double fourM) {
// equation 2.3 in Carlson[1995]
final double lambdaA = sqrtM[0] * sqrtM[1];
final double lambdaB = sqrtM[0] * sqrtM[2];
final double lambdaC = sqrtM[1] * sqrtM[2];
// equations 2.3 and 2.4 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);
// compute elementary symmetric functions (we already know e1 = 0 by construction)
final double e2 = bigX * bigY - bigZ * bigZ;
final double e3 = bigX * bigY * bigZ;
final double e2e2 = e2 * e2;
final double e2e3 = e2 * e3;
final double e3e3 = e3 * e3;
final double e2e2e2 = e2e2 * e2;
// evaluate integral using equation 19.36.1 in DLMF
// (which add more terms than equation 2.7 in Carlson[1995])
final double poly = (e2e2e2 * E2_E2_E2 +
e3e3 * E3_E3 +
e2e3 * E2_E3 +
e2e2 * E2_E2 +
e3 * E3 +
e2 * E2 +
CONSTANT) /
DENOMINATOR;
return poly / FastMath.sqrt(aM);
}
/** {@inheritDoc} */
@Override
public double integral() {
final double x = getVi(0);
final double y = getVi(1);
final double z = getVi(2);
if (x == 0) {
return completeIntegral(y, z);
} else if (y == 0) {
return completeIntegral(x, z);
} else if (z == 0) {
return completeIntegral(x, y);
} else {
return super.integral();
}
}
/** Compute Carlson complete elliptic integral R<sub>F</sub>(u, v, 0).
* @param x first symmetric variable of the integral
* @param y second symmetric variable of the integral
* @return Carlson complete elliptic integral R<sub>F</sub>(u, v, 0)
*/
private double completeIntegral(final double x, final double y) {
double xM = FastMath.sqrt(x);
double yM = FastMath.sqrt(y);
// iterate down
for (int i = 1; i < AGM_MAX; ++i) {
final double xM1 = xM;
final double yM1 = yM;
// arithmetic mean
xM = (xM1 + yM1) * 0.5;
// geometric mean
yM = FastMath.sqrt(xM1 * yM1);
// convergence (by the inequality of arithmetic and geometric means, this is non-negative)
if (FastMath.abs(xM - yM) <= 4 * FastMath.ulp(xM)) {
// convergence has been reached
break;
}
}
return FastMath.PI / (xM + yM);
}
}