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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.math3.special;
import java.util.Arrays;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.exception.ConvergenceException;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.special.Gamma;
import org.apache.commons.math3.util.FastMath;
/**
* This class provides computation methods related to Bessel
* functions of the first kind. Detailed descriptions of these functions are
* available in Wikipedia, Abrabowitz and
* Stegun (Ch. 9-11), and DLMF (Ch. 10).
*
* This implementation is based on the rjbesl Fortran routine at
* Netlib.
*
* From the Fortran code:
*
* This program is based on a program written by David J. Sookne (2) that
* computes values of the Bessel functions J or I of real argument and integer
* order. Modifications include the restriction of the computation to the J
* Bessel function of non-negative real argument, the extension of the
* computation to arbitrary positive order, and the elimination of most
* underflow.
*
* References:
*
* - "A Note on Backward Recurrence Algorithms," Olver, F. W. J., and Sookne,
* D. J., Math. Comp. 26, 1972, pp 941-947.
* - "Bessel Functions of Real Argument and Integer Order," Sookne, D. J., NBS
* Jour. of Res. B. 77B, 1973, pp 125-132.
*
* @since 3.4
*/
public class BesselJ
implements UnivariateFunction {
// ---------------------------------------------------------------------
// Mathematical constants
// ---------------------------------------------------------------------
/** -2 / pi */
private static final double PI2 = 0.636619772367581343075535;
/** first few significant digits of 2pi */
private static final double TOWPI1 = 6.28125;
/** 2pi - TWOPI1 to working precision */
private static final double TWOPI2 = 1.935307179586476925286767e-3;
/** TOWPI1 + TWOPI2 */
private static final double TWOPI = TOWPI1 + TWOPI2;
// ---------------------------------------------------------------------
// Machine-dependent parameters
// ---------------------------------------------------------------------
/**
* 10.0^K, where K is the largest integer such that ENTEN is
* machine-representable in working precision
*/
private static final double ENTEN = 1.0e308;
/**
* Decimal significance desired. Should be set to (INT(log_{10}(2) * (it)+1)).
* Setting NSIG lower will result in decreased accuracy while setting
* NSIG higher will increase CPU time without increasing accuracy.
* The truncation error is limited to a relative error of
* T=.5(10^(-NSIG)).
*/
private static final double ENSIG = 1.0e16;
/** 10.0 ** (-K) for the smallest integer K such that K >= NSIG/4 */
private static final double RTNSIG = 1.0e-4;
/** Smallest ABS(X) such that X/4 does not underflow */
private static final double ENMTEN = 8.90e-308;
/** Minimum acceptable value for x */
private static final double X_MIN = 0.0;
/**
* Upper limit on the magnitude of x. If abs(x) = n, then at least
* n iterations of the backward recursion will be executed. The value of
* 10.0 ** 4 is used on every machine.
*/
private static final double X_MAX = 1.0e4;
/** First 25 factorials as doubles */
private static final double[] FACT = {
1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,
3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,
1.307674368e12, 2.0922789888e13, 3.55687428096e14, 6.402373705728e15,
1.21645100408832e17, 2.43290200817664e18, 5.109094217170944e19,
1.12400072777760768e21, 2.585201673888497664e22,
6.2044840173323943936e23
};
/** Order of the function computed when {@link #value(double)} is used */
private final double order;
/**
* Create a new BesselJ with the given order.
*
* @param order order of the function computed when using {@link #value(double)}.
*/
public BesselJ(double order) {
this.order = order;
}
/**
* Returns the value of the constructed Bessel function of the first kind,
* for the passed argument.
*
* @param x Argument
* @return Value of the Bessel function at x
* @throws MathIllegalArgumentException if {@code x} is too large relative to {@code order}
* @throws ConvergenceException if the algorithm fails to converge
*/
public double value(double x)
throws MathIllegalArgumentException, ConvergenceException {
return BesselJ.value(order, x);
}
/**
* Returns the first Bessel function, \(J_{order}(x)\).
*
* @param order Order of the Bessel function
* @param x Argument
* @return Value of the Bessel function of the first kind, \(J_{order}(x)\)
* @throws MathIllegalArgumentException if {@code x} is too large relative to {@code order}
* @throws ConvergenceException if the algorithm fails to converge
*/
public static double value(double order, double x)
throws MathIllegalArgumentException, ConvergenceException {
final int n = (int) order;
final double alpha = order - n;
final int nb = n + 1;
final BesselJResult res = rjBesl(x, alpha, nb);
if (res.nVals >= nb) {
return res.vals[n];
} else if (res.nVals < 0) {
throw new MathIllegalArgumentException(LocalizedFormats.BESSEL_FUNCTION_BAD_ARGUMENT,order, x);
} else if (FastMath.abs(res.vals[res.nVals - 1]) < 1e-100) {
return res.vals[n]; // underflow; return value (will be zero)
}
throw new ConvergenceException(LocalizedFormats.BESSEL_FUNCTION_FAILED_CONVERGENCE, order, x);
}
/**
* Encapsulates the results returned by {@link BesselJ#rjBesl(double, double, int)}.
*
* {@link #getVals()} returns the computed function values.
* {@link #getnVals()} is the number of values among those returned by {@link #getnVals()}
* that can be considered accurate.
*
*
* - nVals < 0: An argument is out of range. For example, nb <= 0, alpha
* < 0 or > 1, or x is too large. In this case, b(0) is set to zero, the
* remainder of the b-vector is not calculated, and nVals is set to
* MIN(nb,0) - 1 so that nVals != nb.
* - nb > nVals > 0: Not all requested function values could be calculated
* accurately. This usually occurs because nb is much larger than abs(x). In
* this case, b(n) is calculated to the desired accuracy for n < nVals, but
* precision is lost for nVals < n <= nb. If b(n) does not vanish for n >
* nVals (because it is too small to be represented), and b(n)/b(nVals) =
* \(10^{-k}\), then only the first NSIG-k significant figures of b(n) can be
* trusted.
*/
public static class BesselJResult {
/** Bessel function values */
private final double[] vals;
/** Valid value count */
private final int nVals;
/**
* Create a new BesselJResult with the given values and valid value count.
*
* @param b values
* @param n count of valid values
*/
public BesselJResult(double[] b, int n) {
vals = Arrays.copyOf(b, b.length);
nVals = n;
}
/**
* @return the computed function values
*/
public double[] getVals() {
return Arrays.copyOf(vals, vals.length);
}
/**
* @return the number of valid function values (normally the same as the
* length of the array returned by {@link #getnVals()})
*/
public int getnVals() {
return nVals;
}
}
/**
* Calculates Bessel functions \(J_{n+alpha}(x)\) for
* non-negative argument x, and non-negative order n + alpha.
*
* Before using the output vector, the user should check that
* nVals = nb, i.e., all orders have been calculated to the desired accuracy.
* See BesselResult class javadoc for details on return values.
*
* @param x non-negative real argument for which J's are to be calculated
* @param alpha fractional part of order for which J's or exponentially
* scaled J's (\(J\cdot e^{x}\)) are to be calculated. 0 <= alpha < 1.0.
* @param nb integer number of functions to be calculated, nb > 0. The first
* function calculated is of order alpha, and the last is of order
* nb - 1 + alpha.
* @return BesselJResult a vector of the functions
* \(J_{alpha}(x)\) through \(J_{nb-1+alpha}(x)\), or the corresponding exponentially
* scaled functions and an integer output variable indicating possible errors
*/
public static BesselJResult rjBesl(double x, double alpha, int nb) {
final double[] b = new double[nb];
int ncalc = 0;
double alpem = 0;
double alp2em = 0;
// ---------------------------------------------------------------------
// Check for out of range arguments.
// ---------------------------------------------------------------------
final int magx = (int) x;
if ((nb > 0) && (x >= X_MIN) && (x <= X_MAX) && (alpha >= 0) &&
(alpha < 1)) {
// ---------------------------------------------------------------------
// Initialize result array to zero.
// ---------------------------------------------------------------------
ncalc = nb;
for (int i = 0; i < nb; ++i) {
b[i] = 0;
}
// ---------------------------------------------------------------------
// Branch to use 2-term ascending series for small X and asymptotic
// form for large X when NB is not too large.
// ---------------------------------------------------------------------
double tempa;
double tempb;
double tempc;
double tover;
if (x < RTNSIG) {
// ---------------------------------------------------------------------
// Two-term ascending series for small X.
// ---------------------------------------------------------------------
tempa = 1;
alpem = 1 + alpha;
double halfx = 0;
if (x > ENMTEN) {
halfx = 0.5 * x;
}
if (alpha != 0) {
tempa = FastMath.pow(halfx, alpha) /
(alpha * Gamma.gamma(alpha));
}
tempb = 0;
if (x + 1 > 1) {
tempb = -halfx * halfx;
}
b[0] = tempa + (tempa * tempb / alpem);
if ((x != 0) && (b[0] == 0)) {
ncalc = 0;
}
if (nb != 1) {
if (x <= 0) {
for (int n = 1; n < nb; ++n) {
b[n] = 0;
}
} else {
// ---------------------------------------------------------------------
// Calculate higher order functions.
// ---------------------------------------------------------------------
tempc = halfx;
tover = tempb != 0 ? ENMTEN / tempb : 2 * ENMTEN / x;
for (int n = 1; n < nb; ++n) {
tempa /= alpem;
alpem += 1;
tempa *= tempc;
if (tempa <= tover * alpem) {
tempa = 0;
}
b[n] = tempa + (tempa * tempb / alpem);
if ((b[n] == 0) && (ncalc > n)) {
ncalc = n;
}
}
}
}
} else if ((x > 25.0) && (nb <= magx + 1)) {
// ---------------------------------------------------------------------
// Asymptotic series for X > 25
// ---------------------------------------------------------------------
final double xc = FastMath.sqrt(PI2 / x);
final double mul = 0.125 / x;
final double xin = mul * mul;
int m = 0;
if (x >= 130.0) {
m = 4;
} else if (x >= 35.0) {
m = 8;
} else {
m = 11;
}
final double xm = 4.0 * m;
// ---------------------------------------------------------------------
// Argument reduction for SIN and COS routines.
// ---------------------------------------------------------------------
double t = (double) ((int) ((x / TWOPI) + 0.5));
final double z = x - t * TOWPI1 - t * TWOPI2 - (alpha + 0.5) / PI2;
double vsin = FastMath.sin(z);
double vcos = FastMath.cos(z);
double gnu = 2 * alpha;
double capq;
double capp;
double s;
double t1;
double xk;
for (int i = 1; i <= 2; i++) {
s = (xm - 1 - gnu) * (xm - 1 + gnu) * xin * 0.5;
t = (gnu - (xm - 3.0)) * (gnu + (xm - 3.0));
capp = (s * t) / FACT[2 * m];
t1 = (gnu - (xm + 1)) * (gnu + (xm + 1));
capq = (s * t1) / FACT[2 * m + 1];
xk = xm;
int k = 2 * m;
t1 = t;
for (int j = 2; j <= m; j++) {
xk -= 4.0;
s = (xk - 1 - gnu) * (xk - 1 + gnu);
t = (gnu - (xk - 3.0)) * (gnu + (xk - 3.0));
capp = (capp + 1 / FACT[k - 2]) * s * t * xin;
capq = (capq + 1 / FACT[k - 1]) * s * t1 * xin;
k -= 2;
t1 = t;
}
capp += 1;
capq = (capq + 1) * ((gnu * gnu) - 1) * (0.125 / x);
b[i - 1] = xc * (capp * vcos - capq * vsin);
if (nb == 1) {
return new BesselJResult(Arrays.copyOf(b, b.length),
ncalc);
}
t = vsin;
vsin = -vcos;
vcos = t;
gnu += 2.0;
}
// ---------------------------------------------------------------------
// If NB > 2, compute J(X,ORDER+I) I = 2, NB-1
// ---------------------------------------------------------------------
if (nb > 2) {
gnu = 2 * alpha + 2.0;
for (int j = 2; j < nb; ++j) {
b[j] = gnu * b[j - 1] / x - b[j - 2];
gnu += 2.0;
}
}
} else {
// ---------------------------------------------------------------------
// Use recurrence to generate results. First initialize the
// calculation of P*S.
// ---------------------------------------------------------------------
final int nbmx = nb - magx;
int n = magx + 1;
int nstart = 0;
int nend = 0;
double en = 2 * (n + alpha);
double plast = 1;
double p = en / x;
double pold;
// ---------------------------------------------------------------------
// Calculate general significance test.
// ---------------------------------------------------------------------
double test = 2 * ENSIG;
boolean readyToInitialize = false;
if (nbmx >= 3) {
// ---------------------------------------------------------------------
// Calculate P*S until N = NB-1. Check for possible
// overflow.
// ---------------------------------------------------------------------
tover = ENTEN / ENSIG;
nstart = magx + 2;
nend = nb - 1;
en = 2 * (nstart - 1 + alpha);
double psave;
double psavel;
for (int k = nstart; k <= nend; k++) {
n = k;
en += 2.0;
pold = plast;
plast = p;
p = (en * plast / x) - pold;
if (p > tover) {
// ---------------------------------------------------------------------
// To avoid overflow, divide P*S by TOVER. Calculate
// P*S until
// ABS(P) > 1.
// ---------------------------------------------------------------------
tover = ENTEN;
p /= tover;
plast /= tover;
psave = p;
psavel = plast;
nstart = n + 1;
do {
n += 1;
en += 2.0;
pold = plast;
plast = p;
p = (en * plast / x) - pold;
} while (p <= 1);
tempb = en / x;
// ---------------------------------------------------------------------
// Calculate backward test and find NCALC, the
// highest N such that
// the test is passed.
// ---------------------------------------------------------------------
test = pold * plast * (0.5 - 0.5 / (tempb * tempb));
test /= ENSIG;
p = plast * tover;
n -= 1;
en -= 2.0;
nend = FastMath.min(nb, n);
for (int l = nstart; l <= nend; l++) {
pold = psavel;
psavel = psave;
psave = (en * psavel / x) - pold;
if (psave * psavel > test) {
ncalc = l - 1;
readyToInitialize = true;
break;
}
}
ncalc = nend;
readyToInitialize = true;
break;
}
}
if (!readyToInitialize) {
n = nend;
en = 2 * (n + alpha);
// ---------------------------------------------------------------------
// Calculate special significance test for NBMX > 2.
// ---------------------------------------------------------------------
test = FastMath.max(test, FastMath.sqrt(plast * ENSIG) *
FastMath.sqrt(2 * p));
}
}
// ---------------------------------------------------------------------
// Calculate P*S until significance test passes.
// ---------------------------------------------------------------------
if (!readyToInitialize) {
do {
n += 1;
en += 2.0;
pold = plast;
plast = p;
p = (en * plast / x) - pold;
} while (p < test);
}
// ---------------------------------------------------------------------
// Initialize the backward recursion and the normalization sum.
// ---------------------------------------------------------------------
n += 1;
en += 2.0;
tempb = 0;
tempa = 1 / p;
int m = (2 * n) - 4 * (n / 2);
double sum = 0;
double em = (double) (n / 2);
alpem = em - 1 + alpha;
alp2em = 2 * em + alpha;
if (m != 0) {
sum = tempa * alpem * alp2em / em;
}
nend = n - nb;
boolean readyToNormalize = false;
boolean calculatedB0 = false;
// ---------------------------------------------------------------------
// Recur backward via difference equation, calculating (but not
// storing) B(N), until N = NB.
// ---------------------------------------------------------------------
for (int l = 1; l <= nend; l++) {
n -= 1;
en -= 2.0;
tempc = tempb;
tempb = tempa;
tempa = (en * tempb / x) - tempc;
m = 2 - m;
if (m != 0) {
em -= 1;
alp2em = 2 * em + alpha;
if (n == 1) {
break;
}
alpem = em - 1 + alpha;
if (alpem == 0) {
alpem = 1;
}
sum = (sum + tempa * alp2em) * alpem / em;
}
}
// ---------------------------------------------------------------------
// Store B(NB).
// ---------------------------------------------------------------------
b[n - 1] = tempa;
if (nend >= 0) {
if (nb <= 1) {
alp2em = alpha;
if (alpha + 1 == 1) {
alp2em = 1;
}
sum += b[0] * alp2em;
readyToNormalize = true;
} else {
// ---------------------------------------------------------------------
// Calculate and store B(NB-1).
// ---------------------------------------------------------------------
n -= 1;
en -= 2.0;
b[n - 1] = (en * tempa / x) - tempb;
if (n == 1) {
calculatedB0 = true;
} else {
m = 2 - m;
if (m != 0) {
em -= 1;
alp2em = 2 * em + alpha;
alpem = em - 1 + alpha;
if (alpem == 0) {
alpem = 1;
}
sum = (sum + (b[n - 1] * alp2em)) * alpem / em;
}
}
}
}
if (!readyToNormalize && !calculatedB0) {
nend = n - 2;
if (nend != 0) {
// ---------------------------------------------------------------------
// Calculate via difference equation and store B(N),
// until N = 2.
// ---------------------------------------------------------------------
for (int l = 1; l <= nend; l++) {
n -= 1;
en -= 2.0;
b[n - 1] = (en * b[n] / x) - b[n + 1];
m = 2 - m;
if (m != 0) {
em -= 1;
alp2em = 2 * em + alpha;
alpem = em - 1 + alpha;
if (alpem == 0) {
alpem = 1;
}
sum = (sum + b[n - 1] * alp2em) * alpem / em;
}
}
}
}
// ---------------------------------------------------------------------
// Calculate b[0]
// ---------------------------------------------------------------------
if (!readyToNormalize) {
if (!calculatedB0) {
b[0] = 2.0 * (alpha + 1) * b[1] / x - b[2];
}
em -= 1;
alp2em = 2 * em + alpha;
if (alp2em == 0) {
alp2em = 1;
}
sum += b[0] * alp2em;
}
// ---------------------------------------------------------------------
// Normalize. Divide all B(N) by sum.
// ---------------------------------------------------------------------
if (FastMath.abs(alpha) > 1e-16) {
sum *= Gamma.gamma(alpha) * FastMath.pow(x * 0.5, -alpha);
}
tempa = ENMTEN;
if (sum > 1) {
tempa *= sum;
}
for (n = 0; n < nb; n++) {
if (FastMath.abs(b[n]) < tempa) {
b[n] = 0;
}
b[n] /= sum;
}
}
// ---------------------------------------------------------------------
// Error return -- X, NB, or ALPHA is out of range.
// ---------------------------------------------------------------------
} else {
if (b.length > 0) {
b[0] = 0;
}
ncalc = FastMath.min(nb, 0) - 1;
}
return new BesselJResult(Arrays.copyOf(b, b.length), ncalc);
}
}