Package org.apache.commons.math3.special
Class BesselJ
java.lang.Object
org.apache.commons.math3.special.BesselJ
- All Implemented Interfaces:
UnivariateFunction
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
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Nested Class Summary
Nested ClassesModifier and TypeClassDescriptionstatic class
Encapsulates the results returned byrjBesl(double, double, int)
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Field Summary
FieldsModifier and TypeFieldDescriptionprivate static final double
Smallest ABS(X) such that X/4 does not underflowprivate static final double
Decimal significance desired.private static final double
10.0^K, where K is the largest integer such that ENTEN is machine-representable in working precisionprivate static final double[]
First 25 factorials as doublesprivate final double
Order of the function computed whenvalue(double)
is usedprivate static final double
-2 / piprivate static final double
10.0 ** (-K) for the smallest integer K such that K >= NSIG/4private static final double
first few significant digits of 2piprivate static final double
TOWPI1 + TWOPI2private static final double
2pi - TWOPI1 to working precisionprivate static final double
Upper limit on the magnitude of x.private static final double
Minimum acceptable value for x -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptionstatic BesselJ.BesselJResult
rjBesl
(double x, double alpha, int nb) Calculates Bessel functions \(J_{n+alpha}(x)\) for non-negative argument x, and non-negative order n + alpha.double
value
(double x) Returns the value of the constructed Bessel function of the first kind, for the passed argument.static double
value
(double order, double x) Returns the first Bessel function, \(J_{order}(x)\).
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Field Details
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PI2
private static final double PI2-2 / pi- See Also:
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TOWPI1
private static final double TOWPI1first few significant digits of 2pi- See Also:
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TWOPI2
private static final double TWOPI22pi - TWOPI1 to working precision- See Also:
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TWOPI
private static final double TWOPITOWPI1 + TWOPI2- See Also:
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ENTEN
private static final double ENTEN10.0^K, where K is the largest integer such that ENTEN is machine-representable in working precision- See Also:
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ENSIG
private static final double ENSIGDecimal 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)).- See Also:
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RTNSIG
private static final double RTNSIG10.0 ** (-K) for the smallest integer K such that K >= NSIG/4- See Also:
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ENMTEN
private static final double ENMTENSmallest ABS(X) such that X/4 does not underflow- See Also:
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X_MIN
private static final double X_MINMinimum acceptable value for x- See Also:
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X_MAX
private static final double X_MAXUpper 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.- See Also:
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FACT
private static final double[] FACTFirst 25 factorials as doubles -
order
private final double orderOrder of the function computed whenvalue(double)
is used
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Constructor Details
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BesselJ
public BesselJ(double order) Create a new BesselJ with the given order.- Parameters:
order
- order of the function computed when usingvalue(double)
.
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Method Details
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value
Returns the value of the constructed Bessel function of the first kind, for the passed argument.- Specified by:
value
in interfaceUnivariateFunction
- Parameters:
x
- Argument- Returns:
- Value of the Bessel function at x
- Throws:
MathIllegalArgumentException
- ifx
is too large relative toorder
ConvergenceException
- if the algorithm fails to converge
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value
public static double value(double order, double x) throws MathIllegalArgumentException, ConvergenceException Returns the first Bessel function, \(J_{order}(x)\).- Parameters:
order
- Order of the Bessel functionx
- Argument- Returns:
- Value of the Bessel function of the first kind, \(J_{order}(x)\)
- Throws:
MathIllegalArgumentException
- ifx
is too large relative toorder
ConvergenceException
- if the algorithm fails to converge
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rjBesl
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.
- Parameters:
x
- non-negative real argument for which J's are to be calculatedalpha
- fractional part of order for which J's or exponentially scaled J's (\(J\cdot e^{x}\)) are to be calculated. 0 invalid input: '<'= alpha invalid input: '<' 1.0.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.- Returns:
- 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
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