naginterfaces.library.specfun.bessel_​zeros

naginterfaces.library.specfun.bessel_zeros(a, n, mode, rel=1.0536712127723509e-08)[source]

bessel_zeros determines the leading zeros of one of the Bessel functions , , or for real and non-negative .

For full information please refer to the NAG Library document for s17al

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s17alf.html

Parameters
afloat

The order of the function.

nint

The number of zeros required.

modeint

Specifies the form of the function whose zeros are required.

The zeros of are required.

The zeros of are required;

The zeros of are required;

The zeros of are required.

relfloat, optional

The relative accuracy to which the zeros are required.

Returns
xfloat, ndarray, shape

The required zeros of the function specified by .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Notes

bessel_zeros attempts to find the leading zeros of one of the Bessel functions , , or , where is real. When is real, these functions each have an infinite number of real zeros, all of which are simple with the possible exception of . If , the th positive zero is denoted by and , respectively, for , except that is counted as the first zero of when . Since , it, therefore, follows that and for . Further details can be found in Section 9.5 of Abramowitz and Stegun (1972).

bessel_zeros is based on Algol 60 procedures given by Temme (1979). Initial approximations to the zeros are computed from asymptotic expansions. These are then improved by higher-order Newton iteration making use of the differential equation for the Bessel functions.

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Temme, N M, 1976, On the numerical evaluation of the ordinary Bessel function of the second kind, J. Comput. Phys. (21), 343–350

Temme, N M, 1979, An algorithm with Algol 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives, J. Comput. Phys. (32), 270–279