(j3.2006) (SC22WG5.4354) June meeting: draft announcement and agenda

[Van Snyder]

On Thu, 2010-11-11 at 03:25 -0800, John Reid wrote:
> WG5,
> 
> Here are drafts of the announcement and agenda for the June meeting. The item 
> "Consider whether Fortran should have a math functions module, as an optional 
> annex or an optional separate part" has been included following a request by Dan.

> Any comments?

See ftp://ftp.nag.co.uk/sc22wg5/N1651-N1700//N1688.pdf (or .ps.gz),
which I prepared in 2007 in response to the same question.

This is presented in the form of a part of the standard.  Whether it is
a required part, an optional part, a TR, or not further considered, can
be decided in due course.

There is a typo at [15:4+1].  "... the result a processor-dependent
approximation to is the..." should be "... the result is a
processor-dependent approximation to the..." (correct the Yoda syntax).

As we have in 10-007, there should be two functions each to compute the
cylindrical and spherical Bessel function of the first and second kinds,
the latter called the Neumann function in subclauses 2.4.11 and 2.4.23
in N1688.  Recall that in 10-007 we provide procedures to compute
cylindrical Bessel functions of consecutive integer orders, because the
need for such arrays occurs often in mathematical analysis.  The same
argument applies to Bessel functions of arbitrary order (spherical
Bessel functions are Bessel functions of half-integer order).

In 10-007, the cylindrical Bessel function of the second kind is called
Bessel_YN.  In clause 14 of ISO 31 part 11 (hereinafter ISO 31-11-14),
it is called the Neumann function.  In clause 2.4.11 of N1688 the
function to compute it for arbitrary real order is called CYL_NEUMANN.
For consistency with 10-007 this should be CYL_BESSEL_Y, or maybe
BESSEL_YN (with a real instead of integer order, thereby distinguishable
by generic resolution).  In clause 2.4.21 of N1688 the procedure to
compute the spherical Bessel function of the first kind is called
SPH_BESSEL.  For consistency with 10-007 this should be SPH_BESSEL_J.
In clause 2.4.23 of N1688 the procedure to compute spherical the Bessel
function of the second kind is called SPH_NEUMANN.  For consistency with
10-007 this should be SPH_BESSEL_Y.

The additional procedures proposed in subclause 2.5 in N1688 are
important mathematical functions, but do not appear in ISO 31-11-14.
Some or all of those should be moved to clause 2.5, after due
deliberation.

The Voigt functions, which are the real and imaginary parts of the
Fadeeva function (also called the plasma dispersion function) are
important in spectroscopy and radiative transfer (and other
disciplines).  They are not described in ISO 31-11-14, or in N1688, but
should be considered in the present context.

ISO 31-11-14 describes several functions that are not advocated in
N1688.  These include cylindrical and spherical Hankel functions,
associated Legendre functions, spherical harmonics, associated Laguerre
polynomials, hypergeometric functions (2F1), confluent hypergeometric
functions (1F1), and incomplete elliptic integrals of the first, second
and third kinds.

The Hankel functions are not needed, since they are composed of Bessel
functions of the first and second kind and there is no computational
benefit not to compute them as such (the real and imaginary parts cannot
in general be computed together).  Whether any of the others ought to be
included can be decided in due course.  There are three independent
systems for elliptic integrals (Jacobi/Legendre as in ISO 31-11-14,
Carlson, or Bulirsch).  Which (if any) to use can be decided in due
course.

Incomplete beta and incomplete gamma functions are not described in ISO
31-11-14, and not advocated in N1688.  These are important in statistics
(and other disciplines).  Whether these are incorporated acn be decided
in due course.

The NIST Handbook of Mathematical Functions should be included in the
references in clause 1.7 of N1688.

Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W.
Clark, "NIST Handbook of Mathematical Functions," National Institute of
Standards and Technology U.S. Department of Commerce, and Cambridge
University Press (2010) ISBN 978-0-521-19225-5 (hardback) ISBN
978-0-521-14063-8 (paperback).

Perhaps the book on numerical methods for special functions should be
included in the references in clause 1.7 of N1688.

Amparo Gil, Javier Segura and Nico M. Temme, "Numerical Methods for
Special Functions," SIAM (2007) Philadelphia, ISBN 978-0-898716-34-4.

If there is interest I can prepare a revision for June.]

Van.

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