dtorh Module Reference

Calculation of toroidal functions \(P_{n-1/2}^m \left(z\right)\) and \(Q_{n-1/2}^m \left(z\right)\). More...

Functions/Subroutines
integer function, public	dtorh1 (z, m, nmax, pl, ql, newn, mode, ipre)
	Calculates toroidal harmonics of a fixed order m and degrees up to nmax. More...

Detailed Description

Calculation of toroidal functions \(P_{n-1/2}^m \left(z\right)\) and \(Q_{n-1/2}^m \left(z\right)\).

Note

Copied and adapted from the DTORH1 routine by Segura and Gil [14] .

Function/Subroutine Documentation

dtorh1()

integer function, public dtorh::dtorh1	(	real(dp), intent(in)	z,
		integer, intent(in)	m,
		integer, intent(in)	nmax,
		real(dp), dimension(0:nmax), intent(inout)	pl,
		real(dp), dimension(0:nmax), intent(inout)	ql,
		integer, intent(inout)	newn,
		integer, intent(in), optional	mode,
		integer, intent(in), optional	ipre
	)

Calculates toroidal harmonics of a fixed order m and degrees up to nmax.

Optionally, the mode can be specified to be different from 0 [default]

• if mode=1:
  • The set of functions evaluated is:

    \[\frac{P_{n-1/2}^m \left(z\right)}{\Gamma(m+1/2)} \ ,\quad \frac{Q_{n-1/2}^m \left(z\right)}{\Gamma(m+1/2)},\]

    which are respectively stored in the arrays pl(n), ql(n).
  • newn refers to this new set of functions.
  • Note 1. and note 2. also apply in this case.
• if mode=2:
  • The code performs as for mode 1, but the restriction \( z<20 \).
  • For the evaluation of the continued fraction is not considered.

Also, the parameter ipre can be used to select a different precision:

• For ipre=1, the precision is \(10^{-12}\), taking \(\epsilon<10^{-12}\)
• For ipre=2, the precision is \(10^{-8}\), taking \(\epsilon<10^{-8}\)

where \(\epsilon\) controls the accuracy.

Warning

Use mode 2 only if high m 's for \(z>20\) are required. The evaluation of the cf may fail to converge for too high z 's.

Note

1. For a precision of \(10^{-12}\), if \(z>5\) and \(z/m>0.22\), the code uses a series expansion for pl(0).
   When \(z<20\) and \(z/m<0.22\) a continued fraction is applied.
2. For a precision of \(10^{-8}\), if \(z>5\) and \(z/m>0.12\), the code uses a series expansion for pl(0).
   When \(z<20\) and \(z/m<0.12\) a continued fraction is applied.

Returns

ierr

Parameters

[in]	z	(real) point at which toroidal harmonics are evaluated
[in]	m	order of toroidal harmonics (m>0)
[in]	nmax	maximum degree of toroidal harmonics (nmax>0)
[in,out]	pl	toroidal harmonics of first kind \(P_{n-1/2}^m\left(z\right)\)
[in,out]	ql	toroidal harmonics of second kind \(Q_{n-1/2}^m\left(z\right)\)
[in,out]	newn	maximum order of functions calculated when pl(nmax+1)>1/tiny with tiny= \(10^{-290}\)
[in]	mode	mode that controls output
[in]	ipre	precision

Definition at line 75 of file dtorh.f90.

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