Content-type: text/html Manpage of calculateQlm

calculateQlm

Section: User Commands (1)
Updated: Version 1.03: July 2003
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NAME

calculateQlm

SYNOPSIS

#include ccSHT.h

void calculateQlm(int lmax, flt8 x, flt8 *calc1, flt8 *calc2, flt8 *calc3, flt8 *calc4, flt8 *Q );

DESCRIPTION

This function is the member of the ccSHT library which calculates the scaled associated Legendre polynomials. For some background, and more information about the library see the man page for ccSHT.

The associated Legendre polynomials are functions of a real variable, x, and two integer indexes, l and m. calculateQlm() solves for the scaled associated Legendre polynomials for a fixed x, all l in the set {0, 1, ..., lmax}, and all m in the set {0, 1, ..., l}. The scaling of the polynomials is given here:

               ________
              / (l-m)!
Q_{l,m} =   / -------- P_{l,m}
           \/   (l+m)!

where P_{l,m} are the associated Legendre polynomials and Q_{l,m} are the scaled associated Legendre polynomials. The calculation of the polynomials is done with the following three term recursion on l:

2 m/2
Q_{m,m}(x) = c1(m)*( 1 - x )

Q_{m+1,m}(x) = c2(m)*x*Q_{m,m}(x)

Q_{l,m}(x) = c3(l,m)*x*Q_{l-1,m}(x)
- c4(l,m)*Q_{l-2,m}(x)

where the functions of l and m are given below:

                    ____________________
                   /    m               |
              m   /  --------'
c1(m)  = (-1) * /  '  |  |          1
                /      |  |    1 - -----
               /       |  |         2*i
             \/       i = 1

             _____
            /     |
c2(m) =   / 2*m+1
         \/

             ( 2*l-1 )
c3(l,m) = --------------
               2   2  1/2
            ( l - m  )

                _____________
               /      2   2  |
              /  (l-1) - m
c4(l,m) =   / -------------
            /       2   2
          \/       l - m

It can be seen in the above equations that Q_{m,m} gets exponentially small with large m. For x close to -1 or 1 and large m, Q_{m,m} is not representable as a double because the absolute value of Q_{m,m} is less than 2^(-1074). This algorithm rescales Q when it would underflow and keeps track of the scaling. This is because although Q_{m,m}(x) may be virtually zero, Q_{m+500,m} may be substantial but unless the first two numbers in the recursion over l are non-zero the calculation of Q_{l,m}(x) would remain 0 for all l.

Before calling calculateQlm, some preliminary calculations must be done with the function called lmCalculations(). See the man page of lmCalculations() for more information. The preliminary calculations are passed to calculateQlm by way of the inputs calc1, calc2, calc3, and calc4.

Below is a table describing the parameters passed to calculateQlm()

lmax: Maximum l value for the computation
x: The fixed x value (between negative one and one) which determines the position for which the polynomials are computed.
calc1, calc2, calc3, calc4: Calculations which are functions only of l and m. These are to be calculated by the function called lmCalculations(). See man page for lmCalculations() for more details.
Q: This is the vector where the computed scaled associated Legendre polynomials will be stored. There should be space for (lmax*lmax + 3*lmax)/2+1 eight byte floating point numbers in the array. The indexing scheme for this vector is explained in the ccSHT man page.

COPYRIGHT

Version 1.03 July 2003

calculateQlm is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

Christopher Cantalupo <cmc@nersc.gov>

Send bug reports or comments to the above address.