alm2map_der*

This routine is a wrapper to four other routines that synthesize a HEALPix temperature (and polarisation) map(s), its (their) first derivatives, and optionally its (their) second derivatives. The routines accept both single and double precision arrays for alm, map, der1 and der2. The precision of these arrays should match. All maps produced are RING ordered.

See ”Fortran Facilities” Appendix for a note on a bug affecting the calculation of polarisation derivatives on past versions of this routine.

Location in HEALPix directory tree: src/f90/mod/alm_tools.F90 

FORMAT

call alm2map_der*( nsmax, nlmax, nmmax, alm, map, der1[, der2=, zbounds=] )

ARGUMENTS

name & dimensionality	kind	in/out	description
nsmax	I4B	IN	the $N_{\mathrm{side}}$ value of the map to synthesize.
nlmax	I4B	IN	the maximum $\ell$ value used for the $a_{\ell m}$.
nmmax	I4B	IN	the maximum $m$ value used for the $a_{\ell m}$.
alm(1:p, 0:nlmax, 0:nmmax)	SPC/ DPC	IN	The $a_{\ell m}$ values to make the map from. p is either 1 (temperature only) or 3 (temperature+polarisation).
map(0:12*nsmax**2-1) or (0:12*nsmax**2-1,1:3)	SP/ DP	OUT	temperature map $T(p)$ or temperature + polarisation maps $T(p)$, $Q(p)$, $U(p)$ to be synthesized.
der1(0:12*nsmax**2-1, 1:2*p)	SP/ DP	OUT	contains on output the first derivatives of T: $\left({\partial T}/{\partial \theta}, {\partial T}/{\partial \phi}/\sin\theta \right)$ or the interleaved derivatives of T, Q, and U: $\left({\partial T}/{\partial \theta}, {\partial Q}/{\partial \theta}, {\partial U}/{\partial \theta};\right.$ $\left.{\partial T}/{\partial \phi}/\sin\theta, \ldots \right)$
der2(0:12*nsmax**2-1,1:3*p), OPTIONAL	SP/ DP	OUT	If this optional matrix is passed with this rank, it will contain on output the second derivatives $\left({\partial^2 T}/{\partial \theta^2}, {\partial^2 T}/{\partial \theta\partial\phi}/\sin\theta,\right.$ $\left.{\partial^2 T}/{\partial \phi^2}/\sin^2\theta \right)$ or $\left({\partial^2 T}/{\partial \theta^2}, {\partial^2 Q}/{\partial \theta^2}, {\partial^2 Q}/{\partial \theta^2}, \ldots \right)$
zbounds(1:2), OPTIONAL	DP	IN	section of the sphere on which to perform the map synthesis, expressed in terms of $z=\sin(\mathrm{latitude}) = \cos(\theta).$ If zbounds(1)$<$zbounds(2), it is performed on the strip zbounds(1)$<z<$zbounds(2); if not, it is performed outside the strip zbounds(2)$\le z \le$zbounds(1). If absent, the whole map is processed.

EXAMPLE:

use healpix_types
use pix_tools, only : nside2npix
use alm_tools, only : alm2map_der
integer(I4B) :: nside, lmax, mmax, npix
real(SP), dimension(:), allocatable :: map
real(SP), dimension(:,:), allocatable :: der1, der2
complex(SPC), dimension(:,:,:), allocatable :: alm
...
nside=256 ; lmax=512 ; mmax=lmax
npix=nside2npix(nside)
allocate(alm(1:1,0:lmax,0:mmax))
allocate(map(0:npix-1))
allocate(der1(0:npix-1,1:2), der2(0:npix-1,1:3))
...
call alm2map_der(nside, lmax, mmax, alm, map, der1, der2=der2)

Make temperature maps and its derivatives from the $a_{\ell m}$ passed in alm. The maps have $N_{\mathrm{side}}$ of 256, and are constructed from $a_{\ell m}$ values up to 512 in $\ell$ and $m$.

MODULES & ROUTINES

This section lists the modules and routines used by alm2map_der*.

ring_synthesis

Performs FFT over $m$ for synthesis of the rings.

compute_lam_mm, get_pixel_layout,

gen_lamfac_der, gen_mfac,

gen_recfac, init_rescale, l_min_ylm

Ancillary routines used for $\ {_s}Y_{\ell m}$ recursion

misc_utils

module, containing:

assert_alloc

routine to print error message, when an array can not be allocated properly

RELATED ROUTINES

This section lists the routines related to alm2map_der*

alm2map

routine generating maps of temperature and polarisation from their $a_{\ell m}$

alm2map_spin

routine generating maps of arbitrary spin from their ${_s}a_{\ell m}$

synfast

executable using alm2map_der* to synthesize maps.

create_alm

routine to generate randomly distributed $a_{\ell m}$ coefficients according to a given power spectrum