healpy.map2alm computes an array of alm values for an input Healpix map.
The function is
healpy.sphtfunc.map2alm(maps, lmax=None, mmax=None, iter=3, pol=True, use_weights=False, regression=True, datapath=None)`
The parameter lmax determines the number of alm coefficients. One sets the input lmax and the code returns '(lmax+1)**2` number of alm coefficients.
l=0 returns a00.
l=1 returns a11, a10, a1-1 + a00, i.e. 4 coefficients.
l=2 returns a22, a21, a20, a2-1, a2-2, i.e. 5+3+1 = 9 coefficients. That is (lmax+1)**2 = 3**2 = 9 coefficients.
However, using healpy, this is NOT the output I get, irrespective on the map I chose.
import numpy as np
import healpy as hp
nside = 16 # healpix nside parameter
filename = "cmb_fullsky_map.fits" # generic name for sky map
readmap = hp.read_map(filename) # read in the map
# We then use healpy.map2alm(readmap) to generate an array of alm values
ell0 = hp.map2alm(readmap, lmax = 0) # lmax = 0 should output a00
ell0.shape # 1 value
print "l=0 has " + str(len(ell0))
print ell0
ell1 = hp.map2alm(readmap, lmax = 1) # lmax = 1 should outputs
ell1.shape # a11, a10, a1-1
print "l=1 has " + str(len(ell1)) # 3 values
print ell1
ell2 = hp.map2alm(readmap, lmax = 2) # lmax = 2 should output
ell2.shape # a22, a21, a20, a2-1, a2-2
print "l=2 has " + str(len(ell2)) # 5 values
print ell2
ell3 = hp.map2alm(readmap, lmax = 3) # lmax = 3 should output
ell3.shape # a33, a32, a31, a30, a3-1, a3-2, a3-3
print "l=3 has " + str(len(ell3)) # 7 values
print ell3
ell4 = hp.map2alm(readmap, lmax = 4) # lmax = 4 should output
ell4.shape # a44, a43, a42, a41, a40, a4-1, a4-2,
print "l=4 has " + str(len(ell4)) # a4-3 a4-4
print ell4 # 9 values
ell32 = hp.map2alm(readmap, lmax = 32) # lmax = 32 should output
ell32.shape # (lmax+1)**2 = 33**2 = 1089 coefficients
print "l=32 has " + str(len(ell32)) # 65 values
My output is not what I would expect by (lmax+1)**2.
l=0 has 1
[ 2.39883065e-06+0.j]
l=1 has 3
[ 2.39883065e-06 +0.00000000e+00j 4.49747594e-06 +0.00000000e+00j
-5.39401197e-07 +1.07974023e-06j]
l=2 has 6
[ 2.38695037e-06 +0.00000000e+00j 4.49747594e-06 +0.00000000e+00j
-2.93559509e-05 +0.00000000e+00j -5.39401197e-07 +1.07974023e-06j
-1.75256654e-05 -1.57729954e-05j 1.65489261e-05 -1.41571515e-05j]
l=3 has 10
[ 2.38695037e-06 +0.00000000e+00j 4.51148696e-06 +0.00000000e+00j
-2.93559509e-05 +0.00000000e+00j 1.19817810e-05 +0.00000000e+00j
-5.37909950e-07 +1.07783971e-06j -1.75256654e-05 -1.57729954e-05j
-7.17416081e-06 +9.14176685e-06j 1.65489261e-05 -1.41571515e-05j
-2.64725172e-06 -3.09988314e-05j -7.25462692e-06 -6.33251313e-07j]
l=4 has 15
[ 2.38534350e-06 +0.00000000e+00j 4.51148696e-06 +0.00000000e+00j
-2.93586471e-05 +0.00000000e+00j 1.19817810e-05 +0.00000000e+00j
-1.72252485e-06 +0.00000000e+00j -5.37909950e-07 +1.07783971e-06j
-1.75253044e-05 -1.57728790e-05j -7.17416081e-06 +9.14176685e-06j
-1.30606717e-05 -4.21675061e-06j 1.65464103e-05 -1.41578955e-05j
-2.64725172e-06 -3.09988314e-05j 1.09072312e-05 +3.22054569e-06j
-7.25462692e-06 -6.33251313e-07j -1.10079209e-06 +9.39495982e-07j
-8.01588627e-06 -8.03762608e-06j]
l=32 has 561
Do you see the discrepancy?
For lmax=0, I expect a total of 1. healpy outputs 1
For lmax=1, I expect a total of 4. healpy outputs 3
For lmax=2, I expect a total of 9. healpy outputs 6
For lmax=3, I expect a total of 16. healpy outputs 10
For lmax=4, I expect a total of 25. healpy outputs 15
For lmax=32, I expect a total of 1089. healpy outputs 561
Question: Should I act together outputs in order to get the correct total number of arrays?
Please how map2alm executes this operation.
It is due to symmetry. m=-1 and m=1 have the same transform, so HEALPix considers only m>=0.
So for example lmax=2 we have:
6 coefficients total.
The length of the alm array is expected to be: mmax * (2 * lmax + 1 - mmax) / 2 + lmax + 1).