alm2map transforms

On the input each processor, $p$, keeps in the memory a subset of all $a_{\ell m}$s corresponding to a subset of $m$ denoted, ${\cal M}_p$ and all values of $\ell$. On the output - a subset of iso-latitude rings, $r$, denoted by ${\cal R}_p$.

Each processors performs then the transform in two steps (Muciaccia, P.F. et al, Astrophys.J., 488, 63, (1997), Gorski, K.M., et al, Astrophys.J., 622, 759, (2005)) separated by an intermediate products redistribution between processors:

• first step - a sum over the multiple number $\ell$,
  

  $${{\displaystyle \forall} \atop {\displaystyle {\ell\in {\rm ... ...; rings}}\,a_{\ell m}\,{\cal P}_{\ell m}\left(\theta_i\right).$$ (4)

  
  

• global intermediate product redistribution:
  

  $$\left\{\;{{\displaystyle \forall} \atop {\displaystyle {r\in... ..._p}}}\ \ \Delta_{m}\left( \theta \left(r\right)\right)\right\}$$ (5)

  
  

• second step - a sum over multipole number $m$,
  

  $${{\displaystyle \forall} \atop {\displaystyle {m\in {\rm all... ...theta\left(r\right)\right) \, {\exp\left( 2\pi \phi_j\right)};$$ (6)