map2alm transforms

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

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 - an integral over azimuthal angle, $\phi$,
  

  $${{\displaystyle \forall} \atop {\displaystyle {m\in {\rm all... ...left(\theta_i, \phi_j\right) {\exp\left( -2\pi \phi_j\right)};$$ (1)

  
  

• global intermediate product redistribution:
  

  $$\left\{ \; {{\displaystyle \forall} \atop {\displaystyle {m\... ...}_p}}}\ \ \Delta_{m}\left( \theta\left(r\right)\right)\right\}$$ (2)

  
  

• second step - an integral over polar angle, $\theta$,
  

  $${{\displaystyle \forall} \atop {\displaystyle {\ell\in {\rm ... ...\left(\theta_i\right)\,{\cal P}_{\ell m}\left(\theta_i\right).$$ (3)