PCGETRI(l)

📦 version 1.5, in ubuntu jammy

💖 PCGETRI 💖

okay so bestie, this is PCGETRI from the LAPACK routine set — version 1.5 energy, dated 12 May 1997, and she's a whole ScaLAPACK icon, manual section l. she has one job and she ate: ✨ computing the inverse of a distributed matrix using the LU factorization computed by PCGETRF ✨. that's it. that's the vibe. 🎀

🌸 NAME 🌸

PCGETRI — compute the inverse of a distributed matrix using the LU factorization computed by PCGETRF. no cap, that's literally the whole résumé. 💅

✨ SYNOPSIS ✨

SUBROUTINE PCGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK, IWORK,
                    LIWORK, INFO )

    INTEGER         IA, INFO, JA, LIWORK, LWORK, N

    INTEGER         DESCA( * ), IPIV( * ), IWORK( * )

    COMPLEX         A( * ), WORK( * )

🦋 PURPOSE 🦋

PCGETRI computes the inverse of a distributed matrix using the LU factorization computed by PCGETRF. the method is so clean: she inverts U first, and then computes the inverse of sub( A ) = A(IA:IA+N-1,JA:JA+N-1) — which we call InvA — by solving the system InvA*L = inv(U) for InvA. like, she does the algebra and you just get the answer back. obsessed. 😭💜

📝 Notes 📝

every global data object out here is described by its own little description vector. that vector holds all the info you need to map an object element to its corresponding process and memory location. it's giving ✨ backstage pass ✨.

so like, let A be a generic term for any 2D block cyclicly distributed array. that global array has an associated description vector DESCA. and in the comments below, read the character _ as "of the global array". iykyk.

  • DTYPE_A (global) — stored in DESCA( DTYPE_ ): the descriptor type. in this case, DTYPE_A = 1.
  • CTXT_A (global) — stored in DESCA( CTXT_ ): the BLACS context handle, indicating the BLACS process grid A is distributed over. the context itself is global, but the handle (the integer value) may vary. she's global but her number is not, it's giving duality 💅
  • M_A (global) — stored in DESCA( M_ ): the number of rows in the global array A.
  • N_A (global) — stored in DESCA( N_ ): the number of columns in the global array A.
  • MB_A (global) — stored in DESCA( MB_ ): the blocking factor used to distribute the rows of the array.
  • NB_A (global) — stored in DESCA( NB_ ): the blocking factor used to distribute the columns of the array.
  • RSRC_A (global) — stored in DESCA( RSRC_ ): the process row over which the first row of the array A is distributed.
  • CSRC_A (global) — stored in DESCA( CSRC_ ): the process column over which the first column of the array A is distributed.
  • LLD_A (local) — stored in DESCA( LLD_ ): the leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)).

now let K be the number of rows or columns of a distributed matrix, and assume its process grid has dimension p x q.

LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.

similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

the values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

an upper bound for these quantities may be computed by:

LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

👑 ARGUMENTS 👑

  • N — (global input) INTEGER. the number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. periodt.
  • A — (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). on entry, the local pieces of the L and U obtained by the factorization sub( A ) = P*L*U computed by PCGETRF. on exit, if INFO = 0, sub( A ) contains the inverse of the original distributed matrix sub( A ). glow up fr fr 🌈
  • IA — (global input) INTEGER. the row index in the global array A indicating the first row of sub( A ).
  • JA — (global input) INTEGER. the column index in the global array A indicating the first column of sub( A ).
  • DESCA — (global and local input) INTEGER array of dimension DLEN_. the array descriptor for the distributed matrix A.
  • IPIV — (local input) INTEGER array, dimension LOCr(M_A)+MB_A. keeps track of the pivoting information. IPIV(i) is the global row index the local row i was swapped with. this array is tied to the distributed matrix A — they're a package deal, no separating them 💕
  • WORK — (local workspace/local output) COMPLEX array, dimension (LWORK). on exit, WORK(1) returns the minimal and optimal LWORK.
  • LWORK — (local or global input) INTEGER. the dimension of the array WORK. LWORK is local input and must be at least LWORK = LOCr(N+MOD(IA-1,MB_A))*NB_A. WORK is used to keep a copy of at most an entire column block of sub( A ).

if LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA. sneaky little shortcut, she's lowkey efficient like that ✨

  • IWORK — (local workspace/local output) INTEGER array, dimension (LIWORK). on exit, IWORK(1) returns the minimal and optimal LIWORK.
  • LIWORK — (local or global input) INTEGER. the dimension of the array IWORK used as workspace for physically transposing the pivots. LIWORK is local input and must be at least:
if NPROW == NPCOL then
    LIWORK = LOCc( N_A + MOD(JA-1, NB_A) ) + NB_A,
else
    LIWORK =  LOCc( N_A + MOD(JA-1, NB_A) ) + MAX( CEIL(CEIL(LOCr(M_A)/MB_A)/(LCM/NPROW)), NB_A )
end if

where LCM is the least common multiple of process rows and columns (NPROW and NPCOL).

if LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

  • INFO — (global output) INTEGER. the receipts, bestie:
    • = 0: successful exit. slay. 💅
    • < 0: if the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j); if the i-th argument is a scalar and had an illegal value, then INFO = -i.
    • > 0: if INFO = K, U(IA+K-1,IA+K-1) is exactly zero; the matrix is singular and its inverse could not be computed. 💀 not the singular matrix jump scare.
ngl the INFO > 0 case is the dramatic one — if you see it, your matrix is singular and there is genuinely no inverse to be had. no amount of manifestation is fixing that, bestie.

and that's the whole page, no notes left out, no tea spilled that wasn't already in the pot. 🫶 this one's LAPACK version 1.5, dated 12 May 1997, and she's still serving. mwah. 🍓🧚💜

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