PSPOTRI(l)

💅 PSPOTRI(l) 💅

okay bestie, welcome to the PSPOTRI(l) page — she's a LAPACK routine, version 1.5, and she is highkey a ScaLAPACK icon. grab a snack, we're inverting matrices 🍓

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✨ NAME ✨

pspotri — computes the inverse of a real symmetric positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PSPOTRF. periodt.

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🌸 SYNOPSIS 🌸

SUBROUTINE PSPOTRI( UPLO, N, A, IA, JA, DESCA, INFO )

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    REAL            A( * )

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🦋 PURPOSE 🦋

PSPOTRI computes the inverse of a real symmetric positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) using the Cholesky factorization sub( A ) = U**T*U or L*L**T computed by PSPOTRF. literally the glow-up of a Cholesky factor into a full inverse, no cap 💖

📓 Notes (the descriptor lore, do not skip)

Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location.

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

here's the whole notation cheat sheet, bestie, memorise it and you're set 👑

• 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.
• 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)).

Let K be the number of rows or columns of a distributed matrix, and assume that 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

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🎀 ARGUMENTS 🎀

• UPLO (global input) CHARACTER*1
  • = 'U': Upper triangle of sub( A ) is stored;
  • = 'L': Lower triangle of sub( A ) is stored.
• 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. 🫶
• A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the triangular factor U or L from the Cholesky factorization of the distributed matrix sub( A ) = U**T*U or L*L**T, as computed by PSPOTRF. On exit, the local pieces of the upper or lower triangle of the (symmetric) inverse of sub( A ), overwriting the input factor U or L. she eats the factor and leaves an inverse behind, iconic behaviour 💅
• 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.
• INFO (global output) INTEGER — the receipts, bestie:
  • = 0: successful exit 🎉
  • < 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 = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. that's a yikes, the inverse is simply not happening 💀

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ngl the < 0 and > 0 branches are the vibe check at the end — always read INFO before you trust the output, fr fr.

written by the LAPACK girlies, LAPACK version 1.5, dated 12 May 1997 — she's been slaying since '97 and she's still serving 💖✨🎀