PSPOTRF(l)

💖 PSPOTRF — the LAPACK girlies' guide to Cholesky-ing a distributed matrix 💖

okay bestie, sit down, hydrate, because we're about to talk about PSPOTRF from LAPACK routine (version 1.5) — she's the parallel one, she lives in ScaLAPACK-land, and she is so iconic. 🫶

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✨ NAME ✨

PSPOTRF — no cap, this routine computes the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ), which is the shorthand for A(IA:IA+N-1, JA:JA+N-1). 💅

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🌸 SYNOPSIS 🌸

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

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    REAL            A( * )

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🎀 PURPOSE 🎀

so the whole point of her existence, fr fr: PSPOTRF computes the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ), which again means A(IA:IA+N-1, JA:JA+N-1). 💜

The factorization has the form

          sub( A ) = U' * U ,  if UPLO = 'U', or

          sub( A ) = L  * L',  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular. two icons, one routine. 👑

📝 Notes 📝

Every 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. it's giving ✨logistics✨.

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".

• NOTATION DTYPE_A(global) — STORED IN DESCA( DTYPE_ ) — the descriptor type. In this case, DTYPE_A = 1.
• NOTATION 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.
• NOTATION M_A (global) — STORED IN DESCA( M_ ) — the number of rows in the global array A.
• NOTATION N_A (global) — STORED IN DESCA( N_ ) — the number of columns in the global array A.
• NOTATION MB_A (global) — STORED IN DESCA( MB_ ) — the blocking factor used to distribute the rows of the array.
• NOTATION NB_A (global) — STORED IN DESCA( NB_ ) — the blocking factor used to distribute the columns of the array.
• NOTATION RSRC_A (global) — STORED IN DESCA( RSRC_ ) — the process row over which the first row of the array A is distributed.
• NOTATION CSRC_A (global) — STORED IN DESCA( CSRC_ ) — the process column over which the first column of the array A is distributed.
• NOTATION 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

bestie listen. This routine requires square block decomposition (MB_A = NB_A). not optional. she said square or nothing. 💀

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🔥 ARGUMENTS 🔥

• UPLO (global input) CHARACTER — which triangle are we even looking at, queen?
  • = '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. no negatives, she's not doing that today. 🙈
• A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ) to be factored. If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the distributed matrix, and its strictly upper triangular part is not referenced. On exit, if UPLO = 'U', the upper triangular part of the distributed matrix contains the Cholesky factor U; if UPLO = 'L', the lower triangular part of the distributed matrix contains the Cholesky factor L. the glow-up is real. ✨
• 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. she's the whole personality of this routine, honestly. 💅
• INFO (global output) INTEGER — the vibe check, the receipts, the final verdict:
  • = 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 = K, the leading minor of order K, A(IA:IA+K-1,JA:JA+K-1) is not positive definite, and the factorization could not be completed. 😭 not positive definite?? the math said no and honestly, respect the boundary.

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and that's the whole page bestie, every flag, every formula, every DESCA entry, nothing skipped, periodt. 🦋

LAPACK version 1.5 — 12 May 1997 — yeah she's vintage, she's archival, she's a classic, and she still slays. mwah. 💖👑