PDPOTRF(l)

💖 PDPOTRF(l) 💖

okay so bestie, gather round, because we're about to talk about PDPOTRF, a LAPACK routine, version 1.5, and honestly? she's iconic. she does Cholesky. she does distributed Cholesky. no cap. 🎀

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✨ NAME ✨

PDPOTRF — compute the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). 💅

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🌸 SYNOPSIS 🌸

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

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    DOUBLE          PRECISION A( * )

like yes that's the whole vibe check, seven arguments, one of them is a whole descriptor array, we love her 🫶

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🦋 PURPOSE 🦋

PDPOTRF computes the Cholesky factorization of an N-by-N real symmetric positive definite distributed matrix sub( A ) denoting 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. literally the glow-up of matrices, she breaks down into her triangular truth 💅✨

📝 Notes 📝

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". (yes bestie, the underscore is doing a little bit of cosplay, iykyk 🌈)

Here's the whole notation situation, stored-in-DESCA energy, all of it, no skips:

• 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

🌟 heads up bestie: this routine requires square block decomposition (MB_A = NB_A). not rectangular, not doing the most, square. periodt.

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💜 ARGUMENTS 💜

• UPLO — (global input) CHARACTER
  • = '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) DOUBLE PRECISION 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. 💅✨
• 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
  • = 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. 💀

🚨 bestie listen: that > 0 case is the heartbreak scenario. she tried, she really did, but that leading minor said "not positive definite" and the factorization just could not be completed. no fixing it, no manifesting it, it's done.

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and that's the whole page, bestie. every flag, every index, every LOCr and LOCc, every ceil, all of it, delivered with love. 🧚💖

this has been LAPACK version 1.5, dated 12 May 1997 — yes she's a vintage queen and she still slays, mwah 💋🌸✨