PCPOTRF(l)

💖 PCPOTRF(l) 💖

okay bestie, sit down, hydrate, because today we're doing the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix. yes. distributed. as in, the matrix is spread across your whole process grid and everyone's doing their little part. it's giving teamwork, it's giving main character energy, it's giving LAPACK. 🫶

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✨ NAME ✨

PCPOTRF — compute the Cholesky factorization of an N-by-N complex hermitian positive definite distributed matrix sub( A ) denoting A(IA:IA+N-1, JA:JA+N-1). periodt. 💅

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🌸 SYNOPSIS 🌸

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

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    COMPLEX         A( * )

look at her. seven arguments. all of them important. no notes. 😌

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🦋 PURPOSE 🦋

so PCPOTRF computes the Cholesky factorization of an N-by-N complex hermitian 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. that's it, that's the whole vibe. triangle goes brrr. ⭐

Notes 🌈

every global data object is described by an associated description vector. this vector stores the info required to establish the mapping between an object element and its corresponding process and memory location. iykyk.

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        STORED IN      EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
                               DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                               the BLACS process grid A is distribu-
                               ted over. The context itself is glo-
                               bal, but the handle (the integer
                               value) may vary.
M_A    (global) DESCA( M_ )    The number of rows in the global
                               array A.
N_A    (global) DESCA( N_ )    The number of columns in the global
                               array A.
MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                               the rows of the array.
NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                               the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                               row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
                               first column of the array A is
                               distributed.
LLD_A  (local)  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). no rectangles allowed, she's not that kind of girl. 🙈

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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.

so like, which half of the matrix are we actually keeping? you tell her. that's the whole job. 💜

• 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.

square. always square. she doesn't do anything else. ✨

• A — (local input/local output) COMPLEX 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 Hermitian 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.

so she walks in with your matrix and walks out with a triangle. 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. she's the blueprint, the map, the group chat pinned message of where everything lives. 🗺️
• 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.

okay so this one matters: if INFO comes back positive it's telling you the matrix was not positive definite and the factorization just... could not be completed. not a vibe. check your matrix, bestie. 😭

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and that's the whole page, no cap. this one's from LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, yes she still slays, we love a classic. 🧚🌸👑