PCPOTF2(l)

💖 PCPOTF2 💖

okay bestie, buckle up, we're doing LAPACK today — this is the PCPOTF2 routine, straight outta the LAPACK library (version 1.5, she's vintage, she's iconic). it's a subroutine, it's parallel, it does Cholesky, and it's lowkey a whole vibe. let's get into it 🫶

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✨ NAME ✨

PCPOTF2 — computes the Cholesky factorization of a complex hermitian positive definite distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). that's the whole job, and she does it flawlessly. 💅

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🎀 SYNOPSIS 🎀

SUBROUTINE PCPOTF2( UPLO, N, A, IA, JA, DESCA, INFO )

    CHARACTER       UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    COMPLEX         A( * )

that's it. that's the whole signature. seven arguments, zero notes, all business. 💖

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🌸 PURPOSE 🌸

so PCPOTF2 computes the Cholesky factorization of a complex hermitian positive definite distributed matrix sub( A ) = 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. the way she just decomposes you into triangles... main character energy honestly. 👑

💫 Notes 💫

every global data object is described by an associated description vector, bestie. this vector stores the information required to establish the mapping between an object element and its corresponding process and memory location. it's giving phonebook for parallel memory. ✨

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". iykyk 💀

here's the whole lore, one bullet per entry:

• 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)).

okay so now 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

🚨 okay listen up bestie, this is the important part: this routine requires N <= NB_A-MOD(JA-1, NB_A) and square block decomposition (MB_A = NB_A). no square blocks, no vibes. that's just the rules. 💀

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🌈 ARGUMENTS 🌈

• UPLO (global input) CHARACTER — which triangle are we even talking about, bestie?
  • = '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 negative N, that's not a thing, we don't do that here 🫶
• 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 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. she comes in as a matrix and leaves as a factor, glow-up of the century ✨
• 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 (local output) INTEGER — the receipts, the tea, the status report 🍓
  • = 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.

💀 ngl if you get INFO > 0 your matrix said "not positive definite" and the whole factorization just... stopped. no closure, no factor, nothing. go check your matrix, bestie.

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and that's the whole page, you ate it, you understood it, you're literally a ScaLAPACK girlie now 👑

this was PCPOTF2(l) from the LAPACK routine set, LAPACK version 1.5, dated 12 May 1997 — yes she's older than us and still serving 💅 no cap. 🦋