PZPOTRS(l)

💖 PZPOTRS 💖

okay so bestie, welcome to PZPOTRS — she's a LAPACK routine (version 1.5) and she is literally out here solving linear systems for a living. no cap. this is the distributed-computing girlie of the family, she runs on a BLACS process grid and she does NOT do small talk. 🎀

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✨ NAME ✨

PZPOTRS — solve a system of linear equations

sub( A ) * X = sub( B )
A(IA:IA+N-1,JA:JA+N-1)*X = B(IB:IB+N-1,JB:JB+NRHS-1)

slay. that's the whole job description. 💅

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🦋 SYNOPSIS 🦋

SUBROUTINE PZPOTRS( UPLO, N, NRHS, A, IA, JA, DESCA, B, IB, JB, DESCB,
                    INFO )

    CHARACTER       UPLO

    INTEGER         IA, IB, INFO, JA, JB, N, NRHS

    INTEGER         DESCA( * ), DESCB( * )

    COMPLEX*16      A( * ), B( * )

look at her signature. immaculate. every single argument is listed and yes, you have to pass all of them. 🙈

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🌸 PURPOSE 🌸

PZPOTRS solves a system of linear equations

sub( A ) * X = sub( B )

where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a N-by-N hermitian positive definite distributed matrix, using the Cholesky factorization

sub( A ) = U**H*U   or   L*L**H

computed by PZPOTRF. and sub( B ) denotes the distributed matrix B(IB:IB+N-1,JB:JB+NRHS-1).

so like — PZPOTRF does the work, and PZPOTRS comes in after and actually uses it to solve the thing. she's the closer. 👑

📝 Notes

bestie. listen. this part is dense but it's important, don't skip it, i'm begging 🥺

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 breakdown, one bullet per entry, no skipping:

• 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). no rectangles allowed. it's giving very strict. 💀

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💅 ARGUMENTS 💅

every single one, in order, exactly as she wants them:

• 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.
• NRHS — (global input) INTEGER
  • The number of right hand sides, i.e. the number of columns of the distributed submatrix sub( B ). NRHS >= 0.
• A — (local input) COMPLEX*16 pointer into local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the factors L or U from the Cholesky factorization sub( A ) = L*L**H or U**H*U, as computed by PZPOTRF.
• 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.
• B — (local input/local output) COMPLEX*16 pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). On entry, this array contains the the local pieces of the right hand sides sub( B ). On exit, this array contains the local pieces of the solution distributed matrix X.
• IB — (global input) INTEGER
  • The row index in the global array B indicating the first row of sub( B ).
• JB — (global input) INTEGER
  • The column index in the global array B indicating the first column of sub( B ).
• DESCB — (global and local input) INTEGER array of dimension DLEN_.
  • The array descriptor for the distributed matrix B.
• 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.

ngl INFO is the snitch of this whole operation. she will absolutely tell on you if you passed something illegal, and she'll tell you which one, with math. iconic behavior honestly. 🔥

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and that's the whole page bestie, you ate it. 🫶 this was LAPACK version 1.5, dated 12 May 1997 — yeah she's vintage, she's been solving hermitian positive definite systems since before you were born and she's still the main character. periodt. 👑✨