PCPOTRS(l)

💖 PCPOTRS 💖 (the l is for LAPACK, bestie)

okay so. sit down. hydrate. we're doing linear algebra and we're doing it together. this is the LAPACK routine page for PCPOTRS, and yeah it's a mouthful but by the end you're gonna be solving systems like it's nothing. no cap.

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✨ NAME ✨

PCPOTRS — she solves a system of linear equations, two ways, both iconic:

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

that's it. that's the job. 🫶

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🌸 SYNOPSIS 🌸

here's the whole cast list, exactly as it appears on the poster:

SUBROUTINE PCPOTRS( 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         A( * ), B( * )

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🦋 PURPOSE 🦋

PCPOTRS solves 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)

where sub( A ) means 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

as computed by PCPOTRF. and sub( B ) means the distributed matrix B(IB:IB+N-1,JB:JB+NRHS-1). like, she takes the factorization you already did and she finishes the job. main character energy, honestly. 👑

💜 Notes 💜

every global data object out here 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. 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". okay? okay. 🎀

• 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.
• 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 optional. not negotiable. square. 💅

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🌈 ARGUMENTS 🌈

one bullet per argument, all facts intact, let's goooo 🍓

• UPLO — (global input) CHARACTER.
  • = 'U': the Upper triangle of sub( A ) is stored;
  • = 'L': the 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 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 PCPOTRF. 🧚
• 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 pointer into the local memory to an array of local dimension (LLD_B,LOCc(JB+NRHS-1)). on entry, this array contains the local pieces of the right hand sides sub( B ). on exit, this array contains the local pieces of the solution distributed matrix X. the glow-up arc!! she goes in as a right hand side and comes out as a SOLUTION. ✨
• 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. the vibe check, essentially:
  • = 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.

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💀 ngl, INFO coming back negative is not the moment. read the formula, find your culprit, fix your argument, try again. you got this. 💖

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okay that's the whole page, you're officially a ScaLAPACK girlie now, obsessed with you, go solve something 💅🎀🦋

this little love letter to distributed linear algebra is brought to you by LAPACK routine version 1.5, dated 12 May 1997 — and yeah she's been serving since the 90s, periodt. 👑