PSPOTRS(l)

🎀 NAME 🎀

okay bestie, welcome to the PSPOTRS(l) page 💖 this one is, like, the solver — it 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)

periodt. 💅

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✨ SYNOPSIS ✨

SUBROUTINE PSPOTRS( 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( * )

    REAL            A( * ), B( * )

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🌸 PURPOSE 🌸

so PSPOTRS solves a system of linear equations, and here's the vibe: sub( A ) means A(IA:IA+N-1,JA:JA+N-1), and it's a N-by-N symmetric positive definite distributed matrix that's already been Cholesky-factored into sub( A ) = U**T*U or L*L**T by PSPOTRF. and sub( B ) means the distributed matrix B(IB:IB+N-1,JB:JB+NRHS-1). it's giving solver energy fr fr ✨

🧠 Notes 🧠

every global data object is described by an associated description vector, and that vector stores all the info you need 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. and in the following comments, the character _ should be read as "of the global array".

• 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)).

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

🚨 important bestie: this routine requires square block decomposition ( MB_A = NB_A ). no cap.

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💅 ARGUMENTS 💅

• UPLO (global input) CHARACTER — which triangle is living in storage:
  • = '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) REAL 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**T or U**T*U, as computed by PSPOTRF. 🔥
• 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) REAL 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 — the receipts, bestie:
  • = 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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and that's the whole page, no crumbs left 💜 this was the PSPOTRS(l) moment — LAPACK version 1.5, dated 12 May 1997 🌸 slay, go solve something symmetric positive definite, mwah 💖🎀🦋