PZTRTRS(l)

📦 version 1.5, in ubuntu jammy

💅 PZTRTRS 💅

okay bestie, sit down, hydrate, and let me tell you about PZTRTRS — she's a LAPACK auxiliary routine (we're talking version 1.5 energy here) and she solves triangular systems for a living. no cap. 💖

✨ NAME ✨

PZTRTRS — solves a triangular system of the form sub( A ) * X = sub( B ) or sub( A )**T * X = sub( B ) or sub( A )**H * X = sub( B ), 🦋

🎀 SYNOPSIS 🎀

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

    CHARACTER       DIAG, TRANS, UPLO

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

    INTEGER         DESCA( * ), DESCB( * )

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

👑 PURPOSE 👑

so like, PZTRTRS solves a triangular system of the form

where sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1) and is a triangular distributed matrix of order N, and B(IB:IB+N-1,JB:JB+NRHS-1) is an N-by-NRHS distributed matrix denoted by sub( B ). and listen — a check is made to verify that sub( A ) is nonsingular. she checks. she's thorough. periodt. 💅

🌸 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. it's giving ✨organizational queen✨.

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".

  • 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

🔥 ARGUMENTS 🔥

🌈 UPLO — (global input) CHARACTER

which triangle are we working with, bestie:

  • = 'U': sub( A ) is upper triangular;
  • = 'L': sub( A ) is lower triangular.

💜 TRANS — (global input) CHARACTER

specifies the form of the system of equations, iykyk:

  • = 'N': Solve sub( A ) * X = sub( B ) (No transpose)
  • = 'T': Solve sub( A )**T * X = sub( B ) (Transpose)
  • = 'C': Solve sub( A )**H * X = sub( B ) (Conjugate transpose)

🦋 DIAG — (global input) CHARACTER

  • = 'N': sub( A ) is non-unit triangular;
  • = 'U': sub( A ) is unit triangular.

⭐ 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 negativity allowed, she said what she said. 🙈

🍓 NRHS — (global input) INTEGER

the number of right hand sides, i.e., the number of columns of the distributed matrix sub( B ). NRHS >= 0.

🎀 A — (local input) COMPLEX*16 pointer into the local memory

to an array of dimension (LLD_A,LOCc(JA+N-1) ). this array contains the local pieces of the distributed triangular matrix sub( A ).

  • if UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix, and the strictly lower triangular part of sub( A ) is not referenced.
  • if UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced.
  • if DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. she's just out here assuming, and honestly? valid. 💅

💖 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 dimension (LLD_B,LOCc(JB+NRHS-1)).

on entry, this array contains the local pieces of the right hand side distributed matrix sub( B ). on exit, if INFO = 0, sub( B ) is overwritten by the solution matrix X. glow-up moment fr fr. ✨

🧚 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 — (output) INTEGER

the vibes check. listen closely:

  • = 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 = i, the i-th diagonal element of sub( A ) is zero, indicating that the submatrix is singular and the solutions X have not been computed.
okay so that last one? that's the drama. singular matrix means no solution was computed, bestie. don't come for her, she told you upfront. 😭

and that's the whole page, bestie 🫶 — this was PZTRTRS(l) from LAPACK version 1.5, dated 12 May 1997. iconic behavior from a routine older than most of us. mwah. 💋

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