PCTRTRS(l)

📦 version 1.5, in ubuntu jammy

💖 PCTRTRS(l) — the LAPACK auxiliary routine, but make it make sense 💖

okay bestie, sit down, hydrate, because we're about to talk about a LAPACK auxiliary routine and honestly? she's iconic. this is from LAPACK version 1.5 and she's here to solve a triangular system for you, distributed-memory style. no cap. 🦋

✨ NAME ✨

PCTRTRS — 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 PCTRTRS( 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         A( * ), B( * )

🎀 PURPOSE 🎀

So PCTRTRS 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 ). A check is made to verify that sub( A ) is nonsingular. 👑

💅 Notes 💅

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". (yes, the underscore is doing the heavy lifting here. main character energy.)

  • 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. = 'U': sub( A ) is upper triangular; = 'L': sub( A ) is lower triangular.
  • TRANS — (global input) CHARACTER. Specifies the form of the system of equations: = '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.
  • 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 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.
  • 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 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.
  • 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. = 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.
🚨 bestie listen — that INFO > 0 case is NOT a vibe. if it happens, your sub( A ) is singular and the solutions X were never computed. do not go looking for them. they are not there. 💀

and that's the whole page, fr fr 🌈✨

this lil legend was brought to you by LAPACK version 1.5, dated 12 May 1997 — shoutout to the OGs, they ate and left absolutely no crumbs 👑🫶💜

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