PDTRTRS(l)

📦 version 1.5, in ubuntu jammy

💅 PDTRTRS(l) — bestie, it's triangular solve o'clock 💅

okay so, <rainbow>PDTRTRS</rainbow> is a LAPACK auxiliary routine, version 1.5, and she is that girl. She solves a triangular system in distributed-memory land, and honestly? Iconic. Let's go through her whole vibe together, no detail skipped, periodt 🫶

🌸 NAME 🌸

PDTRTRS — solve a triangular system of the form sub( A ) * X = sub( B ) or sub( A )**T * X = sub( B ),

(yes she really does end on a comma, we don't question the queen 💖)

✨ SYNOPSIS ✨

SUBROUTINE PDTRTRS( 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( * )

    DOUBLE          PRECISION A( * ), B( * )

🎀 PURPOSE 🎀

PDTRTRS 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 📝

okay this next bit is the descriptor lore and you need it, so lock in bestie 🔥

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".

  • 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 — tells us which triangle we're 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:
    • = 'N': Solve sub( A ) * X = sub( B ) (No transpose)
    • = 'T': Solve sub( A )**T * X = sub( B ) (Transpose)
    • = 'C': Solve sub( A )**T * X = sub( B ) (Transpose)
  • DIAG (global input) CHARACTER — is the diagonal doing its own thing or nah:
    • = '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) DOUBLE PRECISION 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) DOUBLE PRECISION 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. ✨ she's a glow-up, the answer replaces the question ✨
  • 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 vibe check report:
    • = 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.
🚨 not me warning you but: if INFO > 0, your sub( A ) is singular and the solutions X were not computed. do not go looking for answers in B. they are not there. X said "no thoughts, head empty" 💀

that's the whole page bestie, every flag, every index, every descriptor entry, all of it, fr fr 🦋

signed off in style: this was LAPACK version 1.5, dated 12 May 1997. old money, still slaying 👑

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