PCTRTRI(l)

📦 version 1.5, in ubuntu jammy

PCTRTRI 💖 — the LAPACK baddie that inverts your triangular distributed matrix

okay bestie, gather round, because we're talking about a real one from LAPACK routine (version 1.5) — the PCTRTRI era. this is the parallel-girlie cousin of the classic triangular inverse, and she is serving linear algebra on a process grid. 🫶

✨ NAME ✨

PCTRTRI — compute the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1)

like yes, that's the whole job. she inverts. she distributes. she slays. 💅

🦋 SYNOPSIS 🦋

SUBROUTINE PCTRTRI( UPLO, DIAG, N, A, IA, JA, DESCA, INFO )

    CHARACTER       DIAG, UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    COMPLEX         A( * )

look at her. seven-ish little inputs, one big job. iconic behaviour, no notes. 👑

🌸 PURPOSE 🌸

PCTRTRI computes the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1).

that's it, that's the mission statement. inverse. triangular. distributed. periodt. 💖

Notes

bestie, before you touch anything here, you need to understand the descriptor vector situation, because this is the part where people get humbled. 😭

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".

so like, the cheat sheet of what lives where:

  • 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)).

okay and now the grid math, because this is the part that's giving main character energy but also giving pop quiz. 🧚

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

memorise it or cry about it, those are the options. 🌈

💅 ARGUMENTS 💅

  • UPLO — (global input) CHARACTER. Specifies whether the distributed matrix sub( A ) is upper or lower triangular:
    • = 'U': Upper triangular,
    • = 'L': Lower triangular.
  • DIAG — (global input) CHARACTER. Specifies whether or not the distributed matrix sub( A ) is unit triangular:
    • = 'N': Non-unit triangular,
    • = 'U': 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.
  • A — (local input/local output) COMPLEX pointer into the local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array contains the local pieces of the triangular matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of the matrix sub( A ) contains the upper triangular matrix to be inverted, and the strictly lower triangular part of sub( A ) is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of the matrix sub( A ) contains the lower triangular matrix, and the strictly upper triangular part of sub( A ) is not referenced. On exit, the (triangular) inverse of the original matrix.
  • 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.
  • INFO — (global 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 = K, A(IA+K-1,JA+K-1) is exactly zero. The triangular matrix sub( A ) is singular and its inverse can not be computed.
🚨 ok so real talk: that last one is the villain arc. if INFO comes back > 0, your triangular matrix sub( A ) is singular and its inverse can not be computed. there is no clever workaround hiding in this manpage. she said no. 💀

that's the whole page, bestie. go forth, invert responsibly, and may your process grid always be p x q in your favour. 🎀

sign-off: this was brought to you by LAPACK version 1.5, dated 12 May 1997 — vintage, but the drip is timeless. ⭐🌸💜

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