PSTRTRI(l)

💖 PSTRTRI 💖

okay bestie, sit DOWN, because we're about to talk about PSTRTRI, a routine from LAPACK version 1.5, living its best life in section l of the manual. she computes inverses of triangular distributed matrices and she does NOT play. 🫶

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✨ NAME ✨

PSTRTRI — compute the inverse of a upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). that's the whole job description, no notes. 💅

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🌸 SYNOPSIS 🌸

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

    CHARACTER       DIAG, UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    REAL            A( * )

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🦋 PURPOSE 🦋

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

💜 Notes 💜

every global data object has a lil bestie called a description vector, and that vector is the one holding all the info you need to figure out the mapping between an element of the object and its process + memory location. it's giving matchmaker, honestly. 🎀

so let A be a generic term for any 2D block cyclicly distributed array. a global array like that has an associated description vector DESCA. in the comments below, read the character _ as "of the global array".

here's the whole lore, spelled out:

• 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 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

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👑 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) REAL 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. she ate and left no crumbs. 🔥
• 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. 😭

fr fr, if you see INFO > 0 it means you tried to invert a singular matrix and the math simply said no. not me getting humbled by a triangular matrix 💀

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that's the whole page bestie, you're SO ready for it now. this was LAPACK version 1.5, dated 12 May 1997 — iconic behavior from the 90s honestly, and the PSTRTRI(l) source is still serving. periodt. 🎀💅🌈