PDTRTI2(l)

💖 PDTRTI2 — bestie's guide to the LAPACK routine 💖

okay so this is PDTRTI2, a LAPACK routine (version 1.5) and it's literally the moment where your triangular block matrix gets flipped into its inverse. one and only one process owns this thing, so it's a fully local operation — no drama, no cross-process gossip. 🎀

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✨ NAME ✨

PDTRTI2 — compute the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). slay.

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🌸 SYNOPSIS 🌸

SUBROUTINE PDTRTI2( UPLO, DIAG, N, A, IA, JA, DESCA, INFO )

    CHARACTER       DIAG, UPLO

    INTEGER         IA, INFO, JA, N

    INTEGER         DESCA( * )

    DOUBLE PRECISION A( * )

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🦋 PURPOSE 🦋

PDTRTI2 computes the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). this matrix should be contained in one and only one process memory space (local operation, periodt — it does not go wandering).

💫 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".

so here's the whole rundown of what lives where, no cap:

• 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

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💅 ARGUMENTS 💅

• UPLO — (global input) CHARACTER*1
  • = 'U': sub( A ) is upper triangular;
  • = 'L': sub( A ) is lower triangular.
• DIAG — (global input) CHARACTER*1
  • = '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.
• A — (local input/local output) 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 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, 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. if DIAG = 'U', the diagonal elements of sub( A ) are also not referenced and are assumed to be 1. on exit, the (triangular) inverse of the original matrix, in the same storage format. 🫶
• 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 — (local 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.

ngl if INFO comes back negative, something in your arguments was giving illegal value energy. check the i (and j) and fix it before you blame the math. 💀

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okay that's the whole page, no skips, no summarising, every flag accounted for. mwah. 💖✨

this was PDTRTI2(l) from LAPACK version 1.5, dated 12 May 1997 — iconic behaviour honestly, and that's the sign-off bestie, go invert something triangular. 👑🌈