PSTRTI2(l)

💖 NAME 💖

okay so PSTRTI2 — she computes the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). that's the whole vibe. bestie inverts triangles, no cap. 🦋

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✨ SYNOPSIS ✨

SUBROUTINE PSTRTI2( 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 🎀

So PSTRTI2 computes the inverse of a real upper or lower triangular block matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). iconic behavior.

And listen — this matrix needs to be contained in one and only one process memory space. that means it's a local operation, bestie. no passing notes across the grid, she's doing this solo. 💅

🌸 Notes 🌸

Every global data object has an associated description vector tagging along with it, and that vector is the one holding all the info you need to map an object element to its corresponding process and memory location. it's giving personal assistant energy, honestly. 🫶

Let A be a generic term for any 2D block cyclicly distributed array. A global array like that has an associated description vector DESCA. And in the comments below, whenever you see the character _, read it as "of the global array". iykyk. ⭐

• 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 so 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

math girlies stay winning. 💀✨

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👑 ARGUMENTS 👑

• UPLO (global input) CHARACTER*1 — tells you which triangle you're working with:
  • = 'U': sub( A ) is upper triangular;
  • = 'L': sub( A ) is lower triangular.
• DIAG (global input) CHARACTER*1 — 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. (yes bestie, zero is allowed, she's flexible 💅)
• A (local input/local output) REAL 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. glow up complete. ✨
• 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. the whole lore, in one vector. 📖
• INFO (local output) INTEGER — the receipts:
  • = 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.

heads up bestie: when INFO < 0 she is not being shady, that negative number is literally pointing at which argument had the illegal value. read it like a map, not a mood. 🧚

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that's the whole page, bestie — go invert something triangular and be iconic about it. 🍓

signing off with the receipts: this is LAPACK version 1.5, dated 12 May 1997. vintage, but she still eats. 💜