PDTRTRI(l)

💖 PDTRTRI — bestie edition 💖

okay so this is PDTRTRI(l), a LAPACK routine, and she is here to invert your upper or lower triangular distributed matrix. no cap. it's giving math girlboss energy and we're obsessed 💅

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✨ NAME ✨

PDTRTRI — computes the inverse of an upper or lower triangular distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1)

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🌸 SYNOPSIS 🌸

SUBROUTINE PDTRTRI( 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 👑

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

like. that's the whole job. she inverts it. periodt 💜

🧚 Notes 🧚

ok so listen, every global data object has an associated description vector, and that vector stores all the info you need to establish the mapping between an object element and its corresponding process and memory location. it's the group chat of metadata, fr fr.

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

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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) DOUBLE PRECISION 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.

⚠️ bestie hear me — that last one is the vibe check fail. if INFO = K, that diagonal entry is literally zero, the triangular matrix sub( A ) is singular, and the inverse can not be computed. no inverse for you, sorry not sorry 😭

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so yeah, that's PDTRTRI, and she ate 💅 this page is LAPACK routine version 1.5, dated 12 May 1997. stay singular-free out there, bestie 🦋✨🫶