PDTRCON(l)

📦 version 1.5, in ubuntu jammy

✨ PDTRCON ✨

okay so bestie, welcome to PDTRCON(l) — it's a LAPACK routine, version 1.5, and it's lowkey one of the most useful little gremlins in the whole distributed-matrix lineup. sit down, hydrate, and let's get into it 💖

🎀 NAME 🎀

PDTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. that's it, that's the whole job, and she does it beautifully 💅

🌸 SYNOPSIS 🌸

SUBROUTINE PDTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK,
                    LWORK, IWORK, LIWORK, INFO )

    CHARACTER       DIAG, NORM, UPLO

    INTEGER         IA, JA, INFO, LIWORK, LWORK, N

    DOUBLE          PRECISION RCOND

    INTEGER         DESCA( * ), IWORK( * )

    DOUBLE          PRECISION A( * ), WORK( * )

💫 PURPOSE 💫

so like, PDTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm. both. your choice. iconic range honestly.

the norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and then the reciprocal of the condition number is computed as

RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1)      ) *
              norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).

🧚 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. it's giving ✨address book for your matrix✨.

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". 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. messy but valid, we love her.
  • 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

💜 ARGUMENTS 💜

  • NORM (global input) CHARACTER — specifies whether the 1-norm condition number or the infinity-norm condition number is required:
    • = '1' or 'O': 1-norm;
    • = 'I': Infinity-norm.
  • UPLO (global input) CHARACTER —
    • = 'U': A(IA:IA+N-1,JA:JA+N-1) is upper triangular;
    • = 'L': A(IA:IA+N-1,JA:JA+N-1) is lower triangular.
  • DIAG (global input) CHARACTER —
    • = 'N': A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular;
    • = 'U': A(IA:IA+N-1,JA:JA+N-1) is unit triangular.
  • N (global input) INTEGER — the order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0.
  • A (local input) 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 distributed matrix A(IA:IA+N-1,JA:JA+N-1). if UPLO = 'U', the leading N-by-N upper triangular part of this distributed matrix contains the upper triangular matrix, and its strictly lower triangular part is not referenced. if UPLO = 'L', the leading N-by-N lower triangular part of this ditributed matrix contains the lower triangular matrix, and the strictly upper triangular part is not referenced. if DIAG = 'U', the diagonal elements of A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed to be 1. (she's not reading the parts she doesn't need — boundaries, bestie 💅)
  • 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.
  • RCOND (global output) DOUBLE PRECISION — the reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as
RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1)      ) *
norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).
  • WORK (local workspace/local output) DOUBLE PRECISION array, dimension (LWORK). on exit, WORK(1) returns the minimal and optimal LWORK. ✨
  • LWORK (local or global input) INTEGER — the dimension of the array WORK. LWORK is local input and must be at least
LWORK >= 2*LOCr(N+MOD(IA-1,MB_A)) +
          LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX( NB_A*MAX( 1,
          CEIL(NPROW-1,NPCOL) ), LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX( 1,
          CEIL(NPCOL-1,NPROW) ) ).

if LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

  • IWORK (local workspace/local output) INTEGER array, dimension (LIWORK). on exit, IWORK(1) returns the minimal and optimal LIWORK.
  • LIWORK (local or global input) INTEGER — the dimension of the array IWORK. LIWORK is local input and must be at least
LIWORK >= LOCr(N+MOD(IA-1,MB_A)).

if LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

  • 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.
heads up bestie: INFO < 0 is not a vibe, it's a cry for help. read the formula carefully — array argument problems get the -(i*100+j) treatment, scalar arguments just get -i. no cap.

and that's the whole page, you ate it, periodt 💖 this was LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, yes she still slays, we stan a legacy queen 👑✨💅

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