PCGECON(l)

📦 version 1.5, in ubuntu jammy

🌸 NAME 🌸

bestie, meet PCGECON 💖 — that's the PCGECON(l) page, a LAPACK routine, version 1.5, and she is here to estimate the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PCGETRF. slay.

✨ SYNOPSIS ✨

SUBROUTINE PCGECON( NORM, N, A, IA, JA, DESCA, ANORM, RCOND, WORK,
                    LWORK, RWORK, LRWORK, INFO )

    CHARACTER       NORM

    INTEGER         IA, INFO, JA, LRWORK, LWORK, N

    REAL            ANORM, RCOND

    INTEGER         DESCA( * )

    REAL            RWORK( * )

    COMPLEX         A( * ), WORK( * )

🦋 PURPOSE 🦋

okay so, PCGECON estimates the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PCGETRF. no cap, that's the whole job.

An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and 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

Every single 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 "the receipts for your array" 💅

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

👑 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
  • N — (global input) INTEGER. The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0.
  • A — (local input) COMPLEX 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 factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U; the unit diagonal elements of L are not stored. 🙈
  • 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.
  • ANORM — (global input) REAL. If NORM = '1' or 'O', the 1-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). If NORM = 'I', the infinity-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1).
  • RCOND — (global output) REAL. 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) COMPLEX 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)) + MAX( 2,
MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) +
NB_A*CEIL(NPCOL-1,NPROW)) ).

LOCr and LOCc values can be computed using the ScaLAPACK tool function NUMROC; NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

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.

  • RWORK — (local workspace/local output) REAL array, dimension (LRWORK). On exit, RWORK(1) returns the minimal and optimal LRWORK.
  • LRWORK — (local or global input) INTEGER. The dimension of the array RWORK. LRWORK is local input and must be at least
LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)).

If LRWORK = -1, then LRWORK 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.
🚨 bestie heads up: negative INFO means somebody passed a bad argument, and the formula tells you exactly which one. read it carefully, don't just vibe past it.

and that's the whole page, you ate it 💖 this was PCGECON(l) from LAPACK version 1.5, dated 12 May 1997 — yeah she's a 90s icon and she still runs the grid, periodt. 🎀🦋✨

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