PDGECON(l)

📦 version 1.5, in ubuntu jammy

💖 PDGECON — the LAPACK routine, bestie edition 💖

okay so before we even get into it: this is PDGECON from LAPACK, version 1.5. Yes, the (l) means it's a manual page, we're being fancy today. Sit down, grab a strawberry drink 🍓, and let's get into it.

✨ NAME ✨

PDGECON — she estimates the reciprocal of the condition number of a general distributed real 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 PDGETRF. queen behavior, honestly 👑

🦋 SYNOPSIS 🦋

SUBROUTINE PDGECON( NORM, N, A, IA, JA, DESCA, ANORM, RCOND, WORK,
                    LWORK, IWORK, LIWORK, INFO )

    CHARACTER       NORM

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

    DOUBLE          PRECISION ANORM, RCOND

    INTEGER         DESCA( * ), IWORK( * )

    DOUBLE          PRECISION A( * ), WORK( * )

look at her. iconic signature. no notes. 💅

🌸 PURPOSE 🌸

PDGECON estimates the reciprocal of the condition number of a general distributed real 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 PDGETRF. literally what the NAME said, but with ✨commitment✨.

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)) ) ).

that's the whole formula. she's giving math and she's serving it. 🧮💖

📝 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. (iykyk — it's the little passport that says which process owns which number.)

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) 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 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) DOUBLE PRECISION. 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) 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)) +
         2*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) ) ).

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.

  • 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.
not me reading INFO = -(i*100+j) and having to sit down. it's giving encrypted message from the compiler. it's giving you WILL decode it. 🫶

that's the whole page, bestie — nothing skipped, nothing summarised, every flag and formula accounted for. 🌈

this was LAPACK version 1.5, dated 12 May 1997 — yes she's a vintage icon and she still slays. 💜

kisses, take care of your condition numbers, and remember: RCOND close to zero means your matrix is out here struggling. bye bestie! 💅✨🎀

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