PZLANSY(l)

📦 version 1.5, in ubuntu jammy

💖 PZLANSY 💖

okay so bestie, gather round, because today we're talking about PZLANSY — a LAPACK auxiliary routine, version 1.5, and she is that girl. sit down, get a snack, and let's get into it 🍓

✨ NAME ✨

PZLANSY — return the value of the one norm, or the Frobenius norm, yes that's literally where the sentence ends, and also the infinity norm and the largest absolute value, if you want. she does it all 💅

🌸 SYNOPSIS 🌸

DOUBLE PRECISION FUNCTION PZLANSY( NORM, UPLO, N, A, IA, JA, DESCA, WORK
                 )

    CHARACTER    NORM, UPLO

    INTEGER      IA, JA, N

    INTEGER      DESCA( * )

    DOUBLE       PRECISION WORK( * )

    COMPLEX*16   A( * )

👑 PURPOSE 👑

PZLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1).

so like, which one does she hand back? depends on what you ask for, babe:

( max(abs(A(i,j))),  NORM = 'M' or 'm' with IA <= i <= IA+N-1,
(                                      and  JA <= j <= JA+N-1,
(
( norm1( sub( A ) ), NORM = '1', 'O' or 'o'
(
( normI( sub( A ) ), NORM = 'I' or 'i'
(
( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e'

where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm. it's just... a vibe. iykyk 😭

💫 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.

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 the value to be returned in PZLANSY as described above. yeah, the whole menu up in PURPOSE. pick your fighter 💅
  • UPLO — (global input) CHARACTER. Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is to be referenced.
    • = 'U': Upper triangular part of sub( A ) is referenced,
    • = 'L': Lower triangular part of sub( A ) is referenced.
  • N — (global input) INTEGER. The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PZLANSY is set to zero. N >= 0.
  • A — (local input) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced. so don't stress about the half she's ignoring, she's literally not looking at it 🙈
  • 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.
  • WORK — (local workspace) DOUBLE PRECISION array dimension (LWORK)
LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW
if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF(
NPROW.NE.NPCOL ) THEN LDW =
MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if
NORM = 'F', 'f', 'E' or 'e' (not referenced),

where LCM is the least common multiple of NPROW and NPCOL

LCM = ILCM( NPROW, NPCOL )

and CEIL denotes the ceiling operation (ICEIL).

IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW =
INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL ), Np0 = NUMROC( N+IROFFA, MB_A,
MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL,
IACOL, NPCOL ),

ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

bestie real talk: that WORK sizing is a whole personality. read it twice, count your Np0s, then read it again. no cap. 💀

and that's the whole page, periodt. 🦋 this was LAPACK version 1.5, dated 12 May 1997 — yes she's an elder icon and she still ate, mwah 💖 🎀 ✨

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