PZLANHE(l)

📦 version 1.5, in ubuntu jammy

💖 NAME 💖

PZLANHE — okay so this girl returns the value of the one norm, or the Frobenius norm, bestie, that's literally the whole name. (It's a LAPACK auxiliary routine, manual page PZLANHE(l), slay.)

✨ SYNOPSIS ✨

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

    CHARACTER    NORM, UPLO

    INTEGER      IA, JA, N

    INTEGER      DESCA( * )

    DOUBLE       PRECISION WORK( * )

    COMPLEX*16   A( * )

Not me showing you the whole signature, but yes, every single one of these is real and required. 💅

🎀 PURPOSE 🎀

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

So basically PZLANHE returns the value:

( 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. 🫶

🌸 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".

Here's the whole notation vibe check, one line at a time because we don't do tables in this house:

  • 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)).

Okay now the math part, stay with me 🦋

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 🔥

Every single one, no skipping, we respect the interface in this group chat.

  • NORM (global input) CHARACTER — Specifies the value to be returned in PZLANHE as described above. (yes, look up, it's right there 💖)
  • UPLO (global input) CHARACTER — Specifies whether the upper or lower triangular part of the hermitian 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, PZLANHE 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 hermitian 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.
  • 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) — here's the tea, bestie, the required size is different for every NORM:

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. 🫶

Careful, babe 💀 Get that LWORK wrong for NORM = '1', 'O', 'o', 'I' or 'i' and the workspace just won't be there. The LDW = 0 branch only kicks in when NPROW equals NPCOL. It's giving "read the formula twice".

💖 And that's the whole page, bestie! 💖 This one's from LAPACK version 1.5, dated 12 May 1997, and it lives in the LAPACK auxiliary routine section of the manual. Slay, ate, no cap. ✨👑💅

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