PCLANHE(l)

📦 version 1.5, in ubuntu jammy

💖 PCLANHE 💖

okay so bestie, sit down, hydrate, because today we're learning about PCLANHE from the LAPACK auxiliary routine squad, version 1.5, and it's genuinely iconic. this is a REAL FUNCTION that lives in the .l manual section and she's here to hand you a norm. multiple norms actually. she's flexible like that. 💅

✨ NAME ✨

PCLANHE — returns the value of the one norm, or the Frobenius norm, … (yes the original sentence just trails off like that, we don't gatekeep, we move on 🫶)

🌸 SYNOPSIS 🌸

REAL FUNCTION PCLANHE( NORM, UPLO, N, A, IA, JA, DESCA, WORK )

    CHARACTER NORM, UPLO

    INTEGER   IA, JA, N

    INTEGER   DESCA( * )

    REAL      WORK( * )

    COMPLEX   A( * )

🦋 PURPOSE 🦋

PCLANHE 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). she contains multitudes. 🎀

PCLANHE 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). also, heads up: max(abs(A(i,j))) is not a matrix norm. she's just built different. not me pointing that out in the notes. 💀

📓 Notes 📓

every global data object gets described by an associated description vector, and that vector stores everything you need to map an object element back to its process and memory location. periodt.

let's let A be a generic term for any 2D block cyclicly distributed array. such a global array has an associated description vector DESCA. and in the following comments, read the character _ 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.
  • 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 PCLANHE as described above. she's the vibe-setter fr fr.
  • 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, PCLANHE is set to zero. N >= 0.
  • A — (local input) COMPLEX 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) REAL 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.

ok bestie real talk, this whole thing is a LAPACK auxiliary routine and there is genuinely a LOT of descriptor lore here. read the Notes twice. your future self will thank you. 🫶

LAPACK version 1.5, dated 12 May 1997 — yeah she's vintage, she's timeless, she ate and left no crumbs. that's the whole manual page, bestie, you did it, mwah 💖✨🎀🦋🌸💅🌈⭐👑

📦 where this page comes from