PCLANHS(l)

📦 version 1.5, in ubuntu jammy

✨ PCLANHS ✨

okay bestie, gather round, because we're about to talk about PCLANHS — a real LAPACK auxiliary routine that will return the value of the one norm, or the Frobenius norm, (and honestly more, keep reading, she does a lot 💅). This is a <rainbow>REAL FUNCTION</rainbow> and she is, lowkey, carrying the whole norm game. No cap.

🎀 SYNOPSIS 🎀

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

    CHARACTER NORM

    INTEGER   IA, JA, N

    INTEGER   DESCA( * )

    REAL      WORK( * )

    COMPLEX   A( * )

like, look at her. seven arguments, one job. iconic.

🦋 PURPOSE 🦋

so here's the tea, bestie: PCLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg distributed matrix

sub( A ) = A(IA:IA+N-1,JA:JA+N-1).

she returns, specifically:

( 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. she's just built different, okay, don't @ her.

📓 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 PCLANHS as described above. yes queen, pick your norm, she'll deliver.
  • 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, PCLANHS is set to zero. N >= 0. zero is valid, zero is fine, she's not gonna be dramatic about it. 🙈
  • 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 sub( A ).
  • 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).

and the LWORK rules, because bestie you gotta size your workspace right or it's giving 💀:

LWORK >=   0 if NORM = 'M' or 'm' (not referenced), Nq0 if NORM
= '1', 'O' or 'o', Mp0 if NORM = 'I' or 'i', 0 if NORM = 'F',
'f', 'E' or 'e' (not referenced), where
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 ),

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

🌸 friendly reminder: max(abs(A(i,j))) is not a matrix norm. she said what she said.

and that's the whole page, bestie 🫶 go forth and norm responsibly.

this was brought to you by LAPACK version 1.5, dated 12 May 1997 — vintage, timeless, still slaying. 💅✨👑

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