PCLANTR(l)

📦 version 1.5, in ubuntu jammy

💖 PCLANTR(l) — the LAPACK auxiliary routine that does norm math for you 💖

okay bestie, sit down, hydrate, because we're about to read a whole manpage together and honestly? it's giving iconic. this one's from LAPACK, version 1.5, it's an auxiliary routine, and it's literally here to hand you a norm. no cap.

✨ NAME ✨

PCLANTR — she returns the value of the one norm, or the Frobenius norm, and that's just the beginning of the story, keep reading 🫶

✨ SYNOPSIS ✨

REAL FUNCTION PCLANTR( NORM, UPLO, DIAG, M, N, A, IA, JA, DESCA, WORK )

    CHARACTER DIAG, NORM, UPLO

    INTEGER   IA, JA, M, N

    INTEGER   DESCA( * )

    REAL      WORK( * )

    COMPLEX   A( * )

like, look at that signature. she takes everything and she gives you back a single REAL. main character energy fr fr. 👑

✨ PURPOSE ✨

PCLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular distributed matrix sub( A ) denoting A(IA:IA+M-1, JA:JA+N-1).

so basically she picks her outfit based on what you ask for. and the value she returns is:

( max(abs(A(i,j))),  NORM = 'M' or 'm' with ia <= i <= ia+m-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. 💀 yeah. she said what she said.

🌸 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".

okay so here's the lore table, but make it a vibe:

  • 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

math but make it cute. we love to see it. 🧚

✨ ARGUMENTS ✨

  • NORM — (global input) CHARACTER. Specifies the value to be returned in PCLANTR as described above. she needs to know what kind of norm you're in the mood for, bestie.
  • UPLO — (global input) CHARACTER. Specifies whether the matrix sub( A ) is upper or lower trapezoidal.
    • = 'U': Upper trapezoidal
    • = 'L': Lower trapezoidal

Note that sub( A ) is triangular instead of trapezoidal if M = N. glow-up moment ✨

  • DIAG — (global input) CHARACTER. Specifies whether or not the distributed matrix sub( A ) has unit diagonal.
    • = 'N': Non-unit diagonal
    • = 'U': Unit diagonal
  • M — (global input) INTEGER. The number of rows to be operated on i.e the number of rows of the distributed submatrix sub( A ). When M = 0, PCLANTR is set to zero. M >= 0.
  • N — (global input) INTEGER. The number of columns to be operated on i.e the number of columns of the distributed submatrix sub( A ). When N = 0, PCLANTR 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 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).
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 ), Mp0 = NUMROC( M+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.

also pls note: some of these WORK values are literally marked (not referenced), which is the manpage way of saying "bestie I'm not even looking at that". don't stress about it, just give her the right size. 💅

and that's the whole page, no skips, no summaries, every flag accounted for. you ate that. 🍓

okay so the tea on where this came from: this is LAPACK version 1.5, dated 12 May 1997 — vintage, archival, still serving. 💜 bye bestie, go compute a norm 🌈

📦 where this page comes from