PSLANTR(l)

📦 version 1.5, in ubuntu jammy

💖 PSLANTR 💖 (that's the routine name, bestie — LAPACK auxiliary routine, version 1.5)

okay so. imagine you've got a big distributed matrix and you just wanna know how big it is, number-wise. like, what's the norm. that's PSLANTR. she's a REAL FUNCTION and she will absolutely tell you the vibe of your matrix. no cap.

✨ NAME ✨

PSLANTR — returns the value of the one norm, or the Frobenius norm, … (yes the original NAME line trails off like that, it's giving mysterious 🫶)

🌸 SYNOPSIS 🌸

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

    CHARACTER DIAG, NORM, UPLO

    INTEGER   IA, JA, M, N

    INTEGER   DESCA( * )

    REAL      A( * ), WORK( * )

🦋 PURPOSE 🦋

PSLANTR 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).

like, she returns:

( 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. bestie I need you to sit with that one, it's important. 💅

📝 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 notation tea, no table allowed in this format so we're doing bullets, they hit exactly the same 🎀:

  • 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 PSLANTR as described above.
  • 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.

  • 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, PSLANTR 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, PSLANTR is set to zero. N >= 0.
  • A — (local input) REAL 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.

heads up bestie: if NORM = 'M' or 'm' your WORK array is literally not referenced, and same deal for 'F', 'f', 'E' or 'e' — she simply does not look at it. the other cases DO need it though. don't skip it, that's the whole plot twist 😭

okay that's the whole page!! ily, go compute some norms 💜

LAPACK version 1.5 — dated 12 May 1997. yes ma'am, 1997, and still serving. periodt. ✨

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