PDLANGE(l)

📦 version 1.5, in ubuntu jammy

💖 PDLANGE 💖

okay bestie, sit down, hydrate, and let me tell you about PDLANGE from the LAPACK auxiliary routine lineup (version 1.5, she's an icon, she's been serving since forever). This is a DOUBLE PRECISION FUNCTION and she's here to measure how big your distributed matrix is. no cap. 🌸

✨ NAME ✨

PDLANGE — return the value of the one norm, or the Frobenius norm, (and also the infinity norm, and the biggest absolute element, but the NAME section is being mysterious about it, we don't gatekeep here)

🎀 SYNOPSIS 🎀

DOUBLE PRECISION FUNCTION PDLANGE( NORM, M, N, A, IA, JA, DESCA, WORK )

    CHARACTER    NORM

    INTEGER      IA, JA, M, N

    INTEGER      DESCA( * )

    DOUBLE       PRECISION A( * ), WORK( * )

🦋 PURPOSE 🦋

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

So like, PDLANGE returns the value:

   ( 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. like, it's giving norm but it is literally not one, bestie, don't be fooled 💅

📝 Notes 📝

Every 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) — DESCA( DTYPE_ ) — The descriptor type. In this case, DTYPE_A = 1.
  • CTXT_A (global) — 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) — DESCA( M_ ) — The number of rows in the global array A.
  • N_A (global) — DESCA( N_ ) — The number of columns in the global array A.
  • MB_A (global) — DESCA( MB_ ) — The blocking factor used to distribute the rows of the array.
  • NB_A (global) — DESCA( NB_ ) — The blocking factor used to distribute the columns of the array.
  • RSRC_A (global) — DESCA( RSRC_ ) — The process row over which the first row of the array A is distributed.
  • CSRC_A (global) — DESCA( CSRC_ ) — The process column over which the first column of the array A is distributed.
  • LLD_A (local) — 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 PDLANGE as described above. yes bestie, it's the vibe selector, choose wisely ✨
  • 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, PDLANGE 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, PDLANGE is set to zero. N >= 0.
  • A — (local input) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the distributed matrix 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) DOUBLE PRECISION 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.

🫶 real talk: the WORK sizing is not a suggestion, bestie. do the NUMROC math, don't just vibe-check it. your workspace will not forgive you. 😭

and that's the whole page, no crumbs left 💜 this was the LAPACK auxiliary routine PDLANGE, version 1.5, dated 12 May 1997 — she's vintage, she's iconic, she's still doing the math. slay, periodt. 🎀🌈⭐

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