PZLANGE(l)

✨ NAME ✨

PZLANGE — a LAPACK auxiliary routine, version 1.5, and her whole job description is: return the value of the one norm, or the Frobenius norm,

(yeah the NAME line just... trails off like that, mid-sentence, no cap 😭 the full tea is in PURPOSE, keep scrolling bestie)

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🌸 SYNOPSIS 🌸

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

    CHARACTER    NORM

    INTEGER      IA, JA, M, N

    INTEGER      DESCA( * )

    DOUBLE       PRECISION WORK( * )

    COMPLEX*16   A( * )

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🦋 PURPOSE 🦋

PZLANGE hands you back 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, here's exactly what she returns, depending on the vibe you ask for:

( 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) 👑
• 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 said what she said. It's just the biggest absolute element, don't call it a norm at parties, it's not that girl.

📓 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. (iykyk — it's the little address book that tells every process where its slice of the matrix lives 🏠)

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".

NOTATION        STORED IN      EXPLANATION
--------------- -------------- --------------------------------------
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 distribu-
                               ted over. The context itself is glo-
                               bal, 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

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

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🎀 ARGUMENTS 🎀

• NORM — (global input) CHARACTER — Specifies the value to be returned in PZLANGE as described above. 👆 the whole menu of norm vibes lives up there
• 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, PZLANGE 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, PZLANGE is set to zero. N >= 0.
• A — (local input) COMPLEX*16 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. 🫶

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🌟 and that's the whole page bestie — every flag, every LOCr, every little MOD — nothing skipped, periodt.

LAPACK version 1.5, dated 12 May 1997 — yeah she's a 90s baby and she's still serving 💅 thanks for coming to my TED talk, mwah 🎀💖✨