PZLANHS(l)

💖 NAME 💖

Okay bestie, meet PZLANHS — a LAPACK auxiliary routine (we're on version 1.5, she's vintage, she's iconic) whose whole entire job is to return the value of the one norm, or the Frobenius norm, — yes that sentence just ends like that in the original, we don't ask questions, we respect her 💅

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✨ SYNOPSIS ✨

Look at her signature, she's giving Fortran:

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

    CHARACTER    NORM

    INTEGER      IA, JA, N

    INTEGER      DESCA( * )

    DOUBLE       PRECISION WORK( * )

    COMPLEX*16   A( * )

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🎯 PURPOSE 🎯

So like, PZLANHS 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 Hessenberg distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1). She's a norm girlie with range, fr fr 🦋

Here's exactly what she returns, no cap:

( 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, bestie, to be super clear:

• norm1 is the one norm of a matrix — that's the maximum column sum.
• normI is the infinity norm of a matrix — that's the maximum row sum.
• normF is the Frobenius norm of a matrix — that's the square root of sum of squares.

Real talk though: note that max(abs(A(i,j))) is not a matrix norm. She's just out here vibing with the biggest absolute value, don't @ me 🙈

📝 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" — iykyk 🧚

Here's the whole lore, since I can't do a table for you (terminal said no), but every single row is here:

• 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)).

Now, let K be the number of rows or columns of a distributed matrix, and assume 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

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🫶 ARGUMENTS 🫶

• NORM — (global input) CHARACTER. Specifies the value to be returned in PZLANHS as described up above ☝️ she's the one choosing the vibe.
• 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, PZLANHS 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 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). And the size rules, bestie, pay attention because this is the part people mess up:

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. 🫡

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Go off, you now understand a distributed Hessenberg norm routine. Main character energy, honestly 👑

That's the whole page, bestie — LAPACK version 1.5, dated 12 May 1997, PZLANHS(l). She's been slaying since the nineties and she's not stopping 💖✨🎀