PDLANHS(l)

💖 PDLANHS 💖

okay so bestie, you're reading the man page for PDLANHS — that little (l) after the name just means it lives in the LAPACK corner of the manual universe 🦋 it's a LAPACK auxiliary routine, version 1.5, and no cap, it's a whole vibe. let's get into it 💅

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✨ NAME ✨

PDLANHS — returns the value of the one norm, or the Frobenius norm, … (and lowkey also the infinity norm and the biggest-absolute-value element, keep reading, it's giving range 👑)

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🎀 SYNOPSIS 🎀

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

    CHARACTER    NORM

    INTEGER      IA, JA, N

    INTEGER      DESCA( * )

    DOUBLE       PRECISION A( * ), WORK( * )

like yes it's a function, yes it returns a DOUBLE PRECISION, yes those are all the args, we move 💫

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🌸 PURPOSE 🌸

PDLANHS 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). periodt ✋

so it returns this value:

( 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, real talk:

• norm1 = the one norm of a matrix, which is the maximum column sum
• normI = the infinity norm of a matrix, which is the maximum row sum
• normF = the Frobenius norm of a matrix, which is the square root of sum of squares

🫶 bestie pls note: max(abs(A(i,j))) is not a matrix norm. it's just vibes. don't @ me, that's literally what the page says.

🧚 Notes 🧚

each global data object is described by an associated description vector. this vector stores the info you need to establish the mapping between an object element and its corresponding process and memory location. it's the little map that tells you where everything 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)).

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.
• 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 PDLANHS, as described above. yes queen, you pick the norm 💅
• 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, PDLANHS 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 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. the whole blueprint bestie 🗂️
• 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 ), 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. slay, go fetch them 🌈

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that's the whole page bestie, every last flag, default and formula accounted for 💖

signing off with the receipts: this is from LAPACK version 1.5, dated 12 May 1997 🎀 it's giving vintage but make it iconic 💜✨