PZLATRA(l)

✨ PZLATRA — the LAPACK auxiliary routine, bestie edition 💖

okay so this is PZLATRA(l) from the LAPACK auxiliary lineup, version 1.5, and she is here to hand you the trace of a distributed matrix. pull up a chair, we're doing linear algebra in the group chat 🫶

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🌸 NAME 🌸

PZLATRA — compute the trace of an N-by-N distributed matrix sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ).

that's it, that's the job. she traces. she slays. periodt ✨

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🎀 SYNOPSIS 🎀

COMPLEX*16 FUNCTION PZLATRA( N, A, IA, JA, DESCA )

    INTEGER         IA, JA, N

    INTEGER         DESCA( * )

    COMPLEX*16      A( * )

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💅 PURPOSE 💅

PZLATRA computes the trace of an N-by-N distributed matrix sub( A ), which denotes A( IA:IA+N-1, JA:JA+N-1 ). the result is left on every process of the grid — no cap, everybody eats 🍓

📝 Notes 📝

each global data object is described by an associated description vector. this vector stores all the info you need to establish the mapping between an object element and its corresponding process and memory location. it's giving matchmaker, honestly 💜

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 comments below, read the character _ as "of the global array".

here's the notation lineup, and she is iconic:

• 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 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

lowkey math but make it fashion 🌈

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👑 ARGUMENTS 👑

• N — (global input) INTEGER. the number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0.
• A — (local input) COMPLEX*16 pointer into the local memory to an array of dimension ( LLD_A, LOCc(JA+N-1) ). this array contains the local pieces of the distributed matrix the trace is to be computed. she's carrying the whole squad's data, fr fr 💖
• 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 blueprint, the lore, the source of truth 🦋

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ngl, if you were looking for the trace of a non-distributed matrix you are in the wrong group chat. this one is distributed-only, bestie. DESCA is not optional. 💀

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and that's the whole page!! written by LAPACK, version 1.5, dated 12 May 1997 — a vintage queen who has been serving trace energy since before some of us were born 👑💜 thank you for coming to my TED talk, go compute something cute ✨🎀🫶