PDLATRZ(l)

📦 version 1.5, in ubuntu jammy

✨ PDLATRZ ✨

okay bestie, gather round, because we're about to talk about a LAPACK routine and honestly? it's giving linear algebra main character energy. this is PDLATRZ from LAPACK version 1.5 💅

🌸 NAME 🌸

PDLATRZ — she reduces the M-by-N ( M<=N ) real upper trapezoidal matrix

sub( A ) = [ A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1) ]

to upper triangular form by means of orthogonal transformations. that's the whole vibe. she takes your trapezoid and she makes it a triangle. iconic behavior, no cap 🎀

🦋 SYNOPSIS 🦋

SUBROUTINE PDLATRZ( M, N, L, A, IA, JA, DESCA, TAU, WORK )

    INTEGER         IA, JA, L, M, N

    INTEGER         DESCA( * )

    DOUBLE          PRECISION A( * ), TAU( * ), WORK( * )

💖 PURPOSE 💖

so here's the tea, bestie. PDLATRZ reduces the M-by-N ( M<=N ) real upper trapezoidal matrix

sub( A ) = [ A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1) ]

to upper triangular form by means of orthogonal transformations. periodt. ✨

the upper trapezoidal matrix sub( A ) gets factored as

sub( A ) = ( R  0 ) * Z,

where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix. like... she said "trapezoid? never heard of her." 💀

📝 Notes 📝

bestie this part is important, don't skip it, i will know 🙈

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 breakdown, one bullet per thing so nothing gets lost:

  • 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 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 👑

every single one of them, bestie. no skipping. we respect the whole cast 🫶

  • M — (global input) INTEGER — the number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). 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 ). N >= 0.
  • L — (global input) INTEGER — the columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. L > 0.
  • A — (local input/local output) DOUBLE PRECISION pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper triangular matrix R, and elements N-L+1 to N of the first M rows of sub( A ), with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors.
  • 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.
  • TAU — (local output) DOUBLE PRECISION, array, dimension LOCr(IA+M-1) — this array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A.
  • WORK — (local workspace) DOUBLE PRECISION array, dimension (LWORK). LWORK >= Nq0 + MAX( 1, Mp0 ), where
IROFF = MOD( IA-1, MB_A ), ICOFF = 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+IROFF, MB_A,
MYROW, IAROW, NPROW ), Nq0   = NUMROC( N+ICOFF, NB_A, MYCOL,
IACOL, NPCOL ),

and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

🔥 FURTHER DETAILS 🔥

The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form

Z( k ) = ( I     0   ),
         ( 0  T( k ) )

where

T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                            (   0    )
                                            ( z( k ) )

tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ). she said obliterated. 💅

The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ).

Z is given by

Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

and that's the whole page bestie, you ate it up 💖 this was PDLATRZ from LAPACK version 1.5, dated 12 May 1997 — yeah she's an old queen but she still slays ✨🎀🦋 periodt 💜

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