PCLATRZ(l)

📦 version 1.5, in ubuntu jammy

💖 PCLATRZ 💖

okay bestie, so this one is a LAPACK routine — specifically a PCLATRZ situation, and it's about to reduce an upper trapezoidal matrix to upper triangular form using unitary transformations. it's giving linear algebra main character energy and we're here for it. 💅

✨ NAME ✨

PCLATRZ — reduce the M-by-N ( M<=N ) complex 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)]

🌈 SYNOPSIS 🌈

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

    INTEGER         IA, JA, L, M, N

    INTEGER         DESCA( * )

    COMPLEX         A( * ), TAU( * ), WORK( * )

🦋 PURPOSE 🦋

so basically PCLATRZ takes that M-by-N ( M<=N ) complex 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)]

and reduces it to upper triangular form, using unitary transformations. slay. ✨

the upper trapezoidal matrix sub( A ) gets factored as

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

where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix. iconic behavior honestly.

💜 Notes 💜

okay so this is the part where we talk about descriptors, and it's lowkey a lot, but it's all important bestie, don't skip it. 🫶

every 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.

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".

  • 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 🎀

  • 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) COMPLEX 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 unitary 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) COMPLEX, 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) COMPLEX 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 💀

ngl bestie this is the math part but it's actually kind of gorgeous, stay with me. 🌸

the factorization is obtained by Householder's method. the kth transformation matrix, Z( k ), whose conjugate transpose 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 ).

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, no notes, we ate. 💅

okay so this was LAPACK routine version 1.5, dated 12 May 1997 — vintage but still absolutely serving, periodt. 🎀✨🦋

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