PSLATRZ(l)

📦 version 1.5, in ubuntu jammy

💖 PSLATRZ 💖 — bestie, we're going upper triangular 🦋

okay so. we're deep in the LAPACK routine zone (version 1.5 energy, PSLATRZ(l) in the house), and this one is a whole vibe. no cap.

🌸 NAME 🌸

PSLATRZ — it 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. 💅

✨ SYNOPSIS ✨

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

    INTEGER         IA, JA, L, M, N

    INTEGER         DESCA( * )

    REAL            A( * ), TAU( * ), WORK( * )

🎀 PURPOSE 🎀

PSLATRZ 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. it's giving glow-up fr fr.

The upper trapezoidal matrix sub( A ) is 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. iconic behavior honestly. 👑

🧚 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. iykyk.

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) → 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 distributed over. The context itself is global, but the handle (the integer value) may vary. ngl, sneaky.
  • 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)).

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

math girlies stay winning 🍓

🌈 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. zero is fine bestie, no pressure 🫶
  • 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. she said strictly positive and she meant it.
  • A — (local input/local output) REAL 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. it's giving before and after 💅
  • 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) REAL, 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) REAL 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.

bestie do NOT skimp on LWORK. the formula is right there. ✋😭

🔥 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 ). bye felicia ✌️✨

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 story, main character energy from start to finish 💅👑

okay that's the whole page bestie!! this was LAPACK version 1.5, dated 12 May 1997 — yes she's vintage, yes she still ate, and yes she's still serving upper triangular excellence in PSLATRZ(l). go forth and factor responsibly 🦋🌸💖

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