---
name: "pslatrz"
section: "3"
platform: "linux"
package: "scalapack-doc"
version: "1.5-11"
release: "Debian 11 (Bullseye)"
platforms: ["linux"]
url: "https://new.carta.tech/pslatrz"
---

# pslatrz(3)

reduce 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

## NAME

PSLATRZ - reduce 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

## 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.

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.\

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.\

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)).

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) 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 trian- gular 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) 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.

## 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 ).\

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 ).\