PZLARFG - generate a complex elementary reflector H of order n, such that H * sub( X ) = H * ( x(iax,jax) ) = ( alpha ), H' * H = I

SUBROUTINE PZLARFG(

N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU )

INTEGER IAX, INCX, IX, JAX, JX, N

COMPLEX*16 ALPHA

INTEGER DESCX( * )

COMPLEX*16 TAU( * ), X( * )

PZLARFG generates a complex elementary reflector H of order n, such that ( x ) ( 0 )

where alpha is a real scalar, and sub( X ) is an (N-1)-element complex distributed vector X(IX:IX+N-2,JX) if INCX = 1 and X(IX,JX:JX+N-2) if INCX = DESCX(M_). H is represented in the form

H = I - tau * ( 1 ) * ( 1 v' ) ,

( v )

where tau is a complex scalar and v is a complex (N-1)-element vector. Note that H is not Hermitian.

If the elements of sub( X ) are all zero and X(IAX,JAX) is real, then tau = 0 and H is taken to be the unit matrix.

Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1.

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

Because vectors may be viewed as a subclass of matrices, a distributed vector is considered to be a distributed matrix.

N (global input) INTEGER

The global order of the elementary reflector. N >= 0.

ALPHA (local output) COMPLEX*16

On exit, alpha is computed in the process scope having the vector sub( X ).

IAX (global input) INTEGER

The global row index in X of X(IAX,JAX).

JAX (global input) INTEGER

The global column index in X of X(IAX,JAX).

X (local input/local output) COMPLEX*16, pointer into the

local memory to an array of dimension (LLD_X,*). This array contains the local pieces of the distributed vector sub( X ). Before entry, the incremented array sub( X ) must contain the vector x. On exit, it is overwritten with the vector v.

IX (global input) INTEGER

The row index in the global array X indicating the first row of sub( X ).

JX (global input) INTEGER

The column index in the global array X indicating the first column of sub( X ).

DESCX (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix X.

INCX (global input) INTEGER

The global increment for the elements of X. Only two values of INCX are supported in this version, namely 1 and M_X. INCX must not be zero.

TAU (local output) COMPLEX*16, array, dimension LOCc(JX)

if INCX = 1, and LOCr(IX) otherwise. This array contains the Householder scalars related to the Householder vectors. TAU is tied to the distributed matrix X.