Provided by: scalapack-doc_1.5-11_all 
NAME
PZLABRD - reduce the first NB rows and columns of a complex general M-by-N distributed
matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an unitary
transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the
transfor- mation to the unreduced part of sub( A )
SYNOPSIS
SUBROUTINE PZLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY,
JY, DESCY, WORK )
INTEGER IA, IX, IY, JA, JX, JY, M, N, NB
INTEGER DESCA( * ), DESCX( * ), DESCY( * )
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * )
PURPOSE
PZLABRD reduces the first NB rows and columns of a complex general M-by-N distributed
matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form by an unitary
transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the
transfor- mation to the unreduced part of sub( A ).
If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to lower bidiagonal
form.
This is an auxiliary routine called by PZGEBRD.
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.
NB (global input) INTEGER
The number of leading rows and columns of sub( A ) to be reduced.
A (local input/local output) COMPLEX*16 pointer into the
local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On entry, this array
contains the local pieces of the general distributed matrix sub( A ) to be
reduced. On exit, the first NB rows and columns of the matrix are overwritten; the
rest of the distributed matrix sub( A ) is unchanged. If m >= n, elements on and
below the diagonal in the first NB columns, with the array TAUQ, represent the
unitary matrix Q as a product of elementary reflectors; and elements above the
diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P
as a product of elementary reflectors. If m < n, elements below the diagonal in
the first NB columns, with the array TAUQ, represent the unitary matrix Q as a
product of elementary reflectors, and elements on and above the diagonal in the
first NB rows, with the array TAUP, represent the unitary matrix P as a product of
elementary reflectors. See Further Details. 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.
D (local output) DOUBLE PRECISION array, dimension
LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. The distributed
diagonal elements of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). D is tied
to the distributed matrix A.
E (local output) DOUBLE PRECISION array, dimension
LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise. The distributed
off-diagonal elements of the bidiagonal distributed matrix B: if m >= n, E(i) =
A(ia+i-1,ja+i) for i = 1,2,...,n-1; if m < n, E(i) = A(ia+i,ja+i-1) for i =
1,2,...,m-1. E is tied to the distributed matrix A.
TAUQ (local output) COMPLEX*16 array dimension
LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary reflectors which
represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See
Further Details. TAUP (local output) COMPLEX*16 array, dimension
LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary reflectors which
represent the unitary matrix P. TAUP is tied to the distributed matrix A. See
Further Details. X (local output) COMPLEX*16 pointer into the local memory
to an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed
M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced part of
sub( A ).
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.
Y (local output) COMPLEX*16 pointer into the local memory
to an array of dimension (LLD_Y,NB). On exit, the local pieces of the distributed
N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced part of
sub( A ).
IY (global input) INTEGER
The row index in the global array Y indicating the first row of sub( Y ).
JY (global input) INTEGER
The column index in the global array Y indicating the first column of sub( Y ).
DESCY (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix Y.
WORK (local workspace) COMPLEX*16 array, dimension (LWORK)
LWORK >= NB_A + NQ, with
NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL )
INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can
be determined by calling the subroutine BLACS_GRIDINFO.
FURTHER DETAILS
The matrices Q and P are represented as products of elementary reflectors:
Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are complex scalars, and v and u are complex vectors.
If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).
If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
A(ia+i+1:ia+m-1,ja+i-1); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
A(ia+i-1,ja+i:ja+n-1); tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).
The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n
matrix U' which are needed, with X and Y, to apply the transformation to the unreduced
part of the matrix, using a block update of the form: sub( A ) := sub( A ) - V*Y' - X*U'.
The contents of sub( A ) on exit are illustrated by the following examples with nb = 2:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
( v1 v2 a a a ) ( v1 1 a a a a )
( v1 v2 a a a ) ( v1 v2 a a a a )
( v1 v2 a a a ) ( v1 v2 a a a a )
( v1 v2 a a a )
where a denotes an element of the original matrix which is unchanged, vi denotes an
element of the vector defining H(i), and ui an element of the vector defining G(i).