Provided by: scalapack-doc_1.5-11_all 
NAME
PZGEBRD - reduce 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 B by an unitary transformation
SYNOPSIS
SUBROUTINE PZGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )
INTEGER IA, INFO, JA, LWORK, M, N
INTEGER DESCA( * )
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( * ), TAUP( * ), TAUQ( * ), WORK( * )
PURPOSE
PZGEBRD reduces 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 B by an unitary transformation:
Q' * sub( A ) * P = B.
If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal.
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.
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 ). On exit, if
M >= N, the diagonal and the first superdiagonal of sub( A ) are overwritten with
the upper bidiagonal matrix B; the elements below the diagonal, with the array
TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and
the elements above the first superdiagonal, with the array TAUP, represent the
orthogonal matrix P as a product of elementary reflectors. If M < N, the diagonal
and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ, represent the unitary
matrix Q as a product of elementary reflectors, and the elements above the
diagonal, with the array TAUP, represent the orthogonal 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
LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise. The distributed
diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). 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(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) 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. WORK (local workspace/local output) COMPLEX*16 array,
dimension (LWORK) On exit, WORK( 1 ) returns the minimal and optimal LWORK.
LWORK (local or global input) INTEGER
The dimension of the array WORK. LWORK is local input and must be at least LWORK
>= NB*( MpA0 + NqA0 + 1 ) + NqA0
where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ), IAROW
= INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A,
NPCOL ), MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ), NqA0 = NUMROC(
N+ICOFFA, NB, MYCOL, IACOL, NPCOL ).
INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can
be determined by calling the subroutine BLACS_GRIDINFO.
If LWORK = -1, then LWORK is global input and a workspace query is assumed; the
routine only calculates the minimum and optimal size for all work arrays. Each of
these values is returned in the first entry of the corresponding work array, and
no error message is issued by PXERBLA.
INFO (global output) INTEGER
= 0: successful exit
< 0: If the i-th argument is an array and the j-entry had an illegal value, then
INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then
INFO = -i.
FURTHER DETAILS
The matrices Q and P are represented as products of elementary reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
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;
v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1);
u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1);
tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
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;
v(1:i) = 0, v(i+1) = 1, and v(i+2: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+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).
The contents of sub( A ) on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the
vector defining H(i), and ui an element of the vector defining G(i).
Alignment requirements
======================
The distributed submatrix sub( A ) must verify some alignment proper- ties, namely the
following expressions should be true:
( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )