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
- 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( * )
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
- 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.
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 )