NAME
PDGEBRD - reduce a real 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
orthogonal transformation
SYNOPSIS
SUBROUTINE PDGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK,
LWORK, INFO )
INTEGER IA, INFO, JA, LWORK, M, N
INTEGER DESCA( * )
DOUBLE PRECISION A( * ), D( * ), E( * ), TAUP( * ), TAUQ(
* ), WORK( * )
PURPOSE
PDGEBRD reduces a real 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
orthogonal 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) DOUBLE PRECISION 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 orthogonal 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
orthogonal 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) DOUBLE PRECISION array dimension
LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary
reflectors which represent the orthogonal matrix Q. TAUQ is
tied to the distributed matrix A. See Further Details. TAUP
(local output) DOUBLE PRECISION array, dimension
LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary
reflectors which represent the orthogonal matrix P. TAUP is
tied to the distributed matrix A. See Further Details. WORK
(local workspace/local output) DOUBLE PRECISION 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 real scalars, and v and u are real 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 real scalars, and v and u are real 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 )