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PCLABRD

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 …

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( * ) REAL D( * ), E( * ) COMPLEX A( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * )

PCLABRD 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 PCGEBRD.

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

The number of rows to be operated on, i.e. the number of rows of the distributed submatrix sub( A ). M >= 0.
The number of columns to be operated on, i.e. the number of columns of the distributed submatrix sub( A ). N >= 0.
The number of leading rows and columns of sub( A ) to be reduced.
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 ).
The column index in the global array A indicating the first column of sub( A ).
The array descriptor for the distributed matrix A.
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.
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.
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 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 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 ).
The row index in the global array X indicating the first row of sub( X ).
The column index in the global array X indicating the first column of sub( X ).
The array descriptor for the distributed matrix X.
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 ).
The row index in the global array Y indicating the first row of sub( Y ).
The column index in the global array Y indicating the first column of sub( Y ).
The array descriptor for the distributed matrix Y.
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.

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).