PDLACONSB - look for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and see if this would make a subdiagonal negligible

SUBROUTINE PDLACONSB(

A, DESCA, I, L, M, H44, H33, H43H34, BUF, LWORK )

INTEGER I, L, LWORK, M

DOUBLE PRECISION H33, H43H34, H44

INTEGER DESCA( * )

DOUBLE PRECISION A( * ), BUF( * )

PDLACONSB looks for two consecutive small subdiagonal elements by seeing the effect of starting a double shift QR iteration given by H44, H33, & H43H34 and see if this would make a subdiagonal negligible.

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

A (global input) DOUBLE PRECISION array, dimension

(DESCA(LLD_),*) On entry, the Hessenberg matrix whose tridiagonal part is being scanned. Unchanged on exit.

DESCA (global and local input) INTEGER array of dimension DLEN_.

The array descriptor for the distributed matrix A.

I (global input) INTEGER

The global location of the bottom of the unreduced submatrix of A. Unchanged on exit.

L (global input) INTEGER

The global location of the top of the unreduced submatrix of A. Unchanged on exit.

M (global output) INTEGER

On exit, this yields the starting location of the QR double shift. This will satisfy: L <= M <= I-2.

H44 H33 H43H34 (global input) DOUBLE PRECISION These three values are for the double shift QR iteration.

BUF (local output) DOUBLE PRECISION array of size LWORK.

LWORK (global input) INTEGER

On exit, LWORK is the size of the work buffer. This must be at least 7*Ceil( Ceil( (I-L)/HBL ) / LCM(NPROW,NPCOL) ) Here LCM is least common multiple, and NPROWxNPCOL is the logical grid size.

Logic: ======

Two consecutive small subdiagonal elements will stall convergence of a double shift if their product is small relatively even if each is not very small. Thus it is necessary to scan the "tridiagonal portion of the matrix." In the LAPACK algorithm DLAHQR, a loop of M goes from I-2 down to L and examines H(m,m),H(m+1,m+1),H(m+1,m),H(m,m+1),H(m-1,m-1),H(m,m-1), and H(m+2,m-1). Since these elements may be on separate processors, the first major loop (10) goes over the tridiagonal and has each node store whatever values of the 7 it has that the node owning H(m,m) does not. This will occur on a border and can happen in no more than 3 locations per block assuming square blocks. There are 5 buffers that each node stores these values: a buffer to send diagonally down and right, a buffer to send up, a buffer to send left, a buffer to send diagonally up and left and a buffer to send right. Each of these buffers is actually stored in one buffer BUF where BUF(ISTR1+1) starts the first buffer, BUF(ISTR2+1) starts the second, etc.. After the values are stored, if there are any values that a node needs, they will be sent and received. Then the next major loop passes over the data and searches for two consecutive small subdiagonals.

Notes:

This routine does a global maximum and must be called by all processes.

Implemented by: G. Henry, November 17, 1996