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# NAME

PCTREVC - compute some or all of the right and/or left eigenvectors of a complex upper triangular matrix T in parallel

# SYNOPSIS

SUBROUTINE PCTREVC(
SIDE, HOWMNY, SELECT, N, T, DESCT, VL, DESCVL, VR, DESCVR, MM, M, WORK, RWORK, INFO )
CHARACTER HOWMNY, SIDE INTEGER INFO, M, MM, N LOGICAL SELECT( * ) INTEGER DESCT( * ), DESCVL( * ), DESCVR( * ) REAL RWORK( * ) COMPLEX T( * ), VL( * ), VR( * ), WORK( * )

# PURPOSE

PCTREVC computes some or all of the right and/or left eigenvectors of a complex upper triangular matrix T in parallel. The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:

T*x = w*x, y'*T = w*y'

where y' denotes the conjugate transpose of the vector y.
If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input unitary

matrix. If T was obtained from the Schur factorization of an original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or left eigenvectors of A.

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 r x c.

LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the r 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 c 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

SIDE (global input) CHARACTER*1
= 'R': compute right eigenvectors only;

= 'L': compute left eigenvectors only;

= 'B': compute both right and left eigenvectors.
HOWMNY (global input) CHARACTER*1

= 'A': compute all right and/or left eigenvectors;

= 'B': compute all right and/or left eigenvectors, and backtransform them using the input matrices supplied in VR and/or VL; = 'S': compute selected right and/or left eigenvectors, specified by the logical array SELECT.
SELECT (global input) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be computed. If HOWMNY = 'A' or 'B', SELECT is not referenced. To select the eigenvector corresponding to the j-th eigenvalue, SELECT(j) must be set to .TRUE..
N (global input) INTEGER
The order of the matrix T. N >= 0.
T (global input/output) COMPLEX array, dimension
(DESCT(LLD_),*) The upper triangular matrix T. T is modified, but restored on exit.
DESCT (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix T.
VL (global input/output) COMPLEX array, dimension
(DESCVL(LLD_),MM) On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by CHSEQR). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues. If SIDE = 'R', VL is not referenced.
DESCVL (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix VL.
VR (global input/output) COMPLEX array, dimension
(DESCVR(LLD_),MM) On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by CHSEQR). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S', the right eigenvectors of T specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. If SIDE = 'L', VR is not referenced.
DESCVR (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix VR.
MM (global input) INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M (global output) INTEGER
The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected eigenvector occupies one column.
WORK (local workspace) COMPLEX array,
dimension ( 2*DESCT(LLD_) ) Additional workspace may be required if PCLATTRS is updated to use WORK.
RWORK (local workspace) REAL array,
dimension ( DESCT(LLD_) )
INFO (global output) INTEGER
= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

# FURTHER DETAILS

The algorithm used in this program is basically backward (forward) substitution. It is the hope that scaling would be used to make the the code robust against possible overflow. But scaling has not yet been implemented in PCLATTRS which is called by this routine to solve the triangular systems. PCLATTRS just calls PCTRSV.

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

# FURTHER DETAILS

Implemented by Mark R. Fahey, June, 2000

 13 August 2001 ScaLAPACK version 1.7

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