PZHENGST(3) ScaLAPACK routine of NEC Numeric Library Collection PZHENGST(3)
NAME
PZHEGST - reduce a complex Hermitian-definite generalized eigenproblem
to standard form
SYNOPSIS
SUBROUTINE PZHENGST(
IBTYPE, UPLO, N, A, IA, JA, DESCA, B, IB, JB,
DESCB, SCALE, WORK, LWORK, INFO )
CHARACTER UPLO
INTEGER IA, IB, IBTYPE, INFO, JA, JB, LWORK, N
DOUBLE PRECISION SCALE
INTEGER DESCA( * ), DESCB( * )
COMPLEX*16 A( * ), B( * ), WORK( * )
PURPOSE
PZHENGST reduces a complex Hermitian-definite generalized eigenproblem
to standard form.
PZHENGST performs the same function as PZHEGST, but is based on rank 2K
updates, which are faster and more scalable than triangular solves (the
basis of PZHENGST).
PZHENGST calls PZHEGST when UPLO='U', hence ZDHENGST provides improved
performance only when UPLO='L', IBTYPE=1.
PZHENGST also calls PZHEGST when insufficient workspace is provided,
hence PZHENGST provides improved performance only when LWORK >= 2 * NP0
* NB + NQ0 * NB + NB * NB
In the following sub( A ) denotes A( IA:IA+N-1, JA:JA+N-1 ) and sub( B
) denotes B( IB:IB+N-1, JB:JB+N-1 ).
If IBTYPE = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub(
A ) is overwritten by inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A
)*inv(L**H)
If IBTYPE = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or
sub( B )*sub( A )*x = lambda*x, and sub( A ) is overwritten by U*sub( A
)*U**H or L**H*sub( A )*L.
sub( B ) must have been previously factorized as U**H*U or L*L**H by
PZPOTRF.
Notes
=====
Each global data object is described by an associated description vec-
tor. This vector stores the information required to establish the map-
ping 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 col-
umn.
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
IBTYPE (global input) INTEGER
= 1: compute inv(U**H)*sub( A )*inv(U) or inv(L)*sub( A
)*inv(L**H);
= 2 or 3: compute U*sub( A )*U**H or L**H*sub( A )*L.
UPLO (global input) CHARACTER
= 'U': Upper triangle of sub( A ) is stored and sub( B ) is
factored as U**H*U;
= 'L': Lower triangle of sub( A ) is stored and sub( B ) is
factored as L*L**H.
N (global input) INTEGER
The order of the matrices sub( A ) and sub( B ). 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 N-by-N
Hermitian distributed matrix sub( A ). If UPLO = 'U', the lead-
ing N-by-N upper triangular part of sub( A ) contains the upper
triangular part of the matrix, and its strictly lower triangu-
lar part is not referenced. If UPLO = 'L', the leading N-by-N
lower triangular part of sub( A ) contains the lower triangular
part of the matrix, and its strictly upper triangular part is
not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as sub( A ).
IA (global input) INTEGER
A's global row index, which points to the beginning of the sub-
matrix which is to be operated on.
JA (global input) INTEGER
A's global column index, which points to the beginning of the
submatrix which is to be operated on.
DESCA (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix A.
B (local input) COMPLEX*16 pointer into the local memory
to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this
array contains the local pieces of the triangular factor from
the Cholesky factorization of sub( B ), as returned by PZPOTRF.
IB (global input) INTEGER
B's global row index, which points to the beginning of the sub-
matrix which is to be operated on.
JB (global input) INTEGER
B's global column index, which points to the beginning of the
submatrix which is to be operated on.
DESCB (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix B.
SCALE (global output) DOUBLE PRECISION
Amount by which the eigenvalues should be scaled to compensate
for the scaling performed in this routine. At present, SCALE
is always returned as 1.0, it is returned here to allow for
future enhancement.
WORK (local workspace/local output) COLPLEX*16 array,
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 >= MAX( NB * ( NP0 +1 ), 3 * NB )
When IBTYPE = 1 and UPLO = 'L', PZHENGST provides improved per-
formance when LWORK >= 2 * NP0 * NB + NQ0 * NB + NB * NB
where NB = MB_A = NB_A, NP0 = NUMROC( N, NB, 0, 0, NPROW ), NQ0
= NUMROC( N, NB, 0, 0, NPROW ),
NUMROC ia a ScaLAPACK tool functions MYROW, MYCOL, NPROW and
NPCOL can be determined by calling the subroutine BLACS_GRID-
INFO.
If LWORK = -1, then LWORK is global input and a workspace query
is assumed; the routine only calculates the 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.
ScaLAPACK routine 31 October 2017 PZHENGST(3)