PZGEHRD - reduce a complex general distributed matrix sub( A ) to upper
Hessenberg form H by an unitary similarity transformation
- SUBROUTINE PZGEHRD(
- N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )
INTEGER IA, IHI, ILO, INFO, JA, LWORK, N INTEGER DESCA( * ) COMPLEX*16 A( * ),
TAU( * ), WORK( * )
PZGEHRD reduces a complex general distributed matrix sub( A ) to upper
Hessenberg form H by an unitary similarity transformation: Q' * sub( A ) * Q =
H, where
sub( A ) = A(IA+N-1:IA+N-1,JA+N-1:JA+N-1).
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
- N (global input) INTEGER
- The number of rows and columns to be operated on, i.e. the order of the
distributed submatrix sub( A ). N >= 0.
- ILO (global input) INTEGER
- IHI (global input) INTEGER It is assumed that sub( A ) is already upper
triangular in rows IA:IA+ILO-2 and IA+IHI:IA+N-1 and columns JA:JA+ILO-2
and JA+IHI:JA+N-1. See Further Details. If 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 general distributed matrix
sub( A ) to be reduced. On exit, the upper triangle and the first
subdiagonal of sub( A ) are overwritten with the upper Hessenberg matrix
H, and the ele- ments below the first subdiagonal, with the array TAU,
repre- sent the unitary matrix Q 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.
- TAU (local output) COMPLEX*16 array, dimension LOCc(JA+N-2)
- The scalar factors of the elementary reflectors (see Further Details).
Elements JA:JA+ILO-2 and JA+IHI:JA+N-2 of TAU are set to zero. TAU is tied
to the distributed matrix A.
- WORK (local workspace/local output) COMPLEX*16 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*NB + NB*MAX( IHIP+1, IHLP+INLQ )
where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ),
IOFF = MOD( IA+ILO-2, NB ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW
), IHIP = NUMROC( IHI+IROFFA, NB, MYROW, IAROW, NPROW ), ILROW = INDXG2P(
IA+ILO-1, NB, MYROW, RSRC_A, NPROW ), IHLP = NUMROC( IHI-ILO+IOFF+1, NB,
MYROW, ILROW, NPROW ), ILCOL = INDXG2P( JA+ILO-1, NB, MYCOL, CSRC_A, NPCOL
), INLQ = NUMROC( N-ILO+IOFF+1, NB, MYCOL, ILCOL, 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.
The matrix Q is represented as a product of (ihi-ilo) elementary reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with v(1:I) = 0, v(I+1)
= 1 and v(IHI+1:N) = 0; v(I+2:IHI) is stored on exit in
A(IA+ILO+I:IA+IHI-1,JA+ILO+I-2), and tau in TAU(JA+ILO+I-2).
The contents of A(IA:IA+N-1,JA:JA+N-1) are illustrated by the follow- ing
example, with N = 7, ILO = 2 and IHI = 6:
on entry on exit
( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a
a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h
h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a )
where a denotes an element of the original matrix sub( A ), H denotes a modified
element of the upper Hessenberg matrix H, and vi denotes an element of the
vector defining H(JA+ILO+I-2).
Alignment requirements
======================
The distributed submatrix sub( A ) must verify some alignment proper- ties,
namely the following expression should be true:
( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )