NAME
PSGEHRD - reduce a real general distributed matrix sub( A ) to upper
Hessenberg form H by an orthogonal similarity transforma- tion
SYNOPSIS
SUBROUTINE PSGEHRD( N, ILO, IHI, A, IA, JA, DESCA, TAU, WORK, LWORK,
INFO )
INTEGER IA, IHI, ILO, INFO, JA, LWORK, N
INTEGER DESCA( * )
REAL A( * ), TAU( * ), WORK( * )
PURPOSE
PSGEHRD reduces a real general distributed matrix sub( A ) to upper
Hessenberg form H by an orthogonal similarity transforma- tion: 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
ARGUMENTS
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) REAL 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 orthogonal 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) REAL 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) REAL 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.
FURTHER DETAILS
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 real scalar, and v is a real 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 )