PZLATRD(l) ) PZLATRD(l)NAME
PZLATRD - reduce NB rows and columns of a complex Hermitian distributed
matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by
an unitary similarity transformation Q' * sub( A ) * Q, and returns the
matrices V and W which are needed to apply the transformation to the
unreduced part of sub( A )
SYNOPSIS
SUBROUTINE PZLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW,
JW, DESCW, WORK )
CHARACTER UPLO
INTEGER IA, IW, JA, JW, N, NB
INTEGER DESCA( * ), DESCW( * )
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( * ), TAU( * ), W( * ), WORK( * )
PURPOSE
PZLATRD reduces NB rows and columns of a complex Hermitian distributed
matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by
an unitary similarity transformation Q' * sub( A ) * Q, and returns the
matrices V and W which are needed to apply the transformation to the
unreduced part of sub( A ). If UPLO = 'U', PZLATRD reduces the last NB
rows and columns of a matrix, of which the upper triangle is supplied;
if UPLO = 'L', PZLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by PZHETRD.
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
UPLO (global input) CHARACTER
Specifies whether the upper or lower triangular part of the
Hermitian matrix sub( A ) is stored:
= 'U': Upper triangular
= 'L': Lower triangular
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.
NB (global input) INTEGER
The number of rows and columns to be reduced.
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 Hermitian
distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N
upper triangular part of sub( A ) contains the upper triangular
part of the matrix, and its strictly lower triangular part is
not referenced. If UPLO = 'L', the leading N-by-N lower trian‐
gular part of sub( A ) contains the lower triangular part of
the matrix, and its strictly upper triangular part is not ref‐
erenced. On exit, if UPLO = 'U', the last NB columns have been
reduced to tridiagonal form, with the diagonal elements over‐
writing the diagonal elements of sub( A ); the elements above
the diagonal with the array TAU, represent the unitary matrix Q
as a product of elementary reflectors. If UPLO = 'L', the first
NB columns have been reduced to tridiagonal form, with the
diagonal elements overwriting the diagonal elements of sub( A
); the elements below the diagonal with the array TAU, repre‐
sent the unitary matrix Q as a product of elementary reflec‐
tors; 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.
D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1)
The diagonal elements of the tridiagonal matrix T: D(i) =
A(i,i). D is tied to the distributed matrix A.
E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1)
if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal ele‐
ments of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO =
'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the distrib‐
uted matrix A.
TAU (local output) COMPLEX*16, array, dimension
LOCc(JA+N-1). This array contains the scalar factors TAU of the
elementary reflectors. TAU is tied to the distributed matrix A.
W (local output) COMPLEX*16 pointer into the local memory
to an array of dimension (LLD_W,NB_W), This array contains the
local pieces of the N-by-NB_W matrix W required to update the
unreduced part of sub( A ).
IW (global input) INTEGER
The row index in the global array W indicating the first row of
sub( W ).
JW (global input) INTEGER
The column index in the global array W indicating the first
column of sub( W ).
DESCW (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix W.
WORK (local workspace) COMPLEX*16 array, dimension (NB_A)
FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n) H(n-1) . . . H(n-nb+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(i:n) =
0 and v(i-1) = 1; v(1:i-1) is stored on exit in
A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1)H(2) . . . H(nb).
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 and v(i+1) = 1; v(i+2:n) is stored on exit in
A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).
The elements of the vectors v together form the N-by-NB matrix V which
is needed, with W, to apply the transformation to the unreduced part of
the matrix, using a Hermitian rank-2k update of the form: sub( A ) :=
sub( A ) - V*W' - W*V'.
The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:
if UPLO = 'U': if UPLO = 'L':
( a a a v4 v5 ) ( d )
( a a v4 v5 ) ( 1 d )
( a 1 v5 ) ( v1 1 a )
( d 1 ) ( v1 v2 a a )
( d ) ( v1 v2 a a a )
where d denotes a diagonal element of the reduced matrix, a denotes an
element of the original matrix that is unchanged, and vi denotes an
element of the vector defining H(i).
ScaLAPACK version 1.7 13 August 2001 PZLATRD(l)
