NAME
PDLATRD - reduce NB rows and columns of a real symmetric distributed
matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form
by an orthogonal similarity transformation Q’ * sub( A ) * Q,
SYNOPSIS
SUBROUTINE PDLATRD( 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 A( * ), D( * ), E( * ), TAU( * ), W( * ),
WORK( * )
PURPOSE
PDLATRD reduces NB rows and columns of a real symmetric distributed
matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form
by an orthogonal 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’, PDLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = ’L’, PDLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by PDSYTRD.
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
UPLO (global input) CHARACTER
Specifies whether the upper or lower triangular part of the
symmetric 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) DOUBLE PRECISION 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 symmetric
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
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 UPLO = ’U’, the last NB columns have
been reduced to tridiagonal form, with the diagonal elements
overwriting the diagonal elements of sub( A ); the elements
above the diagonal with the array TAU, represent the orthogonal
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,
represent 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.
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
elements 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
distributed matrix A.
TAU (local output) DOUBLE PRECISION, 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 real scalar, and v is a real 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 real scalar, and v is a real 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 symmetric 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).