PDLATRD(3) ScaLAPACK routine of NEC Numeric Library Collection PDLATRD(3)
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 transforma-
tion 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 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
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 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 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) INTE-
GER 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 dis-
tributed 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).
ScaLAPACK routine 31 October 2017 PDLATRD(3)