PDSYNTRD(3) ScaLAPACK routine of NEC Numeric Library Collection PDSYNTRD(3)
NAME
PDSYNTRD - reduce a real symmetric matrix sub( A ) to symmetric tridi-
agonal form T by an orthogonal similarity transformation
SYNOPSIS
SUBROUTINE PDSYNTRD(
UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK,
INFO )
CHARACTER UPLO
INTEGER IA, INFO, JA, LWORK, N
INTEGER DESCA( * )
DOUBLE PRECISION A( * ), D( * ), E( * ), TAU( * ), WORK( *
)
PURPOSE
PDSYNTRD is a prototype version of PDSYTRD which uses tailored codes
(either the serial, DSYTRD, or the parallel code, PDSYTTRD) when the
workspace provided by the user is adequate.
PDSYNTRD reduces a real symmetric matrix sub( A ) to symmetric tridiag-
onal form T by an orthogonal similarity transformation: Q' * sub( A ) *
Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1).
Features
========
PDSYNTRD is faster than PDSYTRD on almost all matrices, particularly
small ones (i.e. N < 500 * sqrt(P) ), provided that enough workspace is
available to use the tailored codes.
The tailored codes provide performance that is essentially independent
of the input data layout.
The tailored codes place no restrictions on IA, JA, MB or NB. At
present, IA, JA, MB and NB are restricted to those values allowed by
PDSYTRD to keep the interface simple. These restrictions are docu-
mented below. (Search for "restrictions".)
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.
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 diagonal and first super-
diagonal of sub( A ) are over- written by the corresponding
elements of the tridiagonal matrix T, and the elements above
the first superdiagonal, with the array TAU, represent the
orthogonal matrix Q as a product of elementary reflectors; if
UPLO = 'L', the diagonal and first subdiagonal of sub( A ) are
overwritten by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, 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 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.
WORK (local workspace/local output) DOUBLE PRECISION 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 >= MAX( NB * ( NP +1 ), 3 * NB )
For optimal performance, greater workspace is needed, i.e.
LWORK >= 2*( ANB+1 )*( 4*NPS+2 ) + ( NPS + 4 ) * NPS
ICTXT = DESCA( CTXT_ )
ANB = PJLAENV( ICTXT, 3, 'PDSYTTRD', 'L', 0, 0, 0, 0 )
SQNPC = INT( SQRT( DBLE( NPROW * NPCOL ) ) )
NPS = MAX( NUMROC( N, 1, 0, 0, SQNPC ), 2*ANB )
NUMROC is a ScaLAPACK tool functions;
PJLAENV is a ScaLAPACK envionmental inquiry function
MYROW, MYCOL, NPROW and NPCOL can be determined by calling
the subroutine BLACS_GRIDINFO.
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
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 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(n-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 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 contents of sub( A ) on exit are illustrated by the following exam-
ples with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
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 .AND. IROFFA.EQ.0 ) with IROFFA =
MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ).
ScaLAPACK routine 31 October 2017 PDSYNTRD(3)