PSSYNTRD - i a prototype version of PSSYTRD which uses tailored codes (either
the serial, SSYTRD, or the parallel code, PSSYTTRD) when the workspace
provided by the user is adequate
- SUBROUTINE PSSYNTRD(
- UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO )
CHARACTER UPLO INTEGER IA, INFO, JA, LWORK, N INTEGER DESCA( * ) REAL A( * ), D(
* ), E( * ), TAU( * ), WORK( * )
PSSYNTRD is a prototype version of PSSYTRD which uses tailored codes (either the
serial, SSYTRD, or the parallel code, PSSYTTRD) when the workspace provided by
the user is adequate.
PSSYNTRD reduces a real symmetric matrix sub( A ) to symmetric tridiagonal 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
========
PSSYNTRD is faster than PSSYTRD 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 PSSYTRD to keep the
interface simple. These restrictions are documented below. (Search for
"restrictions".)
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
- 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) 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 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 diagonal and first superdiagonal 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) REAL 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) REAL 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) REAL, 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) REAL array,
- dimension (LWORK) On exit, WORK( 1 ) returns the 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, 'PSSYTTRD', 'L', 0, 0, 0, 0 ) SQNPC = INT( SQRT( REAL(
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.
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 examples 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 ).