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Man Pages


Manual Reference Pages  -  PDSYTD2 (l)

NAME

PDSYTD2 - reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation

CONTENTS

Synopsis
Purpose
Arguments
Further Details

SYNOPSIS

SUBROUTINE PDSYTD2( 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

PDSYTD2 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).

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.
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 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) 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.
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 >= 3*N.

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

INFO (local 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 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 ) with
IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ).

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ScaLAPACK version 1.7 PDSYTD2 (l) 13 August 2001

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