Provided by: scalapack-doc_1.5-10_all 
NAME
PDSYTRD - reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an
orthogonal similarity transformation
SYNOPSIS
SUBROUTINE PDSYTRD( 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
PDSYTRD 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
>= MAX( NB * ( NP +1 ), 3 * NB )
where NB = MB_A = NB_A, NP = NUMROC( N, NB, MYROW, IAROW, NPROW ), IAROW =
INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ).
INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can
be determined by calling the subroutine BLACS_GRIDINFO.
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 (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 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 ).