
DSYTRD(l) ) DSYTRD(l)
NAME
DSYTRD  reduce a real symmetric matrix A to real symmetric tridiagonal form T by an
orthogonal similarity transformation
SYNOPSIS
SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), WORK( * )
PURPOSE
DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an
orthogonal similarity transformation: Q**T * A * Q = T.
ARGUMENTS
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading NbyN upper trian
gular part of A contains the upper triangular part of the matrix A, and the
strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading
NbyN lower triangular part of A contains the lower triangular part of the matrix
A, and the strictly upper triangular part of A is not referenced. On exit, if
UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the cor
responding 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 A
are over written 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.
LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N).
D (output) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
E (output) DOUBLE PRECISION array, dimension (N1)
The offdiagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO =
'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU (output) DOUBLE PRECISION array, dimension (N1)
The scalar factors of the elementary reflectors (see Further Details).
WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >=
N*NB, where NB is the optimal blocksize.
If LWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors
Q = H(n1) . . . 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:i1) is stored on exit in
A(1:i1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(n1).
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(i+2:n,i), and tau in TAU(i).
The contents of 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 offdiagonal elements of T, and vi denotes an element of
the vector defining H(i).
LAPACK version 3.0 15 June 2000 DSYTRD(l) 
