DSYTD2 - reduce a real symmetric matrix A to symmetric
tridiagonal form T by an orthogonal similarity transforma
tion
SYNOPSIS
SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
CHARACTER UPLO
INTEGER INFO, LDA, N
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ),
TAU( * )
PURPOSE
DSYTD2 reduces a real symmetric matrix A to symmetric
tridiagonal form T by an orthogonal similarity transforma
tion: Q' * A * Q = T.
ARGUMENTS
UPLO (input) CHARACTER*1
Specifies whether the upper or lower triangular
part of the symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
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 n-by-n upper triangular part of A con
tains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced. If UPLO = 'L', the leading n-by-n
lower triangular part of A contains the lower tri
angular 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 super
diagonal of A are overwritten by the corresponding
elements of the tridiagonal matrix T, and the ele
ments 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).
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E (output) DOUBLE PRECISION array, dimension (N-1)
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'.
TAU (output) DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors
(see Further Details).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value.
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(1:i-1,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(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(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 ) ( 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).
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