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NAME
ZHETRD - reduce a complex Hermitian matrix A to real sym-
metric tridiagonal form T by a unitary similarity transfor-
mation
SYNOPSIS
SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK,
INFO )
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
ZHETRD reduces a complex Hermitian matrix A to real sym-
metric tridiagonal form T by a unitary similarity transfor-
mation: Q**H * 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) COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian 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 triangular
part of the matrix A, and the strictly upper tri-
angular part of A is not referenced. On exit, if
UPLO = 'U', the diagonal and first superdiagonal of
A are overwritten by the corresponding elements of
the tridiagonal matrix T, and the elements above the
first superdiagonal, with the array TAU, represent
the unitary 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 unitary 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 (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) COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see
Further Details).
WORK (workspace) COMPLEX*16 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
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 complex scalar, and v is a complex 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 complex scalar, and v is a complex 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 off-diagonal elements of
T, and vi denotes an element of the vector defining H(i).