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# zhetrd

```
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).
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