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zhptrd

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NAME
ZHPTRD - reduce a complex Hermitian matrix A stored in
packed form to real symmetric tridiagonal form T by a uni-
tary similarity transformation

SYNOPSIS
SUBROUTINE ZHPTRD( UPLO, N, AP, D, E, TAU, INFO )

CHARACTER      UPLO

INTEGER        INFO, N

DOUBLE         PRECISION D( * ), E( * )

COMPLEX*16     AP( * ), TAU( * )

PURPOSE
ZHPTRD reduces a complex Hermitian matrix A stored in packed
form to real symmetric tridiagonal form T by a unitary simi-
larity transformation: 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.

AP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermi-
tian matrix A, packed columnwise in a linear array.
The j-th column of A is stored in the array AP as
follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j)
for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2)
= A(i,j) for j<=i<=n.  On exit, if UPLO = 'U', the
diagonal and first superdiagonal of A are overwrit-
ten by the corresponding elements of the tridiagonal
matrix T, and the elements above the first superdi-
agonal, 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 reflec-
tors. See Further Details.  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).

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 AP, overwriting A(1:i-1,i+1), and tau is stored 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 AP, overwriting A(i+2:n,i), and tau is stored in TAU(i).
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