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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: reduces a complex Hermitian matrix A stored in packed form to real symmetric tridiagonal form T by a unitary similarity

```NAME
LAPACK-3  -  reduces a complex Hermitian matrix A stored in packed form
to  real  symmetric  tridiagonal  form  T  by  a   unitary   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   similarity
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 Hermitian 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 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.
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).

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
```

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