Provided by: liblapack-doc_3.3.1-1_all

#### NAME

```       LAPACK-3  - reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T
by a unitary similarity transformation

```

#### SYNOPSIS

```       SUBROUTINE ZHBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO )

CHARACTER      UPLO, VECT

INTEGER        INFO, KD, LDAB, LDQ, N

DOUBLE         PRECISION D( * ), E( * )

COMPLEX*16     AB( LDAB, * ), Q( LDQ, * ), WORK( * )

```

#### PURPOSE

```       ZHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a
unitary similarity transformation:
Q**H * A * Q = T.

```

#### ARGUMENTS

```        VECT    (input) CHARACTER*1
= 'N':  do not form Q;
= 'V':  form Q;
= 'U':  update a matrix X, by forming X*Q.

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.

KD      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band
matrix A, stored in the first KD+1 rows of the array.  The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
On exit, the diagonal elements of AB are overwritten by the
diagonal elements of the tridiagonal matrix T; if KD > 0, the
elements on the first superdiagonal (if UPLO = 'U') or the
first subdiagonal (if UPLO = 'L') are overwritten by the
off-diagonal elements of T; the rest of AB is overwritten by
values generated during the reduction.

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >= KD+1.

D       (output) DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T.

E       (output) DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

Q       (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if VECT = 'U', then Q must contain an N-by-N
matrix X; if VECT = 'N' or 'V', then Q need not be set.
On exit:
if VECT = 'V', Q contains the N-by-N unitary matrix Q;
if VECT = 'U', Q contains the product X*Q;
if VECT = 'N', the array Q is not referenced.

LDQ     (input) INTEGER
The leading dimension of the array Q.
LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.

WORK    (workspace) COMPLEX*16 array, dimension (N)

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value

```

#### FURTHERDETAILS

```        Modified by Linda Kaufman, Bell Labs.

LAPACK routine (version 3.2)               April 2011                            ZHBTRD(3lapack)
```