 CHBTRD(3) reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation

## SYNOPSIS

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

CHARACTER UPLO, VECT

INTEGER INFO, KD, LDAB, LDQ, N

REAL D( * ), E( * )

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

## PURPOSE

CHBTRD 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 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) REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T.
E (output) REAL 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 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 array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

## FURTHER DETAILS

Modified by Linda Kaufman, Bell Labs.