CHBTRD  - reduce 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  Her­
               mitian  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 ele­
               ments  of  AB are overwritten by the diagonal ele­
               ments 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

               ing 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 uni­
               tary matrix Q; if VECT = 'U', Q contains the prod­
               uct  X*Q; if VECT = 'N', the array Q is not refer­
               enced.

       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 ille­
               gal value


FURTHER DETAILS

       Modified by Linda Kaufman, Bell Labs.


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