DSBTRD  - reduce a real symmetric band matrix A to symmet­
       ric tridiagonal form T by an orthogonal similarity  trans­
       formation


SYNOPSIS

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

           CHARACTER      UPLO, VECT

           INTEGER        INFO, KD, LDAB, LDQ, N

           DOUBLE         PRECISION AB( LDAB, * ), D( * ),  E(  *
                          ), Q( LDQ, * ), WORK( * )


PURPOSE

       DSBTRD reduces a real symmetric band matrix A to symmetric
       tridiagonal form T by an orthogonal similarity transforma­
       tion: Q**T * 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) DOUBLE PRECISION array, dimension
               (LDAB,N)
               On  entry, the upper or lower triangle of the sym­
               metric 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) 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) DOUBLE PRECISION 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
               orthogonal 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) DOUBLE PRECISION 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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