DSBTRD(3)                  MathKeisan LAPACK routine                 DSBTRD(3)



NAME
       DSBTRD - a real symmetric band matrix A to symmetric tridiagonal form T
       by an orthogonal similarity transformation

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 transformation: 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  symmetric  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 overwrit-
               ten 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) 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 illegal value

FURTHER DETAILS
       Modified by Linda Kaufman, Bell Labs.




 LAPACK routine (version 3.1)    November 2006                       DSBTRD(3)