DSBEVD  -  compute  all  the  eigenvalues and, optionally,
       eigenvectors of a real symmetric band matrix A


SYNOPSIS

       SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
                          WORK, LWORK, IWORK, LIWORK, INFO )

           CHARACTER      JOBZ, UPLO

           INTEGER        INFO, KD, LDAB, LDZ, LIWORK, LWORK, N

           INTEGER        IWORK( * )

           DOUBLE         PRECISION  AB( LDAB, * ), W( * ), WORK(
                          * ), Z( LDZ, * )


PURPOSE

       DSBEVD  computes  all  the  eigenvalues  and,  optionally,
       eigenvectors  of a real symmetric band matrix A. If eigen­
       vectors are desired, it uses a divide  and  conquer  algo­
       rithm.

       The  divide  and conquer algorithm makes very mild assump­
       tions about floating point arithmetic.  It  will  work  on
       machines  with  a guard digit in add/subtract, or on those
       binary machines without guard digits which  subtract  like
       the  Cray  X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
       conceivably fail on hexadecimal or decimal machines  with­
       out guard digits, but we know of none.


ARGUMENTS

       JOBZ    (input) CHARACTER*1
               = 'N':  Compute eigenvalues only;
               = 'V':  Compute eigenvalues and eigenvectors.

       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

               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,  AB  is  overwritten by values generated
               during the reduction to tridiagonal form.  If UPLO
               = 'U', the first superdiagonal and the diagonal of
               the tridiagonal matrix T are returned in  rows  KD
               and  KD+1  of  AB, and if UPLO = 'L', the diagonal
               and first subdiagonal of T  are  returned  in  the
               first two rows of AB.

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

       W       (output) DOUBLE PRECISION array, dimension (N)
               If INFO = 0, the eigenvalues in ascending order.

       Z       (output) DOUBLE PRECISION array, dimension (LDZ,
               N)
               If  JOBZ  =  'V', then if INFO = 0, Z contains the
               orthonormal eigenvectors of the matrix A, with the
               i-th  column  of Z holding the eigenvector associ­
               ated with W(i).  If JOBZ = 'N', then Z is not ref­
               erenced.

       LDZ     (input) INTEGER
               The  leading  dimension of the array Z.  LDZ >= 1,
               and if JOBZ = 'V', LDZ >= max(1,N).

       WORK    (workspace/output) DOUBLE PRECISION array,
               dimension (LWORK) On exit, if INFO  =  0,  WORK(1)
               returns the optimal LWORK.

       LWORK   (input) INTEGER
               The  dimension  of  the  array  WORK.   IF N <= 1,
               LWORK must be at least 1.  If JOBZ  = 'N' and N  >
               2, LWORK must be at least 2*N.  If JOBZ  = 'V' and
               N > 2, LWORK must be at least ( 1 + 5*N  +  2*N**2
               ).

               If  LWORK = -1, then a workspace query is assumed;
               the routine only calculates the  optimal  size  of
               the  WORK  array,  returns this value as the first
               entry of the WORK  array,  and  no  error  message
               related to LWORK is issued by XERBLA.

       IWORK   (workspace/output) INTEGER array, dimension
               (LIWORK)
               On exit, if INFO = 0, IWORK(1) returns the optimal
               LIWORK.

               The dimension of the array LIWORK.  If JOBZ  = 'N'
               or N <= 1, LIWORK must be at least 1.  If JOBZ   =
               'V' and N > 2, LIWORK must be at least 3 + 5*N.

               If LIWORK = -1, then a workspace query is assumed;
               the routine only calculates the  optimal  size  of
               the  IWORK  array, returns this value as the first
               entry of the IWORK array,  and  no  error  message
               related to LIWORK is issued by XERBLA.

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value
               > 0:  if INFO = i, the algorithm  failed  to  con­
               verge;  i off-diagonal elements of an intermediate
               tridiagonal form did not converge to zero.


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