DSBGVD(3)      LAPACK routine of NEC Numeric Library Collection      DSBGVD(3)



NAME
       DSBGVD

SYNOPSIS
       SUBROUTINE DSBGVD (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z,
           LDZ, WORK, LWORK, IWORK, LIWORK, INFO)



PURPOSE
            DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
            of a real generalized symmetric-definite banded eigenproblem, of the
            form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and
            banded, and B is also positive definite.  If eigenvectors are
            desired, it uses a divide and conquer algorithm.

            The divide and conquer algorithm makes very mild assumptions 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
            without guard digits, but we know of none.




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

           UPLO      (input)
                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N         (input)
                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           KA        (input)
                     KA is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

           KB        (input)
                     KB is INTEGER
                     The number of superdiagonals of the matrix B if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

           AB        (input/output)
                     AB is DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

                     On exit, the contents of AB are destroyed.

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

           BB        (input/output)
                     BB is DOUBLE PRECISION array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix B, stored in the first kb+1 rows of the array.  The
                     j-th column of B is stored in the j-th column of the array BB
                     as follows:
                     if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
                     if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

                     On exit, the factor S from the split Cholesky factorization
                     B = S**T*S, as returned by DPBSTF.

           LDBB      (input)
                     LDBB is INTEGER
                     The leading dimension of the array BB.  LDBB >= KB+1.

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

           Z         (output)
                     Z is DOUBLE PRECISION array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                     eigenvectors, with the i-th column of Z holding the
                     eigenvector associated with W(i).  The eigenvectors are
                     normalized so Z**T*B*Z = I.
                     If JOBZ = 'N', then Z is not referenced.

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

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

           LWORK     (input)
                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If N <= 1,               LWORK >= 1.
                     If JOBZ = 'N' and N > 1, LWORK >= 3*N.
                     If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK and IWORK
                     arrays, returns these values as the first entries of the WORK
                     and IWORK arrays, and no error message related to LWORK or
                     LIWORK is issued by XERBLA.

           IWORK     (output)
                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.

           LIWORK    (input)
                     LIWORK is INTEGER
                     The dimension of the array IWORK.
                     If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
                     If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK and IWORK arrays, and no error message related to
                     LWORK or LIWORK is issued by XERBLA.

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, and i is:
                        <= N:  the algorithm failed to converge:
                               i off-diagonal elements of an intermediate
                               tridiagonal form did not converge to zero;
                        > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
                               returned INFO = i: B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.



LAPACK routine                  31 October 2017                      DSBGVD(3)