SSBGVD 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.

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangles of A and B are stored;


          = 'L':  Lower triangles of A and B are stored.

N

          N is INTEGER


          The order of the matrices A and B.  N >= 0.

KA

          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

          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

          AB is REAL 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

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= KA+1.

BB

          BB is REAL 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 SPBSTF.

LDBB

          LDBB is INTEGER


          The leading dimension of the array BB.  LDBB >= KB+1.

W

          W is REAL array, dimension (N)


          If INFO = 0, the eigenvalues in ascending order.

Z

          Z is REAL 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

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          JOBZ = 'V', LDZ >= max(1,N).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          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

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          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

          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 SPBSTF


                    returned INFO = i: B is not positive definite.


                    The factorization of B could not be completed and


                    no eigenvalues or eigenvectors were computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA