SSBGVX computes selected eigenvalues, and optionally, 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.  Eigenvalues and


 eigenvectors can be selected by specifying either all eigenvalues,


 a range of values or a range of indices for the desired eigenvalues.

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.

RANGE

          RANGE is CHARACTER*1


          = 'A': all eigenvalues will be found.


          = 'V': all eigenvalues in the half-open interval (VL,VU]


                 will be found.


          = 'I': the IL-th through IU-th eigenvalues will be found.

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.

Q

          Q is REAL array, dimension (LDQ, N)


          If JOBZ = 'V', the n-by-n matrix used in the reduction of


          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,


          and consequently C to tridiagonal form.


          If JOBZ = 'N', the array Q is not referenced.

LDQ

          LDQ is INTEGER


          The leading dimension of the array Q.  If JOBZ = 'N',


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

VL

          VL is REAL


          If RANGE='V', the lower bound of the interval to


          be searched for eigenvalues. VL < VU.


          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is REAL


          If RANGE='V', the upper bound of the interval to


          be searched for eigenvalues. VL < VU.


          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER


          If RANGE='I', the index of the


          smallest eigenvalue to be returned.


          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.


          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER


          If RANGE='I', the index of the


          largest eigenvalue to be returned.


          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.


          Not referenced if RANGE = 'A' or 'V'.

ABSTOL

          ABSTOL is REAL


          The absolute error tolerance for the eigenvalues.


          An approximate eigenvalue is accepted as converged


          when it is determined to lie in an interval [a,b]


          of width less than or equal to


                  ABSTOL + EPS *   max( |a|,|b| ) ,


          where EPS is the machine precision.  If ABSTOL is less than


          or equal to zero, then  EPS*|T|  will be used in its place,


          where |T| is the 1-norm of the tridiagonal matrix obtained


          by reducing A to tridiagonal form.


          Eigenvalues will be computed most accurately when ABSTOL is


          set to twice the underflow threshold 2*SLAMCH('S'), not zero.


          If this routine returns with INFO>0, indicating that some


          eigenvectors did not converge, try setting ABSTOL to


          2*SLAMCH('S').

M

          M is INTEGER


          The total number of eigenvalues found.  0 <= M <= N.


          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+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 (7*N)

IWORK

          IWORK is INTEGER array, dimension (5*N)

IFAIL

          IFAIL is INTEGER array, dimension (M)


          If JOBZ = 'V', then if INFO = 0, the first M elements of


          IFAIL are zero.  If INFO > 0, then IFAIL contains the


          indices of the eigenvalues that failed to converge.


          If JOBZ = 'N', then IFAIL is not referenced.

INFO

          INFO is INTEGER


          = 0 : successful exit


          < 0 : if INFO = -i, the i-th argument had an illegal value


          <= N: if INFO = i, then i eigenvectors failed to converge.


                  Their indices are stored in IFAIL.


          > N : SPBSTF returned an error code; i.e.,


                if INFO = N + i, for 1 <= i <= N, then the leading


                minor of order i of 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.

June 2016

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