DSBEVX computes selected eigenvalues and, optionally, eigenvectors


 of a real symmetric band matrix A.  Eigenvalues and eigenvectors can


 be selected by specifying either 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 triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER


          The number of superdiagonals of the matrix A if UPLO = 'U',


          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          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 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, 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

          LDAB is INTEGER


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

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)


          If JOBZ = 'V', the N-by-N orthogonal matrix used in the


                         reduction 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 = 'V', then


          LDQ >= max(1,N).

VL

          VL is DOUBLE PRECISION

VU

          VU is DOUBLE PRECISION


          If RANGE='V', the lower and upper bounds of the interval to


          be searched for eigenvalues. VL < VU.


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

IL

          IL is INTEGER

IU

          IU is INTEGER


          If RANGE='I', the indices (in ascending order) of the


          smallest and largest eigenvalues 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 DOUBLE PRECISION


          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 AB to tridiagonal form.


          Eigenvalues will be computed most accurately when ABSTOL is


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


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


          eigenvectors did not converge, try setting ABSTOL to


          2*DLAMCH('S').


          See "Computing Small Singular Values of Bidiagonal Matrices


          with Guaranteed High Relative Accuracy," by Demmel and


          Kahan, LAPACK Working Note #3.

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 DOUBLE PRECISION array, dimension (N)


          The first M elements contain the selected eigenvalues in


          ascending order.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))


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


          contain the orthonormal eigenvectors of the matrix A


          corresponding to the selected eigenvalues, with the i-th


          column of Z holding the eigenvector associated with W(i).


          If an eigenvector fails to converge, then that column of Z


          contains the latest approximation to the eigenvector, and the


          index of the eigenvector is returned in IFAIL.


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


          Note: the user must ensure that at least max(1,M) columns are


          supplied in the array Z; if RANGE = 'V', the exact value of M


          is not known in advance and an upper bound must be used.

LDZ

          LDZ is INTEGER


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


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

WORK

          WORK is DOUBLE PRECISION array, dimension (7*N)

IWORK

          IWORK is INTEGER array, dimension (5*N)

IFAIL

          IFAIL is INTEGER array, dimension (N)


          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 eigenvectors 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.


          > 0:  if INFO = i, then i eigenvectors failed to converge.


                Their indices are stored in array IFAIL.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011