!>
!> SSPGVX computes selected eigenvalues, and optionally, eigenvectors
!> of a real generalized symmetric-definite eigenproblem, of the form
!> A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
!> and B are assumed to be symmetric, stored in packed storage, and B
!> is also positive definite.  Eigenvalues and eigenvectors can be
!> selected by specifying either a range of values or a range of indices
!> for the desired eigenvalues.
!> 

ITYPE

!>          ITYPE is INTEGER
!>          Specifies the problem type to be solved:
!>          = 1:  A*x = (lambda)*B*x
!>          = 2:  A*B*x = (lambda)*x
!>          = 3:  B*A*x = (lambda)*x
!> 

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 and B are stored;
!>          = 'L':  Lower triangle of A and B are stored.
!> 

N

!>          N is INTEGER
!>          The order of the matrix pencil (A,B).  N >= 0.
!> 

AP

!>          AP is REAL array, dimension (N*(N+1)/2)
!>          On entry, the upper or lower triangle of the symmetric matrix
!>          A, packed columnwise in a linear array.  The j-th column of A
!>          is stored in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
!>
!>          On exit, the contents of AP are destroyed.
!> 

BP

!>          BP is REAL array, dimension (N*(N+1)/2)
!>          On entry, the upper or lower triangle of the symmetric matrix
!>          B, packed columnwise in a linear array.  The j-th column of B
!>          is stored in the array BP as follows:
!>          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
!>          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
!>
!>          On exit, the triangular factor U or L from the Cholesky
!>          factorization B = U**T*U or B = L*L**T, in the same storage
!>          format as B.
!> 

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)
!>          On normal exit, the first M elements contain the selected
!>          eigenvalues in ascending order.
!> 

Z

!>          Z is REAL array, dimension (LDZ, max(1,M))
!>          If JOBZ = 'N', then Z is not referenced.
!>          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).
!>          The eigenvectors are normalized as follows:
!>          if ITYPE = 1 or 2, Z**T*B*Z = I;
!>          if ITYPE = 3, Z**T*inv(B)*Z = 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.
!>          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 REAL array, dimension (8*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:  SPPTRF or SSPEVX returned an error code:
!>             <= N:  if INFO = i, SSPEVX failed to converge;
!>                    i eigenvectors failed to converge.  Their indices
!>                    are stored in array IFAIL.
!>             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
!>                    principal minor of order i of B is not positive.
!>                    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.

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