DSPGVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION


          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 DOUBLE PRECISION


          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 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 A 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').

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)


          On normal exit, the first M elements contain the selected


          eigenvalues in ascending order.

Z

          Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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:  DPPTRF or DSPEVX returned an error code:


             <= N:  if INFO = i, DSPEVX 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


                    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