SPGVX - 5.2 English - 68552

SPGVX computes all the eigenvalues.

Purpose:

    SPGVX 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.

Parameters:

• ITYPE – [in]

  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 – [in]

  JOBZ is CHARACTER*1

  = ‘N’: Compute eigenvalues only;

  = ‘V’: Compute eigenvalues and eigenvectors.

• RANGE – [in]

  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 – [in]

  UPLO is CHARACTER*1

  = ‘U’: Upper triangle of A and B are stored;

  = ‘L’: Lower triangle of A and B are stored.

• N – [in]

  N is INTEGER

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

• AP – [inout]

  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 – [inout]

  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 – [in]

  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 – [in]

  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 – [in]

  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 – [in]

  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 – [in]

  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 * fla_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 – [out]

  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 – [out]

  W is REAL array, dimension (N)

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

• Z – [out]

  Z is REAL array, dimension (LDZ, fla_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 fla_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 – [in]

  LDZ is INTEGER

  The leading dimension of the array Z. LDZ >= 1, and if JOBZ = ‘V’, LDZ >= fla_max(1,N).

• WORK – [out] WORK is REAL array, dimension (8*N)

• IWORK – [out] IWORK is INTEGER array, dimension (5*N)

• IFAIL – [out]

  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 – [out]

  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 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.