naginterfaces.library.lapackeig.dspevx

naginterfaces.library.lapackeig.dspevx(jobz, erange, uplo, n, ap, vl, vu, il, iu, abstol)[source]

dspevx computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

For full information please refer to the NAG Library document for f08gb

https://www.nag.com/numeric/nl/nagdoc_28.6/flhtml/f08/f08gbf.html

Parameters
jobzstr, length 1

Indicates whether eigenvectors are computed.

Only eigenvalues are computed.

Eigenvalues and eigenvectors are computed.

erangestr, length 1

If , all eigenvalues will be found.

If , all eigenvalues in the half-open interval will be found.

If , the th to th eigenvalues will be found.

uplostr, length 1

If , the upper triangular part of is stored.

If , the lower triangular part of is stored.

nint

, the order of the matrix .

apfloat, array-like, shape

The upper or lower triangle of the symmetric matrix , packed by columns.

vlfloat

If , the lower and upper bounds of the interval to be searched for eigenvalues.

If or , and are not referenced.

vufloat

If , the lower and upper bounds of the interval to be searched for eigenvalues.

If or , and are not referenced.

ilint

If , and specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

If or , and are not referenced.

iuint

If , and specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

If or , and are not referenced.

abstolfloat

The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval of width less than or equal to

where is the machine precision. If is less than or equal to zero, then will be used in its place, where is the tridiagonal matrix obtained by reducing to tridiagonal form. Eigenvalues will be computed most accurately when is set to twice the underflow threshold , not zero. If this function returns with > 0, indicating that some eigenvectors did not converge, try setting to . See Demmel and Kahan (1990).

Returns
apfloat, ndarray, shape

is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of .

mint

The total number of eigenvalues found. .

If , .

If , .

wfloat, ndarray, shape

The selected eigenvalues in ascending order.

zfloat, ndarray, shape

If , then

if no exception or warning is raised, the first columns of contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with ;

if an eigenvector fails to converge ( > 0), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in .

If , is not referenced.

Note: you must ensure that at least columns are supplied in the array ; if , the exact value of is not known in advance and an upper bound of at least must be used.

jfailint, ndarray, shape

If , then

if no exception or warning is raised, the first elements of are zero;

if > 0, contains the indices of the eigenvectors that failed to converge.

If , is not referenced.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: , or .

(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

(errno )

On entry, error in parameter .

Warns
NagAlgorithmicWarning
(errno )

The algorithm failed to converge; eigenvectors did not converge. Their indices are stored in array .

Notes

The symmetric matrix is first reduced to tridiagonal form, using orthogonal similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Demmel, J W and Kahan, W, 1990, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Statist. Comput. (11), 873–912

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore