f08upf computes selected the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form
where and are Hermitian and banded, and is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.
The routine may be called by the names f08upf, nagf_lapackeig_zhbgvx or its LAPACK name zhbgvx.
The generalized Hermitian-definite band problem
is first reduced to a standard band Hermitian problem
where is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that
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
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM16 41–44
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
Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press
1: – Character(1)Input
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
2: – Character(1)Input
On entry: if , all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
On entry: 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 abstol 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 abstol is set to twice the underflow threshold , not zero. If this routine returns with , indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).
18: – IntegerOutput
On exit: the total number of eigenvalues found. .
If , .
If , .
19: – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
20: – Complex (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z
must be at least
if , and at least otherwise.
On exit: if , then
if , the first m columns of contain the eigenvectors corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with . The eigenvectors are normalized so that ;
if an eigenvector fails to converge (), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The algorithm failed to converge; eigenvectors did not converge. Their indices are stored in array jfail.
If , for , f08utf returned : is not positive definite. The factorization of could not be completed and no eigenvalues or eigenvectors were computed.
If is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08upf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08upf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is proportional to if and , and assuming that , is approximately proportional to if . Otherwise the number of floating-point operations depends upon the number of eigenvectors computed.