f08hbf computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix of bandwidth . Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
The routine may be called by the names f08hbf, nagf_lapackeig_dsbevx or its LAPACK name dsbevx.
The symmetric band 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.
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
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).
15: – IntegerOutput
On exit: the total number of eigenvalues found. .
If , .
If , .
16: – Real (Kind=nag_wp) arrayOutput
On exit: the first m elements contain the selected eigenvalues in ascending order.
17: – Real (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 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 (), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
Background information to multithreading can be found in the Multithreading documentation.
f08hbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08hbf 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 is proportional to if and , otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.