The routine may be called by the names f08haf, nagf_lapackeig_dsbev or its LAPACK name dsbev.
The symmetric band matrix is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.
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
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 , the upper triangular part of is stored.
If , the lower triangular part of is stored.
3: – IntegerInput
On entry: , the order of the matrix .
4: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
5: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
On entry: the upper or lower triangle of the symmetric band matrix .
The matrix is stored in rows to , more precisely,
if , the elements of the upper triangle of within the band must be stored with element in ;
if , the elements of the lower triangle of within the band must be stored with element in
On exit: ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix are returned in ab using the same storage format as described above.
6: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f08haf is called.
7: – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
8: – 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 , z contains the orthonormal eigenvectors of the matrix , with the th column of holding the eigenvector associated with .
Background information to multithreading can be found in the Multithreading documentation.
f08haf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08haf 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 otherwise.