The routine may be called by the names f08gnf, nagf_lapackeig_zhpev or its LAPACK name zhpev.
The Hermitian matrix is first reduced to real tridiagonal form, using unitary 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: – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap
must be at least
On entry: the upper or lower triangle of the Hermitian matrix , packed by columns.
if , the upper triangle of must be stored with element in for ;
if , the lower triangle of must be stored with element in for .
On exit: ap 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 .
5: – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
6: – 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 , 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.
f08gnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gnf 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.
Each eigenvector is normalized so that the element of largest absolute value is real.
The total number of floating-point operations is proportional to .