The routine may be called by the names f08fnf, nagf_lapackeig_zheev or its LAPACK name zheev.
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 second dimension of the array a
must be at least
On entry: the Hermitian matrix .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if , a contains the orthonormal eigenvectors of the matrix .
If then on exit the lower triangle (if ) or the upper triangle (if ) of a, including the diagonal, is overwritten.
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08fnf is called.
6: – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
7: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
8: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08fnf is called.
If , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
for optimal performance, , where is the optimal block size for f08fsf.
9: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array rwork
must be at least
10: – IntegerOutput
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; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
The computed eigenvalues and eigenvectors are exact for a nearby matrix , where
Background information to multithreading can be found in the Multithreading documentation.
f08fnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fnf 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 .