The function may be called by the names: f08fnc, nag_lapackeig_zheev or nag_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: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
3: – Nag_UploTypeInput
On entry: if , the upper triangular part of is stored.
If , the lower triangular part of is stored.
4: – IntegerInput
On entry: , the order of the matrix .
5: – ComplexInput/Output
Note: the dimension, dim, of the array a
must be at least
On entry: the Hermitian matrix .
If , is stored in .
If , is stored in .
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.
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
7: – doubleOutput
On exit: the eigenvalues in ascending order.
8: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
On entry, .
On entry, . Constraint: .
On entry, and .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
The computed eigenvalues and eigenvectors are exact for a nearby matrix , where
f08fnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fnc 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 function. 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 .