f08fqc computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix.
If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the or algorithm.
The function may be called by the names: f08fqc, nag_lapackeig_zheevd or nag_zheevd.
f08fqc computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix .
In other words, it can compute the spectral factorization of as
where is a real diagonal matrix whose diagonal elements are the eigenvalues , and is the (complex) unitary matrix whose columns are the eigenvectors . Thus
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: indicates whether the upper or lower triangular part of is stored.
The upper triangular part of is stored.
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 is overwritten by the unitary matrix which contains the eigenvectors of .
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
7: – doubleOutput
Note: the dimension, dim, of the array w
must be at least
On exit: the eigenvalues of the matrix 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.
If and , the algorithm failed to converge; elements of an intermediate tridiagonal form did not converge to zero; if and , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column through .
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
f08fqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fqc 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.