f08wnf computes for a pair of complex nonsymmetric matrices the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the algorithm. f08wnf is marked as deprecated by LAPACK; the replacement routine is f08wqf which makes better use of Level 3 BLAS.
The routine may be called by the names f08wnf, nagf_lapackeig_zggev or its LAPACK name zggev.
3Description
A generalized eigenvalue for a pair of matrices is a scalar or a ratio , such that is singular. It is usually represented as the pair , as there is a reasonable interpretation for , and even for both being zero.
The right generalized eigenvector corresponding to the generalized eigenvalue of satisfies
The left generalized eigenvector corresponding to the generalized eigenvalue of satisfies
where is the conjugate-transpose of .
All the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem , where and are complex, square matrices, are determined using the algorithm. The complex algorithm consists of three stages:
1. is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time is reduced to upper triangular form.
2. is further reduced to triangular form while the triangular form of is maintained and the diagonal elements of are made real and non-negative. This is the generalized Schur form of the pair .
This routine does not actually produce the eigenvalues , but instead returns and such that
The division by becomes your responsibility, since may be zero, indicating an infinite eigenvalue.
3.If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.
4References
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 (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1979) Kronecker's canonical form and the algorithm Linear Algebra Appl.28 285–303
5Arguments
1: – Character(1)Input
On entry: if , do not compute the left generalized eigenvectors.
If , compute the left generalized eigenvectors.
Constraint:
or .
2: – Character(1)Input
On entry: if , do not compute the right generalized eigenvectors.
If , compute the right generalized eigenvectors.
Constraint:
or .
3: – IntegerInput
On entry: , the order of the matrices and .
Constraint:
.
4: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On exit: , for , will be the generalized eigenvalues.
Note: the quotients may easily overflow or underflow, and may even be zero. Thus, you should avoid naively computing the ratio . However, will always be less than and usually comparable with in magnitude, and will always be less than and usually comparable with .
10: – Complex (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vl
must be at least
if , and at least otherwise.
On exit: if , the left generalized eigenvectors are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have .
On entry: the first dimension of the array vl as declared in the (sub)program from which f08wnf is called.
Constraints:
if , ;
otherwise .
12: – Complex (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vr
must be at least
if , and at least otherwise.
On exit: if , the right generalized eigenvectors are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have .
On entry: the first dimension of the array vr as declared in the (sub)program from which f08wnf is called.
Constraints:
if , ;
otherwise .
14: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
15: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08wnf 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.
Suggested value:
for optimal performance, lwork must generally be larger than the minimum; increase workspace by, say, , where is the optimal block size.
Constraint:
or .
16: – Real (Kind=nag_wp) arrayWorkspace
17: – 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 iteration failed. No eigenvectors have been calculated but alpha and beta should be correct from element .
The iteration failed with an unexpected error, please contact NAG.
A failure occurred in f08yxf while computing generalized eigenvectors.
7Accuracy
The computed eigenvalues and eigenvectors are exact for nearby matrices and , where
and is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
Note: interpretation of results obtained with the algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of and . It should be noted that if and are both small for any , it may be that no reliance can be placed on any of the computed eigenvalues . You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08wnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08wnf 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.
9Further Comments
The total number of floating-point operations is proportional to .