f08waf computes for a pair of real nonsymmetric matrices the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the algorithm. f08waf is marked as deprecated by LAPACK; the replacement routine is f08wcf which makes better use of Level 3 BLAS.
The routine may be called by the names f08waf, nagf_lapackeig_dggev or its LAPACK name dggev.
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 eigenvector corresponding to the eigenvalue of satisfies
The left eigenvector corresponding to the eigenvalue of satisfies
where is the conjugate-transpose of .
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem , where and are real, square matrices, are determined using the algorithm. The algorithm consists of four stages:
1. is reduced to upper Hessenberg form and at the same time is reduced to upper triangular form.
2. is further reduced to quasi-triangular form while the triangular form of is maintained. This is the real generalized Schur form of the pair .
3.The quasi-triangular form of is reduced to triangular form and the eigenvalues extracted. 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. Pairs of complex eigenvalues occur with and complex conjugates, even though and are not conjugate.
4.If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.
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
1: – Character(1)Input
On entry: if , do not compute the left generalized eigenvectors.
If , compute the left generalized eigenvectors.
2: – Character(1)Input
On entry: if , do not compute the right generalized eigenvectors.
If , compute the right generalized eigenvectors.
3: – IntegerInput
On entry: , the order of the matrices and .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the first dimension of the array b as declared in the (sub)program from which f08waf is called.
8: – Real (Kind=nag_wp) arrayOutput
On exit: the element contains the real part of .
9: – Real (Kind=nag_wp) arrayOutput
On exit: the element contains the imaginary part of .
10: – Real (Kind=nag_wp) arrayOutput
On exit: , for , will be the generalized eigenvalues.
If is zero, then the th eigenvalue is real; if positive, then the th and st eigenvalues are a complex conjugate pair, with negative.
Note: the quotients and 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 .
11: – Real (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 eigenvectors are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues.
If the th eigenvalue is real, then , the th column of .
If the th and th eigenvalues form a complex conjugate pair, then and . Each eigenvector will be scaled so the largest component has .
On entry: the first dimension of the array vr as declared in the (sub)program from which f08waf is called.
if , ;
15: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
16: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08waf 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, lwork must generally be larger than the minimum; increase workspace by, say, , where is the optimal block size.
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 , and should be correct from element .
The iteration failed with an unexpected error, please contact NAG.
A failure occurred in f08ykf while computing generalized eigenvectors.
The computed eigenvalues and eigenvectors are exact for nearby matrices and , where
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
f08waf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08waf 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.
The total number of floating-point operations is proportional to .