f08xaf computes the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right generalized Schur vectors for a pair of real nonsymmetric matrices . f08xaf is marked as deprecated by LAPACK; the replacement routine is f08xcf which makes better use of Level 3 BLAS.
The routine may be called by the names f08xaf, nagf_lapackeig_dgges or its LAPACK name dgges.
The generalized Schur factorization for a pair of real matrices is given by
where and are orthogonal, is upper triangular and is upper quasi-triangular with and diagonal blocks. The generalized eigenvalues, , of are computed from the diagonals of and and satisfy
where is the corresponding generalized eigenvector. is actually returned as the pair such that
since , or even both and can be zero. The columns of and are the left and right generalized Schur vectors of .
Optionally, f08xaf can order the generalized eigenvalues on the diagonals of so that selected eigenvalues are at the top left. The leading columns of and then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
f08xaf computes to have non-negative diagonal elements, and the blocks of correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the algorithm.
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: if , do not compute the left Schur vectors.
If , compute the left Schur vectors.
2: – Character(1)Input
On entry: if , do not compute the right Schur vectors.
If , compute the right Schur vectors.
3: – Character(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
On exit: , for , will be the generalized eigenvalues.
, and , for , are the diagonals of the complex Schur form that would result if the diagonal blocks of the real Schur form of were further reduced to triangular form using complex unitary transformations.
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, alphar and alphai will always be less than and usually comparable with in magnitude, and beta will always be less than and usually comparable with .
14: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vsl
must be at least
if , and at least otherwise.
On exit: if , vsl will contain the left Schur vectors, .
On entry: the first dimension of the array vsr as declared in the (sub)program from which f08xaf is called.
if , ;
18: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
19: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08xaf 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 mimimum; add, say , where is the optimal block size.
if , ;
20: – Logical arrayWorkspace
Note: the dimension of the array bwork
must be at least
if , and at least otherwise.
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.
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy . This could also be caused by underflow due to scaling.
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
The computed generalized Schur factorization satisfies
f08xaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xaf 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 .