f08xbf 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 .
Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.
The routine may be called by the names f08xbf, nagf_lapackeig_dggesx or its LAPACK name dggesx.
3Description
The generalized real Schur factorization of 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, f08xbf 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.
f08xbf 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.
The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in and respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in and . See Section 4.11 of Anderson et al. (1999) for further information.
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 (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – Character(1)Input
On entry: if , do not compute the left Schur vectors.
If , compute the left Schur vectors.
Constraint:
or .
2: – Character(1)Input
On entry: if , do not compute the right Schur vectors.
If , compute the right Schur vectors.
Constraint:
or .
3: – Character(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
On entry: an eigenvalue is selected if is .TRUE.. If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy after ordering. in this case.
selctg must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f08xbf is called. Arguments denoted as Input must not be changed by this procedure.
5: – Character(1)Input
On entry: determines which reciprocal condition numbers are computed.
None are computed.
Computed for average of selected eigenvalues only.
Computed for selected deflating subspaces only.
Computed for both.
If , or , .
Constraint:
, , or .
6: – IntegerInput
On entry: , the order of the matrices and .
Constraint:
.
7: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the first of the pair of matrices, .
On exit: a has been overwritten by its generalized Schur form .
8: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08xbf is called.
Constraint:
.
9: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
On entry: the second of the pair of matrices, .
On exit: b has been overwritten by its generalized Schur form .
10: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08xbf is called.
Constraint:
.
11: – IntegerOutput
On exit: if , .
If , number of eigenvalues (after sorting) for which selctg is .TRUE.. (Complex conjugate pairs for which selctg is .TRUE. for either eigenvalue count as .)
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 .
15: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vsl
must be at least
if .
On exit: if , vsl will contain the left Schur vectors, .
On entry: the dimension of the array work as declared in the (sub)program from which f08xbf is called.
If , a workspace query is assumed; the routine only calculates the bound on the optimal size of the work array and the minimum size of the iwork array, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraints:
if ,
if , or , ;
otherwise .
Note: that . Note also that an error is only returned if , but if , or this may not be large enough. Consider increasing lwork by , where is the optimal block size.
On entry: the dimension of the array iwork as declared in the (sub)program from which f08xbf is called.
If , a workspace query is assumed; the routine only calculates the bound on the optimal size of the work array and the minimum size of the iwork array, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraints:
if ,
if or , ;
otherwise .
25: – 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).
7Accuracy
The computed generalized Schur factorization satisfies
where
and is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08xbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xbf 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 .
This example finds the generalized Schur factorization of the matrix pair , where
such that the real positive eigenvalues of correspond to the top left diagonal elements of the generalized Schur form, . Estimates of the condition numbers for the selected eigenvalue cluster and corresponding deflating subspaces are also returned.
Note that the block size (NB) of assumed in this example is not realistic for such a small problem, but should be suitable for large problems.