The routine may be called by the names f08pbf, nagf_lapackeig_dgeesx or its LAPACK name dgeesx.
3Description
The real Schur factorization of is given by
where , the matrix of Schur vectors, is orthogonal and is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with and blocks. blocks will be standardized in the form
where . The eigenvalues of such a block are .
Optionally, f08pbf also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of form an orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.8 of Anderson et al. (1999) (where these quantities are called and respectively).
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 , Schur vectors are not computed.
If , Schur vectors are computed.
Constraint:
or .
2: – Character(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
3: – Logical Function, supplied by the user.External Procedure
If , select is used to select eigenvalues to sort to the top left of the Schur form.
If , select is not referenced and f08pbf may be called with the dummy function f08paz.
An eigenvalue is selected if is .TRUE.. If either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case info is set to (see info below).
On entry: the real and imaginary parts of the eigenvalue.
select must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f08pbf is called. Arguments denoted as Input must not be changed by this procedure.
4: – 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 right invariant subspace only.
Computed for both.
If , or , .
Constraint:
, , or .
5: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the matrix .
On exit: a is overwritten by its real Schur form .
7: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08pbf is called.
Constraint:
.
8: – IntegerOutput
On exit: if , .
If , number of eigenvalues (after sorting) for which select is .TRUE.. (Complex conjugate pairs for which select is .TRUE. for either eigenvalue count as .)
9: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array wr
must be at least
.
Note: the dimension of the array wi
must be at least
.
On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form . Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
11: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vs
must be at least
if , and at least otherwise.
On exit: if , vs contains the orthogonal matrix of Schur vectors.
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 f08pbf is called.
If , a workspace query is assumed; the routine only calculates upper bounds on the optimal sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error messages related to lwork or liwork is issued.
If , or , , where sdim is the number of selected eigenvalues computed by this routine. Note that .
Note also that an error is only returned if , but if , or this may not be large enough.
Suggested value:
for optimal performance, lwork must generally be larger than the minimum; increase the workspace by, say, , where is the optimal block size for f08nef.
On entry: the dimension of the array iwork as declared in the (sub)program from which f08pbf is called.
If , a workspace query is assumed; the routine only calculates upper bounds on the optimal sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error messages related to lwork or liwork is issued.
Constraints:
if ,
if or , ;
otherwise .
Note: . Note also that an error is only returned if , but if or this may not be large enough.
19: – 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 algorithm failed to compute all the eigenvalues.
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy . This could also be caused by underflow due to scaling.
7Accuracy
The computed Schur factorization satisfies
where
and is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08pbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pbf 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 Schur factorization of the matrix
such that the real positive eigenvalues of are the top left diagonal elements of the Schur form, . Estimates of the condition numbers for the selected eigenvalue cluster and corresponding invariant subspace 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.