The routine may be called by the names f08ppf, nagf_lapackeig_zgeesx or its LAPACK name zgeesx.
The Schur factorization of is given by
where , the matrix of Schur vectors, is unitary and is the Schur form. A complex matrix is in Schur form if it is upper triangular.
Optionally, f08ppf also orders the eigenvalues on the diagonal of the 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).
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 , Schur vectors are not computed.
If , Schur vectors are computed.
2: – Character(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
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 f08ppf is called.
If , a workspace query is assumed; the routine only calculates an upper bound on 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.
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.
for optimal performance, lwork must generally be larger than the minimum; increase the workspace by, say, , where is the optimal block size for f08nsf.
16: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array rwork
must be at least
17: – Logical arrayWorkspace
Note: the dimension of the array bwork
must be at least
if , and at least otherwise.
f08ppf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ppf 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 .
This example finds the Schur factorization of the matrix
such that the eigenvalues of with positive real part 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.