The function may be called by the names: f08ppc, nag_lapackeig_zgeesx or nag_zgeesx.
3Description
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, f08ppc 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).
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: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_JobTypeInput
On entry: if , Schur vectors are not computed.
If , Schur vectors are computed.
Constraint:
or .
3: – Nag_SortEigValsTypeInput
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_CONVERGENCE
The algorithm failed to compute all the eigenvalues.
NE_ENUM_INT_2
On entry, , and .
Constraint: if , ;
otherwise .
NE_INT
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SCHUR_REORDER
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
NE_SCHUR_REORDER_SELECT
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.
f08ppc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ppc 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 function. 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 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.