The function may be called by the names: f08pnc, nag_lapackeig_zgees or nag_zgees.
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, f08pnc also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.
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: – 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.
2: – Nag_JobTypeInput
On entry: if , Schur vectors are not computed.
If , Schur vectors are computed.
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
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
The algorithm failed to compute all the eigenvalues.
On entry, , and .
Constraint: if , ;
On entry, .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, and .
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.
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.
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.
f08pnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pnc 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.
The total number of floating-point operations is proportional to .