The function may be called by the names: f08pbc, nag_lapackeig_dgeesx or nag_dgeesx.
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, f08pbc 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).
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.
4: – function, supplied by the userExternal Function
If , select is used to select eigenvalues to sort to the top left of the Schur form.
If , select is not referenced and f08pbc may be specified as NULLFN.
An eigenvalue is selected if is Nag_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 is set to .
Note: the dimension, dim, 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.
12: – doubleOutput
Note: the dimension, dim, of the array vs
must be at least
th element of the th vector is stored in
On exit: if , vs contains the orthogonal matrix of Schur vectors.
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.
f08pbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pbc 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 .
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.