f08quc reorders the Schur factorization of a complex general matrix so that a selected cluster of eigenvalues appears in the leading elements on the diagonal of the Schur form. The function also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.
The function may be called by the names: f08quc, nag_lapackeig_ztrsen or nag_ztrsen.
f08quc reorders the Schur factorization of a complex general matrix , so that a selected cluster of eigenvalues appears in the leading diagonal elements of the Schur form.
The reordered Schur form is computed by a unitary similarity transformation: . Optionally the updated matrix of Schur vectors is computed as , giving .
Let , where the selected eigenvalues are precisely the eigenvalues of the leading sub-matrix . Let be correspondingly partitioned as where consists of the first columns of . Then , and so the columns of form an orthonormal basis for the invariant subspace corresponding to the selected cluster of eigenvalues.
Optionally the function also computes estimates of the reciprocal condition numbers of the average of the cluster of eigenvalues and of the invariant subspace.
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: indicates whether condition numbers are required for the cluster of eigenvalues and/or the invariant subspace.
No condition numbers are required.
Only the condition number for the cluster of eigenvalues is computed.
Only the condition number for the invariant subspace is computed.
Condition numbers for both the cluster of eigenvalues and the invariant subspace are computed.
, , or .
3: – Nag_ComputeQTypeInput
On entry: indicates whether the matrix of Schur vectors is to be updated.
The matrix of Schur vectors is updated.
No Schur vectors are updated.
4: – const Nag_BooleanInput
Note: the dimension, dim, of the array select
must be at least
On entry: specifies the eigenvalues in the selected cluster. To select a complex eigenvalue , must be set Nag_TRUE.
5: – IntegerInput
On entry: , the order of the matrix .
6: – ComplexInput/Output
Note: the dimension, dim, of the array t
must be at least
The th element of the matrix is stored in
On entry: the upper triangular matrix , as returned by f08psc.
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.
On entry, , and .
Constraint: if , ;
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 computed matrix is similar to a matrix , where
and is the machine precision.
s cannot underestimate the true reciprocal condition number by more than a factor of . sep may differ from the true value by . The angle between the computed invariant subspace and the true subspace is .
The values of the eigenvalues are never changed by the reordering.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08quc 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.