f08quf 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 routine also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.
The routine may be called by the names f08quf, nagf_lapackeig_ztrsen or its LAPACK name ztrsen.
f08quf 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 routine 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: – Character(1)Input
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 .
2: – Character(1)Input
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.
3: – Logical arrayInput
Note: the dimension 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 .TRUE..
4: – IntegerInput
On entry: , the order of the matrix .
5: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array t
must be at least
On entry: the upper triangular matrix , as returned by f08psf.
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
14: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08quf is called, unless , in which case a workspace query is assumed and the routine only calculates the minimum dimension of work.
if , or ;
if , or ;
if or , or .
The actual amount of workspace required cannot exceed if or if or .
15: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
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.
f08quf 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.