The routine may be called by the names f08qxf, nagf_lapackeig_ztrevc or its LAPACK name ztrevc.
f08qxf computes left and/or right eigenvectors of a complex upper triangular matrix . Such a matrix arises from the Schur factorization of a complex general matrix, as computed by f08psf, for example.
The right eigenvector , and the left eigenvector , corresponding to an eigenvalue , are defined by:
The routine can compute the eigenvectors corresponding to selected eigenvalues, or it can compute all the eigenvectors. In the latter case the eigenvectors may optionally be pre-multiplied by an input matrix . Normally is a unitary matrix from the Schur factorization of a matrix as ; if is a (left or right) eigenvector of , then is an eigenvector of .
The eigenvectors are computed by forward or backward substitution. They are scaled so that
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 left and/or right eigenvectors are to be computed.
Only right eigenvectors are computed.
Only left eigenvectors are computed.
Both left and right eigenvectors are computed.
, or .
2: – Character(1)Input
On entry: indicates how many eigenvectors are to be computed.
All eigenvectors (as specified by job) are computed.
All eigenvectors (as specified by job) are computed and then pre-multiplied by the matrix (which is overwritten).
Selected eigenvectors (as specified by job and select) are computed.
, or .
3: – Logical arrayInput
Note: the dimension of the array select
must be at least
if , and at least otherwise.
On entry: specifies which eigenvectors are to be computed if . To obtain the eigenvector corresponding to the eigenvalue , must be set .TRUE..
On exit: if or , vl contains the computed left eigenvectors (as specified by howmny and select). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.
On exit: if or , vr contains the computed right eigenvectors (as specified by howmny and select). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.
On entry: the first dimension of the array vr as declared in the (sub)program from which f08qxf is called.
if or , ;
if , .
11: – IntegerInput
On entry: the number of columns in the arrays vl and/or vr. The precise number of columns required, , is if or ; if , is the number of selected eigenvectors (see select), in which case .
if or , ;
12: – IntegerOutput
On exit: , the number of selected eigenvectors. If or , m is set to .
13: – Complex (Kind=nag_wp) arrayWorkspace
14: – Real (Kind=nag_wp) arrayWorkspace
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.
If is an exact right eigenvector, and is the corresponding computed eigenvector, then the angle between them is bounded as follows:
where is the reciprocal condition number of .
The condition number may be computed by calling f08qyf.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08qxf 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.