The routine may be called by the names f08qkf, nagf_lapackeig_dtrevc or its LAPACK name dtrevc.
f08qkf computes left and/or right eigenvectors of a real upper quasi-triangular matrix in canonical Schur form. Such a matrix arises from the Schur factorization of a real general matrix, as computed by f08pef, for example.
The right eigenvector , and the left eigenvector , corresponding to an eigenvalue , are defined by:
Note that even though is real, , and may be complex. If is an eigenvector corresponding to a complex eigenvalue , then the complex conjugate vector is the eigenvector corresponding to the complex conjugate eigenvalue .
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 an orthogonal 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, for a real eigenvector ,
and for a complex eigenvector,
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/Output
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 real eigenvector corresponding to the real eigenvalue , must be set .TRUE.. To select the complex eigenvector corresponding to a complex conjugate pair of eigenvalues and , and/or must be set .TRUE.; the eigenvector corresponding to the first eigenvalue in the pair is computed.
On exit: if a complex eigenvector was selected as specified above, is set to .TRUE. and to .FALSE..
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. Corresponding to each real eigenvalue is a real eigenvector, occupying one column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two columns; the first column holds the real part and the second column holds the imaginary part.
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. Corresponding to each real eigenvalue is a real eigenvector, occupying one column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two columns; the first column holds the real part and the second column holds the imaginary part.
On entry: the first dimension of the array vr as declared in the (sub)program from which f08qkf 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 obtained by counting for each selected real eigenvector and for each selected complex eigenvector (see select), in which case .
if or , ;
12: – IntegerOutput
On exit: , the number of columns of vl and/or vr actually used to store the computed eigenvectors. If or , m is set to .
13: – Real (Kind=nag_wp) arrayWorkspace
14: – 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 f08qlf.
8Parallelism and Performance
f08qkf 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.
For a description of canonical Schur form, see the document for f08pef.