The routine may be called by the names f08yxf, nagf_lapackeig_ztgevc or its LAPACK name ztgevc.
f08yxf computes some or all of the right and/or left generalized eigenvectors of the matrix pair which is assumed to be in upper triangular form. If the matrix pair is not upper triangular then the routine f08xsf should be called before invoking f08yxf.
The right generalized eigenvector and the left generalized eigenvector of corresponding to a generalized eigenvalue are defined by
If a generalized eigenvalue is determined as , which is due to zero diagonal elements at the same locations in both and , a unit vector is returned as the corresponding eigenvector.
Note that the generalized eigenvalues are computed using f08xsf but f08yxf does not explicitly require the generalized eigenvalues to compute eigenvectors. The ordering of the eigenvectors is based on the ordering of the eigenvalues as computed by f08yxf.
If all eigenvectors are requested, the routine may either return the matrices and/or of right or left eigenvectors of , or the products and/or , where and are two matrices supplied by you. Usually, and are chosen as the unitary matrices returned by f08xsf. Equivalently, and are the left and right Schur vectors of the matrix pair supplied to f08xsf. In that case, and are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to f08xsf.
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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal.10 241–256
Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London
1: – Character(1)Input
On entry: specifies the required sets of generalized eigenvectors.
Only right eigenvectors are computed.
Only left eigenvectors are computed.
Both left and right eigenvectors are computed.
, or .
2: – Character(1)Input
On entry: specifies further details of the required generalized eigenvectors.
All right and/or left eigenvectors are computed.
All right and/or left eigenvectors are computed; they are backtransformed using the input matrices supplied in arrays vr and/or vl.
Selected right and/or left eigenvectors, defined by the array 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 the eigenvectors to be computed if . To select the generalized eigenvector corresponding to the th generalized eigenvalue, the th element of select should be set to .TRUE..
if , or , for .
4: – IntegerInput
On entry: , the order of the matrices and .
5: – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array a
must be at least
On entry: the matrix must be in upper triangular form. Usually, this is the matrix returned by f08xsf.
6: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08yxf is called.
7: – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
On entry: the matrix must be in upper triangular form with non-negative real diagonal elements. Usually, this is the matrix returned by f08xsf.
8: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08yxf is called.
9: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array vl
must be at least
if or and at least if .
On entry: if and or , vl must be initialized to an matrix . Usually, this is the unitary matrix of left Schur vectors returned by f08xsf.
if , the right eigenvectors of specified by select, stored consecutively in the columns of the array vr, in the same order as their corresponding eigenvalues.
12: – IntegerInput
On entry: the first dimension of the array vr as declared in the (sub)program from which f08yxf is called.
if or , ;
if , .
13: – IntegerInput
On entry: the number of columns in the arrays vl and/or vr.
if or , ;
if , mm must not be less than the number of requested eigenvectors.
14: – IntegerOutput
On exit: the number of columns in the arrays vl and/or vr actually used to store the eigenvectors. If or , m is set to n. Each selected eigenvector occupies one column.
15: – Complex (Kind=nag_wp) arrayWorkspace
16: – Real (Kind=nag_wp) arrayWorkspace
17: – 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.
It is beyond the scope of this manual to summarise the accuracy of the solution of the generalized eigenvalue problem. Interested readers should consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990).
8Parallelism and Performance
f08yxf 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.
f08yxf is the sixth step in the solution of the complex generalized eigenvalue problem and is usually called after f08xsf.
This example computes the and arguments, which defines the generalized eigenvalues and the corresponding left and right eigenvectors, of the matrix pair given by
To compute generalized eigenvalues, it is required to call five routines: f08wvf to balance the matrix, f08asf to perform the factorization of , f08auf to apply to , f08wtf to reduce the matrix pair to the generalized Hessenberg form and f08xsf to compute the eigenvalues via the algorithm.
The computation of generalized eigenvectors is done by calling f08yxf to compute the eigenvectors of the balanced matrix pair. The routine f08wwf is called to backward transform the eigenvectors to the user-supplied matrix pair. If both left and right eigenvectors are required then f08wwf must be called twice.