The routine may be called by the names f08yyf, nagf_lapackeig_ztgsna or its LAPACK name ztgsna.
3Description
f08yyf estimates condition numbers for specified eigenvalues and/or right eigenvectors of an matrix pair in generalized Schur form. The routine actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.
The pair are in generalized Schur form if and are upper triangular as returned, for example, by f08xpforf08xqf, or f08xsf with . The diagonal elements define the generalized eigenvalues , for , of the pair and the eigenvalues are given by
so that
where is the corresponding (right) eigenvector.
If and are the result of a generalized Schur factorization of a matrix pair
then the eigenvalues and condition numbers of the pair are the same as those of the pair .
Let be a simple generalized eigenvalue of . Then the reciprocal of the condition number of the eigenvalue is defined as
where and are the right and left eigenvectors of corresponding to . If both and are zero, then is singular and is returned.
If and are unitary transformations such that
where and are matrices, then the reciprocal condition number is given by
where denotes the smallest singular value of the matrix
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 https://www.netlib.org/lapack/lug
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software22 78–103
5Arguments
1: – Character(1)Input
On entry: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
Condition numbers for eigenvalues only are computed.
Condition numbers for eigenvectors only are computed.
Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint:
, or .
2: – Character(1)Input
On entry: indicates how many condition numbers are to be computed.
Condition numbers for all eigenpairs are computed.
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint:
or .
3: – Logical arrayInput
Note: the dimension of the array select
must be at least
if , and at least otherwise.
On entry: specifies the eigenpairs for which condition numbers are to be computed if . To select condition numbers for the eigenpair corresponding to the eigenvalue , must be set to .TRUE..
Note: the second dimension of the array a
must be at least
.
On entry: the upper triangular matrix .
6: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08yyf is called.
Constraint:
.
7: – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
.
On entry: the upper triangular matrix .
8: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08yyf is called.
Constraint:
.
9: – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array vl
must be at least
if or .
On entry: if or , vl must contain left eigenvectors of , corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by f08wqforf08yxf.
On entry: the first dimension of the array vl as declared in the (sub)program from which f08yyf is called.
Constraints:
if or , ;
otherwise .
11: – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array vr
must be at least
if or .
On entry: if or , vr must contain right eigenvectors of , corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by f08wqforf08yxf.
Note: the dimension of the array dif
must be at least
if or .
On exit: if or , the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute , is set to ; this can only occur when the true value would be very small anyway.
On entry: the number of elements in the arrays s and dif.
Constraints:
if , ;
otherwise .
16: – IntegerOutput
On exit: the number of elements of the arrays s and dif used to store the specified condition numbers; for each selected eigenvalue one element is used.
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
18: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08yyf is called.
If , a workspace query is assumed; the routine only calculates the minimum size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Constraints:
if ,
if or , ;
otherwise .
19: – Integer arrayWorkspace
Note: the dimension of the array iwork
must be at least
.
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.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08yyf 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.
9Further Comments
An approximate asymptotic error bound on the chordal distance between the computed eigenvalue and the corresponding exact eigenvalue is
where is the machine precision.
An approximate asymptotic error bound for the right or left computed eigenvectors or corresponding to the right and left eigenvectors and is given by