The function may be called by the names: f08ylc, nag_lapackeig_dtgsna or nag_dtgsna.
3Description
f08ylc estimates condition numbers for specified eigenvalues and/or right eigenvectors of an matrix pair in real generalized Schur form. The function actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.
The pair are in real generalized Schur form if is block upper triangular with and diagonal blocks and is upper triangular as returned, for example, by f08xbcorf08xcc, or f08xec with . The diagonal elements, or blocks, 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.
The definition of the reciprocal of the estimated condition number of the right eigenvector and the left eigenvector corresponding to the simple eigenvalue depends upon whether is a real eigenvalue, or one of a complex conjugate pair.
If the eigenvalue is real and and are orthogonal transformations such that
where and are matrices, then the reciprocal condition number is given by
where denotes the smallest singular value of the matrix
and is the Kronecker product.
If is part of a complex conjugate pair and and are orthogonal transformations such that
where and are two by two matrices, and are matrices, and corresponds to the complex conjugate eigenvalue pair , , then there exist unitary matrices and such that
The eigenvalues are given by and . Then the Frobenius norm-based, estimated reciprocal condition number is bounded by
where denotes the real part of , , is the complex two by two matrix
and is an upper bound on ; i.e., an upper bound on , where is 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: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_JobTypeInput
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 .
3: – Nag_HowManyTypeInput
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 .
4: – const Nag_BooleanInput
Note: the dimension, dim, of the array select
must be at least
On entry: specifies the eigenpairs for which condition numbers are to be computed if . To select condition numbers for the eigenpair corresponding to the real eigenvalue , must be set Nag_TRUE. To select condition numbers corresponding to a complex conjugate pair of eigenvalues and , and/or must be set to Nag_TRUE.
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 f08wccorf08ykc.
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 f08wccorf08ykc.
On exit: if or , the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of s are set to the same value. Thus , , and the th columns of and all correspond to the same eigenpair (but not in general the th eigenpair, unless all eigenpairs are selected).
On exit: if or , the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of dif are set to the same value. 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 .
17: – Integer *Output
On exit: m, the number of elements of the arrays s and dif used to store the specified condition numbers; for each selected real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If , m is set to n.
18: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_ENUM_INT_2
On entry, , and .
Constraint: if or ,
.
On entry, , and .
Constraint: if or , .
On entry, , and .
Constraint: if or ,
.
On entry, , and .
Constraint: if or , .
NE_ENUM_INT_3
On entry, , , and .
Constraint: if , ;
otherwise .
NE_INT
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08ylc 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 function. 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
This example estimates condition numbers and approximate error estimates for all the eigenvalues and eigenvalues and right eigenvectors of the pair given by
The eigenvalues and eigenvectors are computed by calling f08ykc.