f08xcc computes the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right generalized Schur vectors for a pair of real nonsymmetric matrices .
The function may be called by the names: f08xcc, nag_lapackeig_dgges3 or nag_dgges3.
3Description
The generalized Schur factorization for a pair of real matrices is given by
where and are orthogonal, is upper triangular and is upper quasi-triangular with and diagonal blocks. The generalized eigenvalues, , of are computed from the diagonals of and and satisfy
where is the corresponding generalized eigenvector. is actually returned as the pair such that
since , or even both and can be zero. The columns of and are the left and right generalized Schur vectors of .
Optionally, f08xcc can order the generalized eigenvalues on the diagonals of so that selected eigenvalues are at the top left. The leading columns of and then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
f08xcc computes to have non-negative diagonal elements, and the blocks of correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the algorithm.
4References
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
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
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_LeftVecsTypeInput
On entry: if , do not compute the left Schur vectors.
If , compute the left Schur vectors.
Constraint:
or .
3: – Nag_RightVecsTypeInput
On entry: if , do not compute the right Schur vectors.
If , compute the right Schur vectors.
Constraint:
or .
4: – Nag_SortEigValsTypeInput
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
On entry: an eigenvalue is selected if . If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy after ordering. NE_SCHUR_REORDER_SELECT in this case.
6: – IntegerInput
On entry: , the order of the matrices and .
Constraint:
.
7: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
.
The th element of the matrix is stored in
when ;
when .
On entry: the first of the pair of matrices, .
On exit: a has been overwritten by its generalized Schur form .
8: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint:
.
9: – doubleInput/Output
Note: the dimension, dim, of the array b
must be at least
.
The th element of the matrix is stored in
when ;
when .
On entry: the second of the pair of matrices, .
On exit: b has been overwritten by its generalized Schur form .
10: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint:
.
11: – Integer *Output
On exit: if , .
If , number of eigenvalues (after sorting) for which selctg is Nag_TRUE. (Complex conjugate pairs for which selctg is Nag_TRUE for either eigenvalue count as .)
On exit: , for , will be the generalized eigenvalues.
, and , for , are the diagonals of the complex Schur form that would result if the diagonal blocks of the real Schur form of were further reduced to triangular form using complex unitary transformations.
If is zero, then the th eigenvalue is real; if positive, then the th and st eigenvalues are a complex conjugate pair, with negative.
Note: the quotients and may easily overflow or underflow, and may even be zero. Thus, you should avoid naively computing the ratio . However, alphar and alphai will always be less than and usually comparable with in magnitude, and beta will always be less than and usually comparable with .
15: – doubleOutput
Note: the dimension, dim, of the array vsl
must be at least
when
;
otherwise.
The th element of the matrix is stored in
when ;
when .
On exit: if , vsl will contain the left Schur vectors, .
On entry: the stride separating row or column elements (depending on the value of order) in the array vsr.
Constraints:
if , ;
otherwise .
19: – 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 , ;
otherwise .
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_ITERATION_QZ
The iteration failed. No eigenvectors have been calculated but , and should be correct from element .
The iteration failed with an unexpected error, please contact NAG.
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.
NE_SCHUR_REORDER
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
NE_SCHUR_REORDER_SELECT
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy . This could also be caused by underflow due to scaling.
7Accuracy
The computed generalized Schur factorization satisfies
where
and is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08xcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xcc 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
The total number of floating-point operations is proportional to .