f08wbf computes for a pair of real nonsymmetric matrices the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the algorithm.
Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.
The routine may be called by the names f08wbf, nagf_lapackeig_dggevx or its LAPACK name dggevx.
3Description
A generalized eigenvalue for a pair of matrices is a scalar or a ratio , such that is singular. It is usually represented as the pair , as there is a reasonable interpretation for , and even for both being zero.
The right eigenvector corresponding to the eigenvalue of satisfies
The left eigenvector corresponding to the eigenvalue of satisfies
where is the conjugate-transpose of .
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem , where and are real, square matrices, are determined using the algorithm. The algorithm consists of four stages:
1. is reduced to upper Hessenberg form and at the same time is reduced to upper triangular form.
2. is further reduced to quasi-triangular form while the triangular form of is maintained. This is the real generalized Schur form of the pair .
3.The quasi-triangular form of is reduced to triangular form and the eigenvalues extracted. This routine does not actually produce the eigenvalues , but instead returns and such that
The division by becomes your responsibility, since may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with and complex conjugates, even though and are not conjugate.
4.If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.
For details of the balancing option, see Section 3 in f08whf.
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
Wilkinson J H (1979) Kronecker's canonical form and the algorithm Linear Algebra Appl.28 285–303
5Arguments
1: – Character(1)Input
On entry: specifies the balance option to be performed.
Do not diagonally scale or permute.
Permute only.
Scale only.
Both permute and scale.
Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information, is recommended.
Constraint:
, , or .
2: – Character(1)Input
On entry: if , do not compute the left generalized eigenvectors.
If , compute the left generalized eigenvectors.
Constraint:
or .
3: – Character(1)Input
On entry: if , do not compute the right generalized eigenvectors.
If , compute the right generalized eigenvectors.
Constraint:
or .
4: – Character(1)Input
On entry: determines which reciprocal condition numbers are computed.
None are computed.
Computed for eigenvalues only.
Computed for eigenvectors only.
Computed for eigenvalues and eigenvectors.
Constraint:
, , or .
5: – IntegerInput
On entry: , the order of the matrices and .
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the matrix in the pair .
On exit: a has been overwritten. If or or both, then contains the first part of the real Schur form of the ‘balanced’ versions of the input and .
7: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08wbf is called.
Constraint:
.
8: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
On entry: the first dimension of the array b as declared in the (sub)program from which f08wbf is called.
Constraint:
.
10: – Real (Kind=nag_wp) arrayOutput
On exit: the element contains the real part of .
11: – Real (Kind=nag_wp) arrayOutput
On exit: the element contains the imaginary part of .
12: – Real (Kind=nag_wp) arrayOutput
On exit: , for , will be the generalized eigenvalues.
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, will always be less than and usually comparable with in magnitude, and will always be less than and usually comparable with .
13: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vl
must be at least
if , and at least otherwise.
On exit: if , the left generalized eigenvectors are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have .
On entry: the first dimension of the array vl as declared in the (sub)program from which f08wbf is called.
Constraints:
if , ;
otherwise .
15: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array vr
must be at least
if , and at least otherwise.
On exit: if , the right generalized eigenvectors are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have .
On entry: the first dimension of the array vr as declared in the (sub)program from which f08wbf is called.
Constraints:
if , ;
otherwise .
17: – IntegerOutput
18: – IntegerOutput
On exit: ilo and ihi are integer values such that and if and or .
If or , and .
19: – Real (Kind=nag_wp) arrayOutput
On exit: details of the permutations and scaling factors applied to the left side of and .
If is the index of the row interchanged with row , and is the scaling factor applied to row , then:
, for ;
, for ;
, for .
The order in which the interchanges are made is n to , then to .
20: – Real (Kind=nag_wp) arrayOutput
On exit: details of the permutations and scaling factors applied to the right side of and .
If is the index of the column interchanged with column , and is the scaling factor applied to column , then:
, for ;
if
, for ;
if
, for .
The order in which the interchanges are made is n to , then to .
21: – Real (Kind=nag_wp)Output
On exit: the -norm of the balanced matrix .
22: – Real (Kind=nag_wp)Output
On exit: the -norm of the balanced matrix .
23: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array rconde
must be at least
.
On exit: if or , the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus , , and the th columns of vl and vr all correspond to the th eigenpair.
Note: the dimension of the array rcondv
must be at least
.
On exit: if or , the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.
On exit: if , contains the minimum value of lwork required for optimal performance.
26: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08wbf is called.
If , a workspace query is assumed; the routine only calculates the optimal 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.
Suggested value:
for optimal performance, lwork must generally be larger than the minimum; increase workspace by, say, , where is the optimal block size.
Constraints:
if ,
if ,
if or and and , ;
otherwise ;
if , ;
if or , .
27: – Integer arrayWorkspace
Note: the dimension of the array iwork
must be at least
.
Note: interpretation of results obtained with the algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of and . It should be noted that if and are both small for any , it may be that no reliance can be placed on any of the computed eigenvalues . You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08wbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08wbf 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
The total number of floating-point operations is proportional to .
This example finds all the eigenvalues and right eigenvectors of the matrix pair ,
where
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix pair is used.
Note that the block size (NB) of assumed in this example is not realistic for such a small problem, but should be suitable for large problems.