f08npf computes the eigenvalues and, optionally, the left and/or right eigenvectors for an complex nonsymmetric matrix .
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 f08npf, nagf_lapackeig_zgeevx or its LAPACK name zgeevx.
The right eigenvector of satisfies
where is the th eigenvalue of . The left eigenvector of satisfies
where denotes the conjugate transpose of .
Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation , where is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 184.108.40.206 of Anderson et al. (1999).
Following the optional balancing, the matrix is first reduced to upper Hessenberg form by means of unitary similarity transformations, and the algorithm is then used to further reduce the matrix to upper triangular Schur form, , from which the eigenvalues are computed. Optionally, the eigenvectors of are also computed and backtransformed to those of .
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 (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – Character(1)Input
On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.
Do not diagonally scale or permute.
Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
Diagonally scale the matrix, i.e., replace , where is a diagonal matrix chosen to make the rows and columns of more equal in norm. Do not permute.
Both diagonally scale and permute .
Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
, , or .
2: – Character(1)Input
On entry: if , the left eigenvectors of are not computed.
On entry: the first dimension of the array vr as declared in the (sub)program from which f08npf is called.
if , ;
13: – IntegerOutput
14: – IntegerOutput
On exit: ilo and ihi are integer values determined when was balanced. The balanced has if and or .
15: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array scale
must be at least
On exit: details of the permutations and scaling factors applied when balancing .
If is the index of the row and column interchanged with row and column , and is the scaling factor applied to row and column , then
, for ;
, for ;
, for .
The order in which the interchanges are made is n to , then to .
16: – Real (Kind=nag_wp)Output
On exit: the -norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
17: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array rconde
must be at least
On exit: is the reciprocal condition number of the th eigenvalue.
18: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array rcondv
must be at least
On exit: is the reciprocal condition number of the th right eigenvector.
19: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
20: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08npf 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.
for optimal performance, lwork must generally be larger than the minimum, increase lwork by, say, , where is the optimal block size for f08nef.
if or , ;
if or , .
21: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array rwork
must be at least
22: – 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.
The algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements to and to n of w contain eigenvalues which have converged.
The computed eigenvalues and eigenvectors are exact for a nearby matrix , where
Background information to multithreading can be found in the Multithreading documentation.
f08npf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08npf 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.
Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real.
The total number of floating-point operations is proportional to .
This example finds all the eigenvalues and right eigenvectors of the matrix
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.
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.