The routine may be called by the names f08nnf, nagf_lapackeig_zgeev or its LAPACK name zgeev.
The right eigenvector of satisfies
where is the th eigenvalue of . The left eigenvector of satisfies
where denotes the conjugate transpose of .
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: if , the left eigenvectors of are not computed.
If , the left eigenvectors of are computed.
2: – Character(1)Input
On entry: if , the right eigenvectors of are not computed.
If , the right eigenvectors of are computed.
3: – IntegerInput
On entry: , the order of the matrix .
4: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the first dimension of the array vr as declared in the (sub)program from which f08nnf is called.
if , ;
11: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
12: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08nnf 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 should be generally larger than the minimum, say , where is the optimal block size for f08nsf.
13: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array rwork
must be at least
14: – 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 have been computed; elements 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.
f08nnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nnf 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 .