F08NNF (ZGEEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08NNF (ZGEEV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08NNF (ZGEEV) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n by n complex nonsymmetric matrix A.

2  Specification

SUBROUTINE F08NNF ( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)
INTEGER  N, LDA, LDVL, LDVR, LWORK, INFO
REAL (KIND=nag_wp)  RWORK(*)
COMPLEX (KIND=nag_wp)  A(LDA,*), W(*), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK))
CHARACTER(1)  JOBVL, JOBVR
The routine may be called by its LAPACK name zgeev.

3  Description

The right eigenvector vj of A satisfies
A vj = λj vj
where λj is the jth eigenvalue of A. The left eigenvector uj of A satisfies
ujH A = λj ujH
where ujH denotes the conjugate transpose of uj.
The matrix A is first reduced to upper Hessenberg form by means of unitary similarity transformations, and the QR algorithm is then used to further reduce the matrix to upper triangular Schur form, T, from which the eigenvalues are computed. Optionally, the eigenvectors of T are also computed and backtransformed to those of A.

4  References

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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Parameters

1:     JOBVL – CHARACTER(1)Input
On entry: if JOBVL='N', the left eigenvectors of A are not computed.
If JOBVL='V', the left eigenvectors of A are computed.
Constraint: JOBVL='N' or 'V'.
2:     JOBVR – CHARACTER(1)Input
On entry: if JOBVR='N', the right eigenvectors of A are not computed.
If JOBVR='V', the right eigenvectors of A are computed.
Constraint: JOBVR='N' or 'V'.
3:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
4:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the n by n matrix A.
On exit: A has been overwritten.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08NNF (ZGEEV) is called.
Constraint: LDAmax1,N.
6:     W(*) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the dimension of the array W must be at least max1,N.
On exit: contains the computed eigenvalues.
7:     VL(LDVL,*) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array VL must be at least max1,N.
On exit: if JOBVL='V', the left eigenvectors uj are stored one after another in the columns of VL, in the same order as their corresponding eigenvalues.
If JOBVL='N', VL is not referenced.
uj=VL:,j, the jth column of VL.
8:     LDVL – INTEGERInput
On entry: the first dimension of the array VL as declared in the (sub)program from which F08NNF (ZGEEV) is called.
Constraints:
  • if JOBVL='V', LDVLmax1,N;
  • otherwise LDVL1.
9:     VR(LDVR,*) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array VR must be at least max1,N.
On exit: if JOBVR='V', the right eigenvectors vj are stored one after another in the columns of VR, in the same order as their corresponding eigenvalues.
If JOBVR='N', VR is not referenced.
vj=VR:,j, the jth column of VR.
10:   LDVR – INTEGERInput
On entry: the first dimension of the array VR as declared in the (sub)program from which F08NNF (ZGEEV) is called.
Constraints:
  • if JOBVR='V', LDVRmax1,N;
  • otherwise LDVR1.
11:   WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
12:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08NNF (ZGEEV) is called.
If LWORK=-1, 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 should be generally larger than the minimum, say N+nb×N , where nb  is the optimal block size for F08NSF (ZGEHRD).
Constraint: LWORKmax1,2×N.
13:   RWORK(*) – REAL (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array RWORK must be at least max1,2×N.
14:   INFO – INTEGEROutput

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,
and ε is the machine precision. See Section 4.8 of
Anderson et al. (1999) for further details.

8  Further Comments

Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real and positive.
The total number of floating point operations is proportional to n3.
The real analogue of this routine is F08NAF (DGEEV).

9  Example

This example finds all the eigenvalues and right eigenvectors of the matrix
A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

9.1  Program Text

Program Text (f08nnfe.f90)

9.2  Program Data

Program Data (f08nnfe.d)

9.3  Program Results

Program Results (f08nnfe.r)


F08NNF (ZGEEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011