F08NAF (DGEEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08NAF (DGEEV)

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

F08NAF (DGEEV) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n by n real nonsymmetric matrix A.

2  Specification

SUBROUTINE F08NAF ( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
INTEGER  N, LDA, LDVL, LDVR, LWORK, INFO
REAL (KIND=nag_wp)  A(LDA,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK))
CHARACTER(1)  JOBVL, JOBVR
The routine may be called by its LAPACK name dgeev.

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 orthogonal similarity transformations, and the QR algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form, T, with 1 by 1 and 2 by 2 blocks on the main diagonal. The eigenvalues are computed from T, the 2 by 2 blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of T are computed and backtransformed to the eigenvectors 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  Arguments

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:     ALDA* – REAL (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 F08NAF (DGEEV) is called.
Constraint: LDAmax1,N.
6:     WR* – REAL (KIND=nag_wp) arrayOutput
7:     WI* – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the arrays WR and WI must be at least max1,N.
On exit: WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
8:     VLLDVL* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array VL must be at least max1,N if JOBVL='V', and at least 1 otherwise.
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 the jth eigenvalue is real, then uj=VL:j, the jth column of VL. If the jth and j+1st eigenvalues form a complex conjugate pair, then uj=VL:j+i×VL:j+1 and uj+1=VL:j-i×VL:j+1.
If JOBVL='N', VL is not referenced.
9:     LDVL – INTEGERInput
On entry: the first dimension of the array VL as declared in the (sub)program from which F08NAF (DGEEV) is called.
Constraints:
  • if JOBVL='V', LDVL max1,N ;
  • otherwise LDVL1.
10:   VRLDVR* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array VR must be at least max1,N if JOBVR='V', and at least 1 otherwise.
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 the jth eigenvalue is real, then vj=VR:j, the jth column of VR. If the jth and j+1st eigenvalues form a complex conjugate pair, then vj=VR:j+i×VR:j+1 and vj+1=VR:j-i×VR:j+1.
If JOBVR='N', VR is not referenced.
11:   LDVR – INTEGERInput
On entry: the first dimension of the array VR as declared in the (sub)program from which F08NAF (DGEEV) is called.
Constraints:
  • if JOBVR='V', LDVR max1,N ;
  • otherwise LDVR1.
12:   WORKmax1,LWORK – REAL (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, WORK1 contains the minimum value of LWORK required for optimal performance.
13:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08NAF (DGEEV) 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 must generally be larger than the minimum, say, 4×N+nb×N , where nb  is the optimal block size of F08NEF (DGEHRD).
Constraints:
  • if JOBVL='V' or JOBVR='V', LWORK4×N;
  • otherwise LWORKmax1,3×N.
14:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

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
The QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements value to N of WR and WI contain eigenvalues which have converged.

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  Parallelism and Performance

F08NAF (DGEEV) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08NAF (DGEEV) 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.

9  Further Comments

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 n3.
The complex analogue of this routine is F08NNF (ZGEEV).

10  Example

This example finds all the eigenvalues and right eigenvectors of the matrix
A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .  
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.

10.1  Program Text

Program Text (f08nafe.f90)

10.2  Program Data

Program Data (f08nafe.d)

10.3  Program Results

Program Results (f08nafe.r)


F08NAF (DGEEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016