NAG FL Interface
f08naf (dgeev)

Settings help

FL Name Style:


FL Specification Language:


1 Purpose

f08naf computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n×n real nonsymmetric matrix A.

2 Specification

Fortran Interface
Subroutine f08naf ( jobvl, jobvr, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, work, lwork, info)
Integer, Intent (In) :: n, lda, ldvl, ldvr, lwork
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), wr(*), wi(*), vl(ldvl,*), vr(ldvr,*)
Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
Character (1), Intent (In) :: jobvl, jobvr
C Header Interface
#include <nag.h>
void  f08naf_ (const char *jobvl, const char *jobvr, const Integer *n, double a[], const Integer *lda, double wr[], double wi[], double vl[], const Integer *ldvl, double vr[], const Integer *ldvr, double work[], const Integer *lwork, Integer *info, const Charlen length_jobvl, const Charlen length_jobvr)
The routine may be called by the names f08naf, nagf_lapackeig_dgeev or 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 1×1 and 2×2 blocks on the main diagonal. The eigenvalues are computed from T, the 2×2 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 https://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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the n×n matrix A.
On exit: a has been overwritten.
5: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08naf is called.
Constraint: ldamax(1,n).
6: wr(*) Real (Kind=nag_wp) array Output
7: wi(*) Real (Kind=nag_wp) array Output
Note: the dimension of the arrays wr and wi must be at least max(1,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: vl(ldvl,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vl must be at least max(1,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+1)st 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 Integer Input
On entry: the first dimension of the array vl as declared in the (sub)program from which f08naf is called.
Constraints:
  • if jobvl='V', ldvl max(1,n) ;
  • otherwise ldvl1.
10: vr(ldvr,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vr must be at least max(1,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+1)st 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 Integer Input
On entry: the first dimension of the array vr as declared in the (sub)program from which f08naf is called.
Constraints:
  • if jobvr='V', ldvr max(1,n) ;
  • otherwise ldvr1.
12: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
13: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08naf 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.
Constraints:
if lwork−1,
  • if jobvl='V' or jobvr='V', lwork4×n;
  • otherwise lworkmax(1,3×n).
14: info Integer Output
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 = O(ε) A2 ,  
and ε is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08naf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08naf 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.

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)