NAG Library Routine Document

F08NAF (DGEEV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08NAF (DGEEV) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an

n

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

v_{j}

A

satisfies

A v_{j} = λ_{j} v_{j}

where

λ_{j}

is the

j

th eigenvalue of

A

. The left eigenvector

u_{j}

A

satisfies

u_{j}^{H} A = λ_{j} u_{j}^{H}

where

u_{j}^{H}

denotes the conjugate transpose of

u_{j}

The matrix

A

is first reduced to upper Hessenberg form by means of orthogonal similarity transformations, and the

Q R

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 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.

JOBVL ='V'

, the left eigenvectors of

A

are computed.

Constraint:

JOBVL ='N'

'V'

2: JOBVR – CHARACTER(1)Input

On entry: if

JOBVR ='N'

, the right eigenvectors of

A

are not computed.

JOBVR ='V'

, the right eigenvectors of

A

are computed.

Constraint:

JOBVR ='N'

'V'

3: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

4: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput/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 – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08NAF (DGEEV) is called.

Constraint:

LDA \geq \max (1, 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

\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) arrayOutput

Note: the second dimension of the array VL must be at least

\max (1, N)

JOBVL ='V'

, and at least

1

otherwise.

On exit: if

JOBVL ='V'

, the left eigenvectors

u_{j}

are stored one after another in the columns of VL, in the same order as their corresponding eigenvalues. If the

j

th eigenvalue is real, then

u_{j} = VL (: j)

, the

j

th column of VL. If the

j

th and

(j + 1)

st eigenvalues form a complex conjugate pair, then

u_{j} = VL (: j) + i \times VL (: j + 1)

and

u_{j + 1} = VL (: j) - i \times VL (: j + 1)

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 $JOBVL ='V'$ , $LDVL \geq \max (1, N)$ ;
otherwise $LDVL \geq 1$ .

10: VR(LDVR, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array VR must be at least

\max (1, N)

JOBVR ='V'

, and at least

1

otherwise.

On exit: if

JOBVR ='V'

, the right eigenvectors

v_{j}

are stored one after another in the columns of VR, in the same order as their corresponding eigenvalues. If the

j

th eigenvalue is real, then

v_{j} = VR (: j)

, the

j

th column of VR. If the

j

th and

(j + 1)

st eigenvalues form a complex conjugate pair, then

v_{j} = VR (: j) + i \times VR (: j + 1)

and

v_{j + 1} = VR (: j) - i \times VR (: j + 1)

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 $JOBVR ='V'$ , $LDVR \geq \max (1, N)$ ;
otherwise $LDVR \geq 1$ .

12: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if

INFO = 0

WORK (1)

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.

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 \times N + nb \times N

, where

nb

is the optimal block size of F08NEF (DGEHRD).

Constraints:

if $JOBVL ='V'$ or $JOBVR ='V'$ , $LWORK \geq 4 \times N$ ;
otherwise $LWORK \geq \max (1, 3 \times N)$ .

14: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

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.

$INFO > 0$: If $INFO = i$ , the $Q R$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $i + 1 : N$ of WR WI contain eigenvalues which have converged.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(A + E)

, where

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

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

n^{3}

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

9 Example

This example finds all the eigenvalues and right eigenvectors of the matrix

A = (\begin{array}{r} 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 \end{array}) .

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.

NAG Library Routine DocumentF08NAF (DGEEV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F08NAF (DGEEV)