NAG Library Routine Document

F08NBF (DGEEVX)

Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

2 Specification

SUBROUTINE F08NBF (

BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)

INTEGER	N, LDA, LDVL, LDVR, ILO, IHI, LWORK, IWORK(*), INFO
REAL (KIND=nag_wp)	A(LDA,), WR(), WI(), VL(LDVL,), VR(LDVR,), SCALE(), ABNRM, RCONDE(), RCONDV(), WORK(max(1,LWORK))
CHARACTER(1)	BALANC, JOBVL, JOBVR, SENSE

The routine may be called by its LAPACK name dgeevx.

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}

Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation

D A D^{- 1}

, where

D

is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 4.8.1.2 of Anderson et al. (1999).

Following the optional balancing, the matrix

A

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

Q R

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: $BALANC$ – CHARACTER(1)Input

On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.

$BALANC ='N'$: Do not diagonally scale or permute.
$BALANC ='P'$: Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
$BALANC ='S'$: Diagonally scale the matrix, i.e., replace $A$ by $D A D^{- 1}$ , where $D$ is a diagonal matrix chosen to make the rows and columns of $A$ more equal in norm. Do not permute.
$BALANC ='B'$: Both diagonally scale and permute $A$ .

Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

Constraint:

BALANC ='N'

'P'

'S'

'B'

2: $JOBVL$ – 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.

SENSE ='E'

'B'

, JOBVL must be set to

JOBVL ='V'

Constraint:

JOBVL ='N'

'V'

3: $JOBVR$ – 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.

SENSE ='E'

'B'

, JOBVR must be set to

JOBVR ='V'

Constraint:

JOBVR ='N'

'V'

4: $SENSE$ – CHARACTER(1)Input

On entry: determines which reciprocal condition numbers are computed.

$SENSE ='N'$: None are computed.
$SENSE ='E'$: Computed for eigenvalues only.
$SENSE ='V'$: Computed for right eigenvectors only.
$SENSE ='B'$: Computed for eigenvalues and right eigenvectors.

SENSE ='E'

'B'

, both left and right eigenvectors must also be computed (

JOBVL ='V'

and

JOBVR ='V'

Constraint:

SENSE ='N'

'E'

'V'

'B'

5: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

6: $A (LDA *)$ – 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. If

JOBVL ='V'

JOBVR ='V'

A

contains the real Schur form of the balanced version of the input matrix

A

7: $LDA$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08NBF (DGEEVX) is called.

Constraint:

LDA \geq \max (1, N)

8: $WR (*)$ – REAL (KIND=nag_wp) arrayOutput

9: $WI (*)$ – 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.

10: $VL (LDVL *)$ – 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.

11: $LDVL$ – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08NBF (DGEEVX) is called.

Constraints:

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

12: $VR (LDVR *)$ – 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.

13: $LDVR$ – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08NBF (DGEEVX) is called.

Constraints:

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

14: $ILO$ – INTEGEROutput

15: $IHI$ – INTEGEROutput

On exit: ILO and IHI are integer values determined when

A

was balanced. The balanced

A

has

a_{i j} = 0

i > j

and

j = 1, 2, \dots, ILO - 1

i = IHI + 1, \dots, N

16: $SCALE (*)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array SCALE must be at least

\max (1, N)

On exit: details of the permutations and scaling factors applied when balancing

A

p_{j}

is the index of the row and column interchanged with row and column

j

, and

d_{j}

is the scaling factor applied to row and column

j

, then

$SCALE (j) = p_{j}$ , for $j = 1, 2, \dots, ILO - 1$ ;
$SCALE (j) = d_{j}$ , for $j = ILO, \dots, IHI$ ;
$SCALE (j) = p_{j}$ , for $j = IHI + 1, \dots, N$ .

The order in which the interchanges are made is N to

IHI + 1

, then

1

ILO - 1

17: $ABNRM$ – REAL (KIND=nag_wp)Output

On exit: the

1

-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).

18: $RCONDE (*)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array RCONDE must be at least

\max (1, N)

On exit:

RCONDE (j)

is the reciprocal condition number of the

j

th eigenvalue.

19: $RCONDV (*)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array RCONDV must be at least

\max (1, N)

On exit:

RCONDV (j)

is the reciprocal condition number of the

j

th right eigenvector.

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

On exit: if

INFO = 0

WORK (1)

contains the minimum value of LWORK required for optimal performance.

21: $LWORK$ – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08NBF (DGEEVX) 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, increase LWORK by, say,

N \times nb

, where

nb

is the optimal block size for F08NEF (DGEHRD).

Constraints:

if JOBVL='N' and JOBVR='N',
- if $SENSE ='N'$ , $LWORK \geq \max (1, 2 \times N)$ ;
- otherwise $LWORK \geq \max (1, N^{2} + 6 \times N)$ ;
if JOBVL='V' or JOBVR='V',
- if $SENSE ='N'$ or $'E'$ , $LWORK \geq \max (1, 3 \times N)$ ;
- otherwise $LWORK \geq \max (1, N^{2} + 6 \times N)$ .

22: $IWORK (*)$ – INTEGER arrayWorkspace

Note: the dimension of the array IWORK must be at least

\max (1, 2 \times N - 1)

SENSE ='N'

'E'

, IWORK is not referenced.

23: $INFO$ – 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.

$INFO > 0$: If $INFO = i$ , the $Q R$ algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements $1 : ILO - 1$ and $i + 1 : N$ 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

{‖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 Parallelism and Performance

F08NBF (DGEEVX) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F08NBF (DGEEVX) 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 and positive.

The total number of floating-point operations is proportional to

n^{3}

The complex analogue of this routine is F08NPF (ZGEEVX).

10 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}),

together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.

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 DocumentF08NBF (DGEEVX)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

F08NBF (DGEEVX)