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

Fortran Interface

Subroutine f08wbf (

balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info)

Integer, Intent (In)	::	n, lda, ldb, ldvl, ldvr, lwork
Integer, Intent (Inout)	::	iwork(*)
Integer, Intent (Out)	::	ilo, ihi, info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,), vl(ldvl,), vr(ldvr,), rconde(), rcondv()
Real (Kind=nag_wp), Intent (Out)	::	alphar(n), alphai(n), beta(n), lscale(n), rscale(n), abnrm, bbnrm, work(max(1,lwork))
Logical, Intent (Inout)	::	bwork(*)
Character (1), Intent (In)	::	balanc, jobvl, jobvr, sense

C Header Interface

#include <nag.h>

void

f08wbf_ (const char *balanc, const char *jobvl, const char *jobvr, const char *sense, const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, double alphar[], double alphai[], double beta[], double vl[], const Integer *ldvl, double vr[], const Integer *ldvr, Integer *ilo, Integer *ihi, double lscale[], double rscale[], double *abnrm, double *bbnrm, double rconde[], double rcondv[], double work[], const Integer *lwork, Integer iwork[], logical bwork[], Integer *info, const Charlen length_balanc, const Charlen length_jobvl, const Charlen length_jobvr, const Charlen length_sense)

The routine may be called by the names f08wbf, nagf_lapackeig_dggevx or its LAPACK name dggevx.

3 Description

A generalized eigenvalue for a pair of matrices

(A, B)

is a scalar

λ

or a ratio

α / β = λ

, such that

A - λ B

is singular. It is usually represented as the pair

(α, β)

, as there is a reasonable interpretation for

β = 0

, and even for both being zero.

The right eigenvector

v_{j}

corresponding to the eigenvalue

λ_{j}

(A, B)

satisfies

A v_{j} = λ_{j} B v_{j} .

The left eigenvector

u_{j}

corresponding to the eigenvalue

λ_{j}

(A, B)

satisfies

u_{j}^{H} A = λ_{j} u_{j}^{H} B,

where

u_{j}^{H}

is the conjugate-transpose of

u_{j}

All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem

A x = λ B x

, where

A

and

B

are real, square matrices, are determined using the

Q Z

algorithm. The

Q Z

algorithm consists of four stages:

1. $A$ is reduced to upper Hessenberg form and at the same time $B$ is reduced to upper triangular form.
2. $A$ is further reduced to quasi-triangular form while the triangular form of $B$ is maintained. This is the real generalized Schur form of the pair $(A, B)$ .
3.The quasi-triangular form of $A$ is reduced to triangular form and the eigenvalues extracted. This routine does not actually produce the eigenvalues $λ_{j}$ , but instead returns $α_{j}$ and $β_{j}$ such that

$λ_{j} = α_{j} / β_{j}, j = 1, 2, \dots, n .$

The division by $β_{j}$ becomes your responsibility, since $β_{j}$ may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with $α_{j} / β_{j}$ and $α_{j + 1} / β_{j + 1}$ complex conjugates, even though $α_{j}$ and $α_{j + 1}$ are not conjugate.
4.If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.

For details of the balancing option, see Section 3 in f08whf.

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 (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

Wilkinson J H (1979) Kronecker's canonical form and the

Q Z

algorithm Linear Algebra Appl. 28 285–303

5 Arguments

1: $balanc$ – Character(1) Input

On entry: specifies the balance option to be performed.

$balanc ='N'$: Do not diagonally scale or permute.
$balanc ='P'$: Permute only.
$balanc ='S'$: Scale only.
$balanc ='B'$: Both permute and scale.

Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information,

balanc ='B'

is recommended.

Constraint:

balanc ='N'

'P'

'S'

'B'

2: $jobvl$ – Character(1) Input

On entry: if

jobvl ='N'

, do not compute the left generalized eigenvectors.

jobvl ='V'

, compute the left generalized eigenvectors.

Constraint:

jobvl ='N'

'V'

3: $jobvr$ – Character(1) Input

On entry: if

jobvr ='N'

, do not compute the right generalized eigenvectors.

jobvr ='V'

, compute the right generalized eigenvectors.

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 eigenvectors only.
$sense ='B'$: Computed for eigenvalues and eigenvectors.

Constraint:

sense ='N'

'E'

'V'

'B'

5: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

6: $a (lda, *)$ – 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 matrix

A

in the pair

(A, B)

On exit: a has been overwritten. If

jobvl ='V'

jobvr ='V'

or both, then

A

contains the first part of the real Schur form of the ‘balanced’ versions of the input

A

and

B

7: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08wbf is called.

Constraint:

lda \geq \max (1, n)

8: $b (ldb, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the matrix

B

in the pair

(A, B)

On exit: b has been overwritten.

9: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08wbf is called.

Constraint:

ldb \geq \max (1, n)

10: $alphar (n)$ – Real (Kind=nag_wp) array Output

On exit: the element

alphar (j)

contains the real part of

α_{j}

11: $alphai (n)$ – Real (Kind=nag_wp) array Output

On exit: the element

alphai (j)

contains the imaginary part of

α_{j}

12: $beta (n)$ – Real (Kind=nag_wp) array Output

On exit:

(alphar (j) + alphai (j) \times i) / beta (j)

, for

j = 1, 2, \dots, n

, will be the generalized eigenvalues.

alphai (j)

is zero, then the

j

th eigenvalue is real; if positive, then the

j

th and

(j + 1)

st eigenvalues are a complex conjugate pair, with

alphai (j + 1)

negative.

Note: the quotients

alphar (j) / beta (j)

and

alphai (j) / beta (j)

may easily overflow or underflow, and

beta (j)

may even be zero. Thus, you should avoid naively computing the ratio

α_{j} / β_{j}

. However,

\max | α_{j} |

will always be less than and usually comparable with

{‖ A ‖}_{2}

in magnitude, and

\max | β_{j} |

will always be less than and usually comparable with

{‖ B ‖}_{2}

13: $vl (ldvl, *)$ – Real (Kind=nag_wp) array Output

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 generalized eigenvectors

u_{j}

are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have

| real part | + | imag. part | = 1

jobvl ='N'

, vl is not referenced.

14: $ldvl$ – Integer Input

On entry: the first dimension of the array vl as declared in the (sub)program from which f08wbf is called.

Constraints:

if $jobvl ='V'$ , $ldvl \geq \max (1, n)$ ;
otherwise $ldvl \geq 1$ .

15: $vr (ldvr, *)$ – Real (Kind=nag_wp) array Output

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 generalized eigenvectors

v_{j}

are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have

| real part | + | imag. part | = 1

jobvr ='N'

, vr is not referenced.

16: $ldvr$ – Integer Input

On entry: the first dimension of the array vr as declared in the (sub)program from which f08wbf is called.

Constraints:

if $jobvr ='V'$ , $ldvr \geq \max (1, n)$ ;
otherwise $ldvr \geq 1$ .

17: $ilo$ – Integer Output

18: $ihi$ – Integer Output

On exit: ilo and ihi are integer values such that

a (i, j) = 0

and

b (i, j) = 0

i > j

and

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

i = ihi + 1, \dots, n

balanc ='N'

'S'

ilo = 1

and

ihi = n

19: $lscale (n)$ – Real (Kind=nag_wp) array Output

On exit: details of the permutations and scaling factors applied to the left side of

A

and

B

{pl}_{j}

is the index of the row interchanged with row

j

, and

{dl}_{j}

is the scaling factor applied to row

j

, then:

$lscale (j) = {pl}_{j}$ , for $j = 1, 2, \dots, ilo - 1$ ;
$lscale = {dl}_{j}$ , for $j = ilo, \dots, ihi$ ;
$lscale = {pl}_{j}$ , for $j = ihi + 1, \dots, n$ .

The order in which the interchanges are made is n to

ihi + 1

, then

1

ilo - 1

20: $rscale (n)$ – Real (Kind=nag_wp) array Output

On exit: details of the permutations and scaling factors applied to the right side of

A

and

B

{pr}_{j}

is the index of the column interchanged with column

j

, and

{dr}_{j}

is the scaling factor applied to column

j

, then:

$rscale (j) = {pr}_{j}$ , for $j = 1, 2, \dots, ilo - 1$ ;
if $rscale = {dr}_{j}$ , for $j = ilo, \dots, ihi$ ;
if $rscale = {pr}_{j}$ , for $j = ihi + 1, \dots, n$ .

The order in which the interchanges are made is n to

ihi + 1

, then

1

ilo - 1

21: $abnrm$ – Real (Kind=nag_wp) Output

On exit: the

1

-norm of the balanced matrix

A

22: $bbnrm$ – Real (Kind=nag_wp) Output

On exit: the

1

-norm of the balanced matrix

B

23: $rconde (*)$ – Real (Kind=nag_wp) array Output

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

\max (1, n)

On exit: if

sense ='E'

'B'

, the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus

rconde (j)

rcondv (j)

, and the

j

th columns of vl and vr all correspond to the

j

th eigenpair.

sense ='V'

, rconde is not referenced.

24: $rcondv (*)$ – Real (Kind=nag_wp) array Output

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

\max (1, n)

On exit: if

sense ='V'

'B'

, the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.

sense ='E'

, rcondv is not referenced.

25: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

26: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08wbf 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 workspace by, say,

nb \times n

, where

nb

is the optimal block size.

Constraints:

if $sense ='N'$ ,
- if $balanc ='N'$ or $'P'$ and $jobvl ='N'$ and $jobvr ='N'$ , $lwork \geq \max (1, 2 \times n)$ ;
- otherwise $lwork \geq \max (1, 6 \times n)$ ;
if $sense ='E'$ , $lwork \geq \max (1, 10 \times n)$ ;
if $sense ='B'$ or $sense ='V'$ , $lwork \geq \max (10 \times n, 2 \times n \times (n + 4) + 16)$ .

27: $iwork (*)$ – Integer array Workspace

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

n + 6

sense ='E'

, iwork is not referenced.

28: $bwork (*)$ – Logical array Workspace

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

\max (1, n)

sense ='N'

, bwork is not referenced.

29: $info$ – 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.

$info = 1, \dots, n$: The $Q Z$ iteration failed. No eigenvectors have been calculated but $alphar (j)$ , $alphai (j)$ and $beta (j)$ should be correct from element $⟨ value ⟩$ .

$info = n + 1$: The $Q Z$ iteration failed with an unexpected error, please contact NAG.

$info = n + 2$: A failure occurred in f08ykf while computing generalized eigenvectors.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for nearby matrices

(A + E)

and

(B + F)

, where

{‖ (E, F) ‖}_{F} = O (ε) {‖ (A, B) ‖}_{F},

and

ε

is the machine precision.

An approximate error bound on the chordal distance between the

i

th computed generalized eigenvalue

w

and the corresponding exact eigenvalue

λ

ε \times {‖ abnrm, bbnrm ‖}_{2} / rconde (i) .

An approximate error bound for the angle between the

i

th computed eigenvector

u_{j}

v_{j}

is given by

ε \times {‖ abnrm, bbnrm ‖}_{2} / rcondv (i) .

For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.11 of Anderson et al. (1999).

Note: interpretation of results obtained with the

Q Z

algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of

α_{j}

and

β_{j}

. It should be noted that if

α_{j}

and

β_{j}

are both small for any

j

, it may be that no reliance can be placed on any of the computed eigenvalues

λ_{i} = α_{i} / β_{i}

. You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.

8 Parallelism and Performance

f08wbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08wbf 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

The total number of floating-point operations is proportional to

n^{3}

The complex analogue of this routine is f08wpf.

10 Example

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

(A, B)

, where

A = (\begin{array}{r} 3.9 & 12.5 & - 34.5 & - 0.5 \\ 4.3 & 21.5 & - 47.5 & 7.5 \\ 4.3 & 21.5 & - 43.5 & 3.5 \\ 4.4 & 26.0 & - 46.0 & 6.0 \end{array}) and B = (\begin{array}{r} 1.0 & 2.0 & - 3.0 & 1.0 \\ 1.0 & 3.0 & - 5.0 & 4.0 \\ 1.0 & 3.0 & - 4.0 & 3.0 \\ 1.0 & 3.0 & - 4.0 & 4.0 \end{array}),

together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix pair is used.

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.

f08wb: FL CL CPP AD

NAG FL Interfacef08wbf (dggevx)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 FL Interface
f08wbf (dggevx)