Integer, Intent (In)	::	n, lda, ldb, ldvl, ldvr, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,), vl(ldvl,), vr(ldvr,)
Real (Kind=nag_wp), Intent (Out)	::	alphar(n), alphai(n), beta(n), work(max(1,lwork))
Character (1), Intent (In)	::	jobvl, jobvr

C Header Interface

#include nagmk26.h

void

f08waf_ ( const char *jobvl, const char *jobvr, 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, double work[], const Integer *lwork, Integer *info, const Charlen length_jobvl, const Charlen length_jobvr)

The routine may be called by its LAPACK name dggev.

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:

A

is reduced to upper Hessenberg form and at the same time

B

is reduced to upper triangular form.

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)

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.

If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.

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 (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: $jobvl$ – 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'

2: $jobvr$ – 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'

3: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

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 matrix

A

in the pair

(A, B)

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 f08waf (dggev) is called.

Constraint:

lda \geq \max (1, n)

6: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/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.

7: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08waf (dggev) is called.

Constraint:

ldb \geq \max (1, n)

8: $alphar (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: the element

alphar (j)

contains the real part of

α_{j}

9: $alphai (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: the element

alphai (j)

contains the imaginary part of

α_{j}

10: $beta (n)$ – Real (Kind=nag_wp) arrayOutput

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}

11: $vl (ldvl *)$ – 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 the 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)

th 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)

. Each eigenvector will be scaled so the largest component has

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

jobvl ='N'

, vl is not referenced.

12: $ldvl$ – IntegerInput

On entry: the first dimension of the array vl as declared in the (sub)program from which f08waf (dggev) is called.

Constraints:

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

13: $vr (ldvr *)$ – 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 the 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)

th 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)

. Each eigenvector will be scaled so the largest component has

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

jobvr ='N'

, vr is not referenced.

14: $ldvr$ – IntegerInput

On entry: the first dimension of the array vr as declared in the (sub)program from which f08waf (dggev) is called.

Constraints:

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

15: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

16: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08waf (dggev) 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.

Constraint:

lwork \geq \max (1, 8 \times n)

17: $info$ – 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.

$info = 1 to n$: The $Q Z$ iteration failed. No eigenvectors have been calculated, but $alphar (j)$ , $alphai (j)$ , and $beta (j)$ should be correct for $j = info + 1, \dots, n$ .

$info = n + 1$: Unexpected error returned from f08xef (dhgeqz).

$info = n + 2$: Error returned from f08ykf (dtgevc).

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. See Section 4.11 of Anderson et al. (1999) for further details.

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

f08waf (dggev) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08waf (dggev) 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 f08wnf (zggev).

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}) .

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 Document

f08waf (dggev)

▸▿ 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

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