void	f08wac (Nag_OrderType order, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Integer n, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double vl[], Integer pdvl, double vr[], Integer pdvr, NagError *fail)

The function may be called by the names: f08wac, nag_lapackeig_dggev or nag_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:

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 function 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.

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: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $jobvl$ – Nag_LeftVecsType Input

On entry: if

jobvl = Nag_NotLeftVecs

, do not compute the left generalized eigenvectors.

jobvl = Nag_LeftVecs

, compute the left generalized eigenvectors.

Constraint:

jobvl = Nag_NotLeftVecs

Nag_LeftVecs

3: $jobvr$ – Nag_RightVecsType Input

On entry: if

jobvr = Nag_NotRightVecs

, do not compute the right generalized eigenvectors.

jobvr = Nag_RightVecs

, compute the right generalized eigenvectors.

Constraint:

jobvr = Nag_NotRightVecs

Nag_RightVecs

4: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times n)

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the matrix

A

in the pair

(A, B)

On exit: a has been overwritten.

6: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, n)

7: $b [\dim]$ – double Input/Output

Note: the dimension, dim, of the array b must be at least

\max (1, pdb \times n)

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the matrix

B

in the pair

(A, B)

On exit: b has been overwritten.

8: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraint:

pdb \geq \max (1, n)

9: $alphar [n]$ – double Output

On exit: the element

alphar [j - 1]

contains the real part of

α_{j}

10: $alphai [n]$ – double Output

On exit: the element

alphai [j - 1]

contains the imaginary part of

α_{j}

11: $beta [n]$ – double Output

On exit:

(alphar [j - 1] + alphai [j - 1] \times i) / beta [j - 1]

, for

j = 1, 2, \dots, n

, will be the generalized eigenvalues.

alphai [j - 1]

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]

negative.

Note: the quotients

alphar [j - 1] / beta [j - 1]

and

alphai [j - 1] / beta [j - 1]

may easily overflow or underflow, and

beta [j - 1]

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}

12: $vl [\dim]$ – double Output

Note: the dimension, dim, of the array vl must be at least

$\max (1, pdvl \times n)$ when $jobvl = Nag_LeftVecs$ ;
$1$ otherwise.

where

VL (i, j)

appears in this document, it refers to the array element

$vl [(j - 1) \times pdvl + i - 1]$ when $order = Nag_ColMajor$ ;
$vl [(i - 1) \times pdvl + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

jobvl = Nag_LeftVecs

, 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 = Nag_NotLeftVecs

, vl is not referenced.

13: $pdvl$ – Integer Input

On entry: the stride used in the array vl.

Constraints:

if $jobvl = Nag_LeftVecs$ , $pdvl \geq \max (1, n)$ ;
otherwise $pdvl \geq 1$ .

14: $vr [\dim]$ – double Output

Note: the dimension, dim, of the array vr must be at least

$\max (1, pdvr \times n)$ when $jobvr = Nag_RightVecs$ ;
$1$ otherwise.

where

VR (i, j)

appears in this document, it refers to the array element

$vr [(j - 1) \times pdvr + i - 1]$ when $order = Nag_ColMajor$ ;
$vr [(i - 1) \times pdvr + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

jobvr = Nag_RightVecs

, 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 = Nag_NotRightVecs

, vr is not referenced.

15: $pdvr$ – Integer Input

On entry: the stride used in the array vr.

Constraints:

if $jobvr = Nag_RightVecs$ , $pdvr \geq \max (1, n)$ ;
otherwise $pdvr \geq 1$ .

16: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_EIGENVECTORS: A failure occurred in f08ykc while computing generalized eigenvectors.
NE_ENUM_INT_2: On entry, $jobvl = ⟨ value ⟩$ , $pdvl = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvl = Nag_LeftVecs$ , $pdvl \geq \max (1, n)$ ;
otherwise $pdvl \geq 1$ .

On entry, $jobvr = ⟨ value ⟩$ , $pdvr = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvr = Nag_RightVecs$ , $pdvr \geq \max (1, n)$ ;
otherwise $pdvr \geq 1$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .

On entry, $pdvl = ⟨ value ⟩$ .
Constraint: $pdvl > 0$ .

On entry, $pdvr = ⟨ value ⟩$ .
Constraint: $pdvr > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_ITERATION_QZ: The $Q Z$ iteration failed. No eigenvectors have been calculated but $alphar [j]$ , $alphai [j]$ and $beta [j]$ should be correct from element $⟨ value ⟩$ .

The $Q Z$ iteration failed with an unexpected error, please contact NAG.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

Background information to multithreading can be found in the Multithreading documentation.

f08wac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08wac 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 function. 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 function is f08wnc.

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}) .

f08wa: FL CL CPP AD PY MB

NAG CL Interfacef08wac (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

NAG CL Interface
f08wac (dggev)