void	f08wqc (Nag_OrderType order, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Integer n, Complex a[], Integer pda, Complex b[], Integer pdb, Complex alpha[], Complex beta[], Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, NagError *fail)

The function may be called by the names: f08wqc, nag_lapackeig_zggev3 or nag_zggev3.

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 generalized eigenvector

v_{j}

corresponding to the generalized eigenvalue

λ_{j}

(A, B)

satisfies

A v_{j} = λ_{j} B v_{j} .

The left generalized eigenvector

u_{j}

corresponding to the generalized 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 complex generalized eigenproblem

A x = λ B x

, where

A

and

B

are complex, square matrices, are determined using the

Q Z

algorithm. The complex

Q Z

algorithm consists of three stages:

1. $A$ is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time $B$ is reduced to upper triangular form.
2. $A$ is further reduced to triangular form while the triangular form of $B$ is maintained and the diagonal elements of $B$ are made real and non-negative. This is the generalized Schur form of the pair $(A, B)$ .
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.
3.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: $a [\dim]$ – Complex 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: $b [\dim]$ – Complex 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: $alpha [n]$ – Complex Output

On exit: see the description of beta.

10: $beta [n]$ – Complex Output

On exit:

alpha [j - 1] / beta [j - 1]

, for

j = 1, 2, \dots, n

, will be the generalized eigenvalues.

Note: the quotients

alpha [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}

11: $vl [\dim]$ – Complex Output

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

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

i

th element of the

j

th vector is stored in

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

On exit: if

jobvl = Nag_LeftVecs

, the left 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 = Nag_NotLeftVecs

, vl is not referenced.

12: $pdvl$ – Integer Input

On entry: the stride used in the array vl.

Constraints:

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

13: $vr [\dim]$ – Complex Output

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

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

i

th element of the

j

th vector is stored in

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

On exit: if

jobvr = Nag_RightVecs

, the right 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 = Nag_NotRightVecs

, vr is not referenced.

14: $pdvr$ – Integer Input

On entry: the stride used in the array vr.

Constraints:

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

15: $fail$ – 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 f08yxc 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 alpha and beta 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.

f08wqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08wqc 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 real analogue of this function is f08wcc.

10 Example

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

(A, B)

, where

A = (\begin{array}{r} - 21.10 - 22.50 i & 53.50 - 50.50 i & - 34.50 + 127.50 i & 7.50 + 0.50 i \\ - 0.46 - 7.78 i & - 3.50 - 37.50 i & - 15.50 + 58.50 i & - 10.50 - 1.50 i \\ 4.30 - 5.50 i & 39.70 - 17.10 i & - 68.50 + 12.50 i & - 7.50 - 3.50 i \\ 5.50 + 4.40 i & 14.40 + 43.30 i & - 32.50 - 46.00 i & - 19.00 - 32.50 i \end{array})

and

B = (\begin{array}{r} 1.00 - 5.00 i & 1.60 + 1.20 i & - 3.00 + 0.00 i & 0.00 - 1.00 i \\ 0.80 - 0.60 i & 3.00 - 5.00 i & - 4.00 + 3.00 i & - 2.40 - 3.20 i \\ 1.00 + 0.00 i & 2.40 + 1.80 i & - 4.00 - 5.00 i & 0.00 - 3.00 i \\ 0.00 + 1.00 i & - 1.80 + 2.40 i & 0.00 - 4.00 i & 4.00 - 5.00 i \end{array}) .

f08wq: FL CL CPP AD PY MB

NAG CL Interfacef08wqc (zggev3)

▸▿ 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
f08wqc (zggev3)