algorithm. nag_dggev (f08wac) is marked as deprecated by LAPACK; the replacement routine is nag_dggev3 (f08wcc) which makes better use of level 3 BLAS.

2

Specification

#include <nag.h>

#include <nagf08.h>

void	nag_dggev (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)

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 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.

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: $order$ – Nag_OrderTypeInput

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $jobvl$ – Nag_LeftVecsTypeInput

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_RightVecsTypeInput

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: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $a [\dim]$ – doubleInput/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: $pda$ – IntegerInput

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: $b [\dim]$ – doubleInput/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: $pdb$ – IntegerInput

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: $alphar [n]$ – doubleOutput

On exit: the element

alphar [j - 1]

contains the real part of

α_{j}

10: $alphai [n]$ – doubleOutput

On exit: the element

alphai [j - 1]

contains the imaginary part of

α_{j}

11: $beta [n]$ – doubleOutput

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: $vl [\dim]$ – doubleOutput

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: $pdvl$ – IntegerInput

On entry: the stride used in the array vl.

Constraints:

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

14: $vr [\dim]$ – doubleOutput

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: $pdvr$ – IntegerInput

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_EIGENVECTORS: A failure occurred in nag_dtgevc (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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation 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

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

nag_dggev (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 nag_zggev (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}) .

NAG Library Function Document

nag_dggev (f08wac)

▸▿ 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