Integer n, double a[], Integer pda, double b[], Integer pdb, Integer *sdim, double alphar[], double alphai[], double beta[], double vsl[], Integer pdvsl, double vsr[], Integer pdvsr, NagError *fail)

The function may be called by the names: f08xcc, nag_lapackeig_dgges3 or nag_dgges3.

3 Description

The generalized Schur factorization for a pair of real matrices

(A, B)

is given by

A = Q S Z^{T}, B = Q T Z^{T},

where

Q

and

Z

are orthogonal,

T

is upper triangular and

S

is upper quasi-triangular with

1 \times 1

and

2 \times 2

diagonal blocks. The generalized eigenvalues,

λ

, of

(A, B)

are computed from the diagonals of

S

and

T

and satisfy

A z = λ B z,

where

z

is the corresponding generalized eigenvector.

λ

is actually returned as the pair

(α, β)

such that

λ = α / β

since

β

, or even both

α

and

β

can be zero. The columns of

Q

and

Z

are the left and right generalized Schur vectors of

(A, B)

Optionally, f08xcc can order the generalized eigenvalues on the diagonals of

(S, T)

so that selected eigenvalues are at the top left. The leading columns of

Q

and

Z

then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.

f08xcc computes

T

to have non-negative diagonal elements, and the

2 \times 2

blocks of

S

correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the

Q Z

algorithm.

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

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: $jobvsl$ – Nag_LeftVecsType Input

On entry: if

jobvsl = Nag_NotLeftVecs

, do not compute the left Schur vectors.

jobvsl = Nag_LeftVecs

, compute the left Schur vectors.

Constraint:

jobvsl = Nag_NotLeftVecs

Nag_LeftVecs

3: $jobvsr$ – Nag_RightVecsType Input

On entry: if

jobvsr = Nag_NotRightVecs

, do not compute the right Schur vectors.

jobvsr = Nag_RightVecs

, compute the right Schur vectors.

Constraint:

jobvsr = Nag_NotRightVecs

Nag_RightVecs

4: $sort$ – Nag_SortEigValsType Input

On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.

$sort = Nag_NoSortEigVals$: Eigenvalues are not ordered.
$sort = Nag_SortEigVals$: Eigenvalues are ordered (see selctg).

Constraint:

sort = Nag_NoSortEigVals

Nag_SortEigVals

5: $selctg$ – function, supplied by the user External Function

sort = Nag_SortEigVals

, selctg is used to select generalized eigenvalues to be moved to the top left of the generalized Schur form.

sort = Nag_NoSortEigVals

, selctg is not referenced by f08xcc, and may be specified as NULLFN.

The specification of selctg is:

Nag_Boolean

selctg (double ar, double ai, double b)

1: $ar$ – double Input
2: $ai$ – double Input
3: $b$ – double Input: On entry: an eigenvalue $(ar [j - 1] + \sqrt{- 1} \times ai [j - 1]) / b [j - 1]$ is selected if $selctg (ar [j - 1], ai [j - 1], b [j - 1]) = Nag_TRUE$ . If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy $selctg (ar [j - 1], ai [j - 1], b [j - 1]) = Nag_TRUE$ after ordering. $fail . code =$ NE_SCHUR_REORDER_SELECT in this case.

6: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

7: $a [\dim]$ – 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 first of the pair of matrices,

A

On exit: a has been overwritten by its generalized Schur form

S

8: $pda$ – 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)

9: $b [\dim]$ – 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 second of the pair of matrices,

B

On exit: b has been overwritten by its generalized Schur form

T

10: $pdb$ – 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)

11: $sdim$ – Integer * Output

On exit: if

sort = Nag_NoSortEigVals

sdim = 0

sort = Nag_SortEigVals

sdim =

number of eigenvalues (after sorting) for which selctg is Nag_TRUE. (Complex conjugate pairs for which selctg is Nag_TRUE for either eigenvalue count as

2

12: $alphar [n]$ – double Output

On exit: see the description of beta.

13: $alphai [n]$ – double Output

On exit: see the description of beta.

14: $beta [n]$ – 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.

alphar [j - 1] + alphai [j - 1] \times i

, and

beta [j - 1]

, for

j = 1, 2, \dots, n

, are the diagonals of the complex Schur form

(S, T)

that would result if the

2 \times 2

diagonal blocks of the real Schur form of

(A, B)

were further reduced to triangular form using

2 \times 2

complex unitary transformations.

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 / β

. However, alphar and alphai will always be less than and usually comparable with

{‖ A ‖}_{2}

in magnitude, and beta will always be less than and usually comparable with

{‖ B ‖}_{2}

15: $vsl [\dim]$ – double Output

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

$\max (1, pdvsl \times n)$ when $jobvsl = Nag_LeftVecs$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix is stored in

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

On exit: if

jobvsl = Nag_LeftVecs

, vsl will contain the left Schur vectors,

Q

jobvsl = Nag_NotLeftVecs

, vsl is not referenced.

16: $pdvsl$ – Integer Input

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

Constraints:

if $jobvsl = Nag_LeftVecs$ , $pdvsl \geq \max (1, n)$ ;
otherwise $pdvsl \geq 1$ .

17: $vsr [\dim]$ – double Output

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

$\max (1, pdvsr \times n)$ when $jobvsr = Nag_RightVecs$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix is stored in

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

On exit: if

jobvsr = Nag_RightVecs

, vsr will contain the right Schur vectors,

Z

jobvsr = Nag_NotRightVecs

, vsr is not referenced.

18: $pdvsr$ – Integer Input

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

Constraints:

if $jobvsr = Nag_RightVecs$ , $pdvsr \geq \max (1, n)$ ;
otherwise $pdvsr \geq 1$ .

19: $fail$ – 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_ENUM_INT_2: On entry, $jobvsl = ⟨ value ⟩$ , $pdvsl = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvsl = Nag_LeftVecs$ , $pdvsl \geq \max (1, n)$ ;
otherwise $pdvsl \geq 1$ .

On entry, $jobvsr = ⟨ value ⟩$ , $pdvsr = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvsr = Nag_RightVecs$ , $pdvsr \geq \max (1, n)$ ;
otherwise $pdvsr \geq 1$ .
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, $pdvsl = ⟨ value ⟩$ .
Constraint: $pdvsl > 0$ .

On entry, $pdvsr = ⟨ value ⟩$ .
Constraint: $pdvsr > 0$ .
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.
NE_SCHUR_REORDER: The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
NE_SCHUR_REORDER_SELECT: After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy $selctg = Nag_TRUE$ . This could also be caused by underflow due to scaling.

7 Accuracy

The computed generalized Schur factorization satisfies

A + E = Q S Z^{T}, B + F = Q T Z^{T},

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.

8 Parallelism and Performance

f08xcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08xcc 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 f08xqc.

10 Example

This example finds the generalized Schur factorization 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}),

such that the real positive eigenvalues of

(A, B)

correspond to the top left diagonal elements of the generalized Schur form,

(S, T)

f08xc: FL CL CPP AD

NAG CL Interfacef08xcc (dgges3)

▸▿ 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
f08xcc (dgges3)