void	f08yfc (Nag_OrderType order, Nag_Boolean wantq, Nag_Boolean wantz, Integer n, double a[], Integer pda, double b[], Integer pdb, double q[], Integer pdq, double z[], Integer pdz, Integer ifst, Integer ilst, NagError *fail)

The function may be called by the names: f08yfc, nag_lapackeig_dtgexc or nag_dtgexc.

3 Description

f08yfc reorders the generalized real

n \times n

matrix pair

(S, T)

in real generalized Schur form, so that the diagonal element or block of

(S, T)

with row index

i_{1}

is moved to row

i_{2}

, using an orthogonal equivalence transformation. That is,

S

and

T

are factorized as

S = \hat{Q} \hat{S} {\hat{Z}}^{T}, T = \hat{Q} \hat{T} {\hat{Z}}^{T},

where

(\hat{S}, \hat{T})

are also in real generalized Schur form.

The pair

(S, T)

are in real generalized Schur form if

S

is block upper triangular with

1 \times 1

and

2 \times 2

diagonal blocks and

T

is upper triangular as returned, for example, by f08xcc, or f08xec with

job = Nag_Schur

S

and

T

are the result of a generalized Schur factorization of a matrix pair

(A, B)

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

then, optionally, the matrices

Q

and

Z

can be updated as

Q \hat{Q}

and

Z \hat{Z}

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

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: $wantq$ – Nag_Boolean Input

On entry: if

wantq = Nag_TRUE

, update the left transformation matrix

Q

wantq = Nag_FALSE

, do not update

Q

3: $wantz$ – Nag_Boolean Input

On entry: if

wantz = Nag_TRUE

, update the right transformation matrix

Z

wantz = Nag_FALSE

, do not update

Z

4: $n$ – Integer Input

On entry:

n

, the order of the matrices

S

and

T

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

S

in the pair

(S, T)

On exit: the updated matrix

\hat{S}

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

T

, in the pair

(S, T)

On exit: the updated matrix

\hat{T}

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: $q [\dim]$ – double Input/Output

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

$\max (1, pdq \times n)$ when $wantq = Nag_TRUE$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Q

is stored in

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

On entry: if

wantq = Nag_TRUE

, the orthogonal matrix

Q

On exit: if

wantq = Nag_TRUE

, the updated matrix

Q \hat{Q}

wantq = Nag_FALSE

, q is not referenced.

10: $pdq$ – Integer Input

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

Constraints:

if $wantq = Nag_TRUE$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

11: $z [\dim]$ – double Input/Output

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

$\max (1, pdz \times n)$ when $wantz = Nag_TRUE$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Z

is stored in

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

On entry: if

wantz = Nag_TRUE

, the orthogonal matrix

Z

On exit: if

wantz = Nag_TRUE

, the updated matrix

Z \hat{Z}

wantz = Nag_FALSE

, z is not referenced.

12: $pdz$ – Integer Input

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

Constraints:

if $wantz = Nag_TRUE$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

13: $ifst$ – Integer * Input/Output

14: $ilst$ – Integer * Input/Output

On entry: the indices

i_{1}

and

i_{2}

that specify the reordering of the diagonal blocks of

(S, T)

. The block with row index ifst is moved to row ilst, by a sequence of swapping between adjacent blocks.

On exit: if ifst pointed on entry to the second row of a

2 \times 2

block, it is changed to point to the first row; ilst always points to the first row of the block in its final position (which may differ from its input value by

+ 1

−1

Constraint:

1 \leq ifst \leq n

and

1 \leq ilst \leq n

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_CONSTRAINT: Constraint: if $wantq = Nag_TRUE$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

Constraint: if $wantz = Nag_TRUE$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
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, $pdq = ⟨ value ⟩$ .
Constraint: $pdq > 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
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_INT_3: On entry, $ifst = ⟨ value ⟩$ , $ilst = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq ifst \leq n$ and $1 \leq ilst \leq n$ .
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_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: The transformed matrix pair would be too far from generalized Schur form; the problem is ill-conditioned. $(S, T)$ may have been partially reordered, and ilst points to the first row of the current position of the block being moved.

7 Accuracy

The computed generalized Schur form is nearly the exact generalized Schur form for nearby matrices

(S + E)

and

(T + F)

, where

{‖ E ‖}_{2} = O ε {‖ S ‖}_{2} and {‖ F ‖}_{2} = O ε {‖ T ‖}_{2},

and

ε

is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details of error bounds for the generalized nonsymmetric eigenproblem.

8 Parallelism and Performance

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

f08yfc 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 complex analogue of this function is f08ytc.

10 Example

This example exchanges blocks

2

and

1

of the matrix pair

(S, T)

, where

S = (\begin{array}{r} 4.0 & 1.0 & 1.0 & 2.0 \\ 0 & 3.0 & 4.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 6.0 \end{array}) and T = (\begin{array}{r} 2.0 & 1.0 & 1.0 & 3.0 \\ 0 & 1.0 & 2.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 2.0 \end{array}) .

f08yf: FL CL CPP AD PY MB

NAG CL Interfacef08yfc (dtgexc)

▸▿ 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
f08yfc (dtgexc)