void	nag_dtgexc (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)

3

Description

nag_dtgexc (f08yfc) reorders the generalized real

n

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

1

and

2

2

diagonal blocks and

T

is upper triangular as returned, for example, by nag_dgges (f08xac), or nag_dhgeqz (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 http://www.netlib.org/lapack/lug

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: $wantq$ – Nag_BooleanInput

On entry: if

wantq = Nag_TRUE

, update the left transformation matrix

Q

wantq = Nag_FALSE

, do not update

Q

3: $wantz$ – Nag_BooleanInput

On entry: if

wantz = Nag_TRUE

, update the right transformation matrix

Z

wantz = Nag_FALSE

, do not update

Z

4: $n$ – IntegerInput

On entry:

n

, the order of the matrices

S

and

T

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

S

in the pair

(S, T)

On exit: the updated matrix

\hat{S}

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

T

, in the pair

(S, T)

On exit: the updated matrix

\hat{T}

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: $q [\dim]$ – doubleInput/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: $pdq$ – IntegerInput

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: $z [\dim]$ – doubleInput/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: $pdz$ – IntegerInput

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

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 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_CONSTRAINT: On entry, $wantq = 〈value〉$ , $pdq = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $wantq = Nag_TRUE$ , $pdq \geq \max (1, n)$ ;
otherwise $pdq \geq 1$ .

On entry, $wantz = 〈value〉$ , $pdz = 〈value〉$ and $n = 〈value〉$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.
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

nag_dtgexc (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 nag_ztgexc (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}) .

NAG Library Function Document

nag_dtgexc (f08yfc)

▸▿ 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