NAG Library Routine Document

Integer, Intent (In)	::	n, lda, ldb, ldq, ldz, lwork
Integer, Intent (Inout)	::	ifst, ilst
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,), q(ldq,), z(ldz,)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Logical, Intent (In)	::	wantq, wantz

C Header Interface

#include <nagmk26.h>

void	f08yff_ (const logical wantq, const logical wantz, const Integer n, double a[], const Integer lda, double b[], const Integer ldb, double q[], const Integer ldq, double z[], const Integer ldz, Integer ifst, Integer ilst, double work[], const Integer lwork, Integer *info)

The routine may be called by its LAPACK name dtgexc.

3

Description

f08yff (dtgexc) 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 f08xaf (dgges), or f08xef (dhgeqz) with

job ='S'

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: $wantq$ – LogicalInput

On entry: if

wantq = .TRUE.

, update the left transformation matrix

Q

wantq = .FALSE.

, do not update

Q

2: $wantz$ – LogicalInput

On entry: if

wantz = .TRUE.

, update the right transformation matrix

Z

wantz = .FALSE.

, do not update

Z

3: $n$ – IntegerInput

On entry:

n

, the order of the matrices

S

and

T

Constraint:

n \geq 0

4: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: the matrix

S

in the pair

(S, T)

On exit: the updated matrix

\hat{S}

5: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08yff (dtgexc) is called.

Constraint:

lda \geq \max (1, n)

6: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array b must be at least

\max (1, n)

On entry: the matrix

T

, in the pair

(S, T)

On exit: the updated matrix

\hat{T}

7: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08yff (dtgexc) is called.

Constraint:

ldb \geq \max (1, n)

8: $q (ldq *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array q must be at least

\max (1, n)

wantq = .TRUE.

, and at least

1

otherwise.

On entry: if

wantq = .TRUE.

, the orthogonal matrix

Q

On exit: if

wantq = .TRUE.

, the updated matrix

Q \hat{Q}

wantq = .FALSE.

, q is not referenced.

9: $ldq$ – IntegerInput

On entry: the first dimension of the array q as declared in the (sub)program from which f08yff (dtgexc) is called.

Constraints:

if $wantq = .TRUE.$ , $ldq \geq \max (1, n)$ ;
otherwise $ldq \geq 1$ .

10: $z (ldz *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array z must be at least

\max (1, n)

wantz = .TRUE.

, and at least

1

otherwise.

On entry: if

wantz = .TRUE.

, the orthogonal matrix

Z

On exit: if

wantz = .TRUE.

, the updated matrix

Z \hat{Z}

wantz = .FALSE.

, z is not referenced.

11: $ldz$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08yff (dtgexc) is called.

Constraints:

if $wantz = .TRUE.$ , $ldz \geq \max (1, n)$ ;
otherwise $ldz \geq 1$ .

12: $ifst$ – IntegerInput/Output

13: $ilst$ – IntegerInput/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

14: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

15: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08yff (dtgexc) is called.

lwork = - 1

, a workspace query is assumed; the routine only calculates the minimum size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Constraints:

lwork \neq - 1

if $n \leq 1$ , $lwork \geq 1$ ;
otherwise $lwork \geq 4 \times n + 16$ .

16: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info = 1$: 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

f08yff (dtgexc) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The complex analogue of this routine is f08ytf (ztgexc).

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 Routine Document

f08yff (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

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