# dtgex2.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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dtgex2.f

## SYNOPSIS

### Functions/Subroutines

subroutine dtgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO)
DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

## Function/Subroutine Documentation

### subroutine dtgex2 (logical WANTQ, logical WANTZ, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ, integer J1, integer N1, integer N2, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

Purpose:

``` DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
(A, B) by an orthogonal equivalence transformation.

(A, B) must be in generalized real Schur canonical form (as returned
by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
diagonal blocks. B is upper triangular.

Optionally, the matrices Q and Z of generalized Schur vectors are
updated.

Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
```

Parameters:

WANTQ

```          WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
```

WANTZ

```          WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
```

N

```          N is INTEGER
The order of the matrices A and B. N >= 0.
```

A

```          A is DOUBLE PRECISION array, dimensions (LDA,N)
On entry, the matrix A in the pair (A, B).
On exit, the updated matrix A.
```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
```

B

```          B is DOUBLE PRECISION array, dimensions (LDB,N)
On entry, the matrix B in the pair (A, B).
On exit, the updated matrix B.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
```

Q

```          Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
On exit, the updated matrix Q.
Not referenced if WANTQ = .FALSE..
```

LDQ

```          LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N.
```

Z

```          Z is DOUBLE PRECISION array, dimension (LDZ,N)
On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
On exit, the updated matrix Z.
Not referenced if WANTZ = .FALSE..
```

LDZ

```          LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N.
```

J1

```          J1 is INTEGER
The index to the first block (A11, B11). 1 <= J1 <= N.
```

N1

```          N1 is INTEGER
The order of the first block (A11, B11). N1 = 0, 1 or 2.
```

N2

```          N2 is INTEGER
The order of the second block (A22, B22). N2 = 0, 1 or 2.
```

WORK

```          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
LWORK >=  MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
```

INFO

```          INFO is INTEGER
=0: Successful exit
>0: If INFO = 1, the transformed matrix (A, B) would be
too far from generalized Schur form; the blocks are
not swapped and (A, B) and (Q, Z) are unchanged.
The problem of swapping is too ill-conditioned.
<0: If INFO = -16: LWORK is too small. Appropriate value
for LWORK is returned in WORK(1).
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

December 2016

Further Details:

In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

```  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software,
Report UMINF - 94.04, Department of Computing Science, Umea
University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.
```

Definition at line 223 of file dtgex2.f.

## Author

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