## MB03DZ

### Exchanging eigenvalues of a complex 2-by-2 upper triangular pencil

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute unitary matrices Q1 and Q2 for a complex 2-by-2 regular
pencil aA - bB with A, B upper triangular, such that
Q2' (aA - bB) Q1 is still upper triangular but the eigenvalues are
in reversed order. The matrices Q1 and Q2 are represented by

(  CO1  SI1  )       (  CO2  SI2  )
Q1 = (            ), Q2 = (            ).
( -SI1' CO1  )       ( -SI2' CO2  )

The notation M' denotes the conjugate transpose of the matrix M.

```
Specification
```      SUBROUTINE MB03DZ( A, LDA, B, LDB, CO1, SI1, CO2, SI2 )
C     .. Scalar Arguments ..
INTEGER            LDA, LDB
DOUBLE PRECISION   CO1, CO2
COMPLEX*16         SI1, SI2
C     .. Array Arguments ..
COMPLEX*16         A( LDA, * ), B( LDB, * )

```
Arguments

Input/Output Parameters

```  A       (input) COMPLEX*16 array, dimension (LDA, 2)
On entry, the leading 2-by-2 upper triangular part of
this array must contain the matrix A of the pencil.
The (2,1) entry is not referenced.

LDA     INTEGER
The leading dimension of the array A.  LDA >= 2.

B       (input) COMPLEX*16 array, dimension (LDB, 2)
On entry, the leading 2-by-2 upper triangular part of
this array must contain the matrix B of the pencil.
The (2,1) entry is not referenced.

LDB     INTEGER
The leading dimension of the array B.  LDB >= 2.

CO1     (output) DOUBLE PRECISION
The upper left element of the unitary matrix Q1.

SI1     (output) COMPLEX*16
The upper right element of the unitary matrix Q1.

CO2     (output) DOUBLE PRECISION
The upper left element of the unitary matrix Q2.

SI2     (output) COMPLEX*16
The upper right element of the unitary matrix Q2.

```
Method
```  The algorithm uses unitary transformations as described on page 42
in .

```
References
```   Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.

```
Numerical Aspects
```  The algorithm is numerically backward stable.

```
```  None
```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```