## MB03ZA

### Reordering a selected cluster of eigenvalues of a given matrix pair in periodic Schur form

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

Purpose

```  1. To compute, for a given matrix pair (A,B) in periodic Schur
form, orthogonal matrices Ur and Vr so that

T           [ A11  A12 ]     T           [ B11  B12 ]
Vr * A * Ur = [          ],  Ur * B * Vr = [          ], (1)
[  0   A22 ]                 [  0   B22 ]

is in periodic Schur form, and the eigenvalues of A11*B11
form a selected cluster of eigenvalues.

2. To compute an orthogonal matrix W so that

T  [  0  -A11 ]       [  R11   R12 ]
W * [          ] * W = [            ],           (2)
[ B11   0  ]       [   0    R22 ]

where the eigenvalues of R11 and -R22 coincide and have
positive real part.

Optionally, the matrix C is overwritten by Ur'*C*Vr.

All eigenvalues of A11*B11 must either be complex or real and
negative.

```
Specification
```      SUBROUTINE MB03ZA( COMPC, COMPU, COMPV, COMPW, WHICH, SELECT, N,
\$                   A, LDA, B, LDB, C, LDC, U1, LDU1, U2, LDU2, V1,
\$                   LDV1, V2, LDV2, W, LDW, WR, WI, M, DWORK,
\$                   LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         COMPC, COMPU, COMPV, COMPW, WHICH
INTEGER           INFO, LDA, LDB, LDC, LDU1, LDU2, LDV1, LDV2,
\$                  LDW, LDWORK, M, N
C     .. Array Arguments ..
LOGICAL           SELECT(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
\$                  U1(LDU1,*), U2(LDU2,*), V1(LDV1,*), V2(LDV2,*),
\$                  W(LDW,*), WI(*), WR(*)

```
Arguments

Mode Parameters

```  COMPC   CHARACTER*1
= 'U':  update the matrix C;
= 'N':  do not update C.

COMPU   CHARACTER*1
= 'U':  update the matrices U1 and U2;
= 'N':  do not update U1 and U2.
See the description of U1 and U2.

COMPV   CHARACTER*1
= 'U':  update the matrices V1 and V2;
= 'N':  do not update V1 and V2.
See the description of V1 and V2.

COMPW   CHARACTER*1
Indicates whether or not the user wishes to accumulate
the matrix W as follows:
= 'N':  the matrix W is not required;
= 'I':  W is initialized to the unit matrix and the
orthogonal transformation matrix W is returned;
= 'V':  W must contain an orthogonal matrix Q on entry,
and the product Q*W is returned.

WHICH   CHARACTER*1
= 'A':  select all eigenvalues, this effectively means
that Ur and Vr are identity matrices and A11 = A,
B11 = B;
= 'S':  select a cluster of eigenvalues specified by
SELECT.

SELECT  LOGICAL array, dimension (N)
If WHICH = 'S', then SELECT specifies the eigenvalues of
A*B in the selected cluster. To select a real eigenvalue
w(j), SELECT(j) must be set to .TRUE.. To select a complex
conjugate pair of eigenvalues w(j) and w(j+1),
corresponding to a 2-by-2 diagonal block in A, both
SELECT(j) and SELECT(j+1) must be set to .TRUE.; a complex
conjugate pair of eigenvalues must be either both included
in the cluster or both excluded.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix A. N >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A of the matrix
pair (A,B) in periodic Schur form.
On exit, the leading M-by-M part of this array contains
the matrix R22 in (2).

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

B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix B of the matrix pair
(A,B) in periodic Schur form.
On exit, the leading N-by-N part of this array is
overwritten.

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

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, if COMPC = 'U', the leading N-by-N part of this
array must contain a general matrix C.
On exit, if COMPC = 'U', the leading N-by-N part of this
array contains the updated matrix Ur'*C*Vr.
If COMPC = 'N' or WHICH = 'A', this array is not
referenced.

LDC     INTEGER
The leading dimension of the array C.  LDC >= 1.
LDC >= N,  if COMPC = 'U' and WHICH = 'S'.

U1      (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
On entry, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain U1, the (1,1)
block of an orthogonal symplectic matrix
U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains U1*Ur.
If COMPU = 'N' or WHICH = 'A', this array is not
referenced.

LDU1    INTEGER
The leading dimension of the array U1.  LDU1 >= 1.
LDU1 >= N,  if COMPU = 'U' and WHICH = 'S'.

U2      (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
On entry, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain U2, the (1,2)
block of an orthogonal symplectic matrix
U = [ U1, U2; -U2, U1 ].
On exit, if COMPU = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains U2*Ur.
If COMPU = 'N' or WHICH = 'A', this array is not
referenced.

LDU2    INTEGER
The leading dimension of the array U2.  LDU2 >= 1.
LDU2 >= N,  if COMPU = 'U' and WHICH = 'S'.

V1      (input/output) DOUBLE PRECISION array, dimension (LDV1,N)
On entry, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain V1, the (1,1)
block of an orthogonal symplectic matrix
V = [ V1, V2; -V2, V1 ].
On exit, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains V1*Vr.
If COMPV = 'N' or WHICH = 'A', this array is not
referenced.

LDV1    INTEGER
The leading dimension of the array V1.  LDV1 >= 1.
LDV1 >= N,  if COMPV = 'U' and WHICH = 'S'.

V2      (input/output) DOUBLE PRECISION array, dimension (LDV2,N)
On entry, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array must contain V2, the (1,2)
block of an orthogonal symplectic matrix
V = [ V1, V2; -V2, V1 ].
On exit, if COMPV = 'U' and WHICH = 'S', the leading
N-by-N part of this array contains V2*Vr.
If COMPV = 'N' or WHICH = 'A', this array is not
referenced.

LDV2    INTEGER
The leading dimension of the array V2.  LDV2 >= 1.
LDV2 >= N,  if COMPV = 'U' and WHICH = 'S'.

W       (input/output) DOUBLE PRECISION array, dimension (LDW,2*M)
On entry, if COMPW = 'V', then the leading 2*M-by-2*M part
of this array must contain a matrix W.
If COMPW = 'I', then W need not be set on entry, W is set
to the identity matrix.
On exit, if COMPW = 'I' or 'V' the leading 2*M-by-2*M part
of this array is post-multiplied by the transformation
matrix that produced (2).
If COMPW = 'N', this array is not referenced.

LDW     INTEGER
The leading dimension of the array W.  LDW >= 1.
LDW >= 2*M,  if COMPW = 'I' or COMPW = 'V'.

WR      (output) DOUBLE PRECISION array, dimension (M)
WI      (output) DOUBLE PRECISION array, dimension (M)
The real and imaginary parts, respectively, of the
eigenvalues of R11. The eigenvalues are stored in the same
order as on the diagonal of R22, with
WR(i) = -R22(i,i) and, if R22(i:i+1,i:i+1) is a 2-by-2
diagonal block, WI(i) > 0 and WI(i+1) = -WI(i).
In exact arithmetic, these eigenvalue are the positive
square roots of the selected eigenvalues of the product
A*B. However, if an eigenvalue is sufficiently
ill-conditioned, then its value may differ significantly.

M       (output) INTEGER
The number of selected eigenvalues.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if  INFO = -28,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= MAX( 1, 4*N, 8*M ).

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  reordering of the product A*B in Step 1 failed
because some eigenvalues are too close to separate;
= 2:  reordering of some submatrix in Step 2 failed
because some eigenvalues are too close to separate;
= 3:  the QR algorithm failed to compute the Schur form
of some submatrix in Step 2;
= 4:  the condition that all eigenvalues of A11*B11 must
either be complex or real and negative is
numerically violated.

```
Method
```  Step 1 is performed using a reordering technique analogous to the
LAPACK routine DTGSEN for reordering matrix pencils [1,2]. Step 2
is an implementation of Algorithm 2 in [3]. It requires O(M*N*N)
floating point operations.

```
References
```  [1] Kagstrom, B.
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] Kagstrom, B. and Poromaa P.:
Computing eigenspaces with specified eigenvalues of a regular
matrix pair (A, B) and condition estimation: Theory,
algorithms and software, Numer. Algorithms, 1996, vol. 12,
pp. 369-407.

[3] Benner, P., Mehrmann, V., and Xu, H.
A new method for computing the stable invariant subspace of a
real Hamiltonian matrix,  J. Comput. Appl. Math., 86,
pp. 17-43, 1997.

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Example

Program Text

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Program Data
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Program Results
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