## MB03RD

### Reduction of a real Schur form matrix to a block-diagonal form

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

Purpose

```  To reduce a matrix A in real Schur form to a block-diagonal form
using well-conditioned non-orthogonal similarity transformations.
The condition numbers of the transformations used for reduction
are roughly bounded by PMAX*PMAX, where PMAX is a given value.
The transformations are optionally postmultiplied in a given
matrix X. The real Schur form is optionally ordered, so that
clustered eigenvalues are grouped in the same block.

```
Specification
```      SUBROUTINE MB03RD( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS,
\$                   BLSIZE, WR, WI, TOL, DWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         JOBX, SORT
INTEGER           INFO, LDA, LDX, N, NBLCKS
DOUBLE PRECISION  PMAX, TOL
C     .. Array Arguments ..
INTEGER           BLSIZE(*)
DOUBLE PRECISION  A(LDA,*), DWORK(*), WI(*), WR(*), X(LDX,*)

```
Arguments

Mode Parameters

```  JOBX    CHARACTER*1
Specifies whether or not the transformations are
accumulated, as follows:
= 'N':  The transformations are not accumulated;
= 'U':  The transformations are accumulated in X (the
given matrix X is updated).

SORT    CHARACTER*1
Specifies whether or not the diagonal blocks of the real
Schur form are reordered, as follows:
= 'N':  The diagonal blocks are not reordered;
= 'S':  The diagonal blocks are reordered before each
step of reduction, so that clustered eigenvalues
appear in the same block;
= 'C':  The diagonal blocks are not reordered, but the
"closest-neighbour" strategy is used instead of
the standard "closest to the mean" strategy
(see METHOD);
= 'B':  The diagonal blocks are reordered before each
step of reduction, and the "closest-neighbour"
strategy is used (see METHOD).

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

PMAX    (input) DOUBLE PRECISION
An upper bound for the infinity norm of elementary
submatrices of the individual transformations used for
reduction (see METHOD).  PMAX >= 1.0D0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the matrix A to be block-diagonalized, in real
Schur form.
On exit, the leading N-by-N part of this array contains
the computed block-diagonal matrix, in real Schur
canonical form. The non-diagonal blocks are set to zero.

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

X       (input/output) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if JOBX = 'U', the leading N-by-N part of this
array must contain a given matrix X.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
transformation matrix that reduced A to block-diagonal
form. The transformation matrix is itself a product of
non-orthogonal similarity transformations having elements
with magnitude less than or equal to PMAX.
If JOBX = 'N', this array is not referenced.

LDX     INTEGER
The leading dimension of array X.
LDX >= 1,        if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.

NBLCKS  (output) INTEGER
The number of diagonal blocks of the matrix A.

BLSIZE  (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrix A.

WR,     (output) DOUBLE PRECISION arrays, dimension (N)
WI      These arrays contain the real and imaginary parts,
respectively, of the eigenvalues of the matrix A.

```
Tolerances
```  TOL     DOUBLE PRECISION
The tolerance to be used in the ordering of the diagonal
blocks of the real Schur form matrix.
If the user sets TOL > 0, then the given value of TOL is
used as an absolute tolerance: a block i and a temporarily
fixed block 1 (the first block of the current trailing
submatrix to be reduced) are considered to belong to the
same cluster if their eigenvalues satisfy

| lambda_1 - lambda_i | <= TOL.

If the user sets TOL < 0, then the given value of TOL is
used as a relative tolerance: a block i and a temporarily
fixed block 1 are considered to belong to the same cluster
if their eigenvalues satisfy, for j = 1, ..., N,

| lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.

If the user sets TOL = 0, then an implicitly computed,
default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
is used instead, as a relative tolerance, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
If SORT = 'N' or 'C', this parameter is not referenced.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (N)

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  Consider first that SORT = 'N'. Let

( A    A   )
(  11   12 )
A = (          ),
( 0    A   )
(       22 )

be the given matrix in real Schur form, where initially A   is the
11
first diagonal block of dimension 1-by-1 or 2-by-2. An attempt is
made to compute a transformation matrix X of the form

( I   P )
X = (       )                                               (1)
( 0   I )

(partitioned as A), so that

( A     0  )
-1      (  11      )
X  A X = (          ),
( 0    A   )
(       22 )

and the elements of P do not exceed the value PMAX in magnitude.
An adaptation of the standard method for solving Sylvester
equations [1], which controls the magnitude of the individual
elements of the computed solution [2], is used to obtain matrix P.
When this attempt failed, an 1-by-1 (or 2-by-2) diagonal block of
A  , whose eigenvalue(s) is (are) the closest to the mean of those
22
of A   is selected, and moved by orthogonal similarity
11
transformations in the leading position of A  ; the moved diagonal
22
block is then added to the block A  , increasing its order by 1
11
(or 2). Another attempt is made to compute a suitable
transformation matrix X with the new definitions of the blocks A
11
and A  . After a successful transformation matrix X has been
22
obtained, it postmultiplies the current transformation matrix
(if JOBX = 'U'), and the whole procedure is repeated for the
matrix A  .
22

When SORT = 'S', the diagonal blocks of the real Schur form are
reordered before each step of the reduction, so that each cluster
of eigenvalues, defined as specified in the definition of TOL,
appears in adjacent blocks. The blocks for each cluster are merged
together, and the procedure described above is applied to the
larger blocks. Using the option SORT = 'S' will usually provide
better efficiency than the standard option (SORT = 'N'), proposed
in [2], because there could be no or few unsuccessful attempts
to compute individual transformation matrices X of the form (1).
However, the resulting dimensions of the blocks are usually
larger; this could make subsequent calculations less efficient.

When SORT = 'C' or 'B', the procedure is similar to that for
SORT = 'N' or 'S', respectively, but the block of A   whose
22
eigenvalue(s) is (are) the closest to those of A   (not to their
11
mean) is selected and moved to the leading position of A  . This
22
is called the "closest-neighbour" strategy.

```
References
```  [1] Bartels, R.H. and Stewart, G.W.  T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.

[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.

[3] Demmel, J.
The Condition Number of Equivalence Transformations that
Block Diagonalize Matrix Pencils.
SIAM J. Numer. Anal., 20, pp. 599-610, 1983.

```
Numerical Aspects
```                                    3                     4
The algorithm usually requires 0(N ) operations, but 0(N ) are
possible in the worst case, when all diagonal blocks in the real
Schur form of A are 1-by-1, and the matrix cannot be diagonalized
by well-conditioned transformations.

```
```  The individual non-orthogonal transformation matrices used in the
reduction of A to a block-diagonal form have condition numbers
of the order PMAX*PMAX. This does not guarantee that their product
is well-conditioned enough. The routine can be easily modified to
provide estimates for the condition numbers of the clusters of
eigenvalues.

```
Example

Program Text

```*     MB03RD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX
PARAMETER        ( NMAX = 10 )
INTEGER          LDA, LDX
PARAMETER        ( LDA = NMAX, LDX = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = 3*NMAX )
*     .. Local Scalars ..
CHARACTER*1      JOBX, SORT
INTEGER          I, INFO, J, N, NBLCKS, SDIM
DOUBLE PRECISION PMAX, TOL
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), WI(NMAX), WR(NMAX),
\$                 X(LDX,NMAX)
INTEGER          BLSIZE(NMAX)
LOGICAL          BWORK(NMAX)
*     .. External Functions ..
LOGICAL          SELECT
*     .. External Subroutines ..
EXTERNAL         DGEES, MB03RD
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, PMAX, TOL, JOBX, SORT
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
*        Compute Schur form, eigenvalues and Schur vectors.
CALL DGEES( 'Vectors', 'Not sorted', SELECT, N, A, LDA, SDIM,
\$               WR, WI, X, LDX, DWORK, LDWORK, BWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
*           Block-diagonalization.
CALL MB03RD( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS,
\$                   BLSIZE, WR, WI, TOL, DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99995 ) NBLCKS
WRITE ( NOUT, FMT = 99994 ) ( BLSIZE(I), I = 1,NBLCKS )
WRITE ( NOUT, FMT = 99993 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( A(I,J), J = 1,N )
10          CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99992 ) ( X(I,J), J = 1,N )
20          CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from DGEES  = ',I2)
99997 FORMAT (' INFO on exit from MB03RD = ',I2)
99995 FORMAT (' The number of blocks is ',I5)
99994 FORMAT (' The orders of blocks are ',/(20(I3,2X)))
99993 FORMAT (' The block-diagonal matrix is ')
99992 FORMAT (8X,20(1X,F8.4))
99991 FORMAT (' The transformation matrix is ')
99972 FORMAT (/' N is out of range.',/' N = ',I5)
END
```
Program Data
``` MB03RD EXAMPLE PROGRAM DATA
8   1.D03   1.D-2     U     S
1.   -1.    1.    2.    3.    1.    2.    3.
1.    1.    3.    4.    2.    3.    4.    2.
0.    0.    1.   -1.    1.    5.    4.    1.
0.    0.    0.    1.   -1.    3.    1.    2.
0.    0.    0.    1.    1.    2.    3.   -1.
0.    0.    0.    0.    0.    1.    5.    1.
0.    0.    0.    0.    0.    0.    0.99999999   -0.99999999
0.    0.    0.    0.    0.    0.    0.99999999    0.99999999
```
Program Results
``` MB03RD EXAMPLE PROGRAM RESULTS

The number of blocks is     2
The orders of blocks are
6    2
The block-diagonal matrix is
1.0000  -1.0000  -1.2247  -0.7071  -3.4186   1.4577   0.0000   0.0000
1.0000   1.0000   0.0000   1.4142  -5.1390   3.1637   0.0000   0.0000
0.0000   0.0000   1.0000  -1.7321  -0.0016   2.0701   0.0000   0.0000
0.0000   0.0000   0.5774   1.0000   0.7516   1.1379   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   1.0000  -5.8606   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   0.1706   1.0000   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.0000  -0.8850
0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.0000
The transformation matrix is
1.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.9045   0.1957
0.0000   1.0000   0.0000   0.0000   0.0000   0.0000  -0.3015   0.9755
0.0000   0.0000   0.8165   0.0000  -0.5768  -0.0156  -0.3015   0.0148
0.0000   0.0000  -0.4082   0.7071  -0.5768  -0.0156   0.0000  -0.0534
0.0000   0.0000  -0.4082  -0.7071  -0.5768  -0.0156   0.0000   0.0801
0.0000   0.0000   0.0000   0.0000  -0.0276   0.9805   0.0000   0.0267
0.0000   0.0000   0.0000   0.0000   0.0332  -0.0066   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   0.0011   0.1948   0.0000   0.0000
```