### Eigenvalues of a real skew-Hamiltonian/Hamiltonian pencil in factored form

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

Purpose

```  To compute the eigenvalues of a real N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH with

(  0  I  )
S = T Z = J Z' J' Z, where J = (        ),                   (1)
( -I  0  )

via generalized symplectic URV decomposition. That is, orthogonal
matrices Q1 and Q2 and orthogonal symplectic matrices U1 and U2
are computed such that

( T11  T12 )
Q1' T U1 = Q1' J Z' J' U1 = (          ) = Tout,
(  0   T22 )

( Z11  Z12 )
U2' Z Q2 = (          ) = Zout,                              (2)
(  0   Z22 )

( H11  H12 )
Q1' H Q2 = (          ) = Hout,
(  0   H22 )

where T11, T22', Z11, Z22', H11 are upper triangular and H22' is
upper quasi-triangular. The notation M' denotes the transpose of
the matrix M.
Optionally, if COMPQ1 = 'I' or COMPQ1 = 'U', the orthogonal
transformation matrix Q1 will be computed.
Optionally, if COMPQ2 = 'I' or COMPQ2 = 'U', the orthogonal
transformation matrix Q2 will be computed.
Optionally, if COMPU1 = 'I' or COMPU1 = 'U', the orthogonal
symplectic transformation matrix

(  U11  U12  )
U1 = (            )
( -U12  U11  )

will be computed.
Optionally, if COMPU2 = 'I' or COMPU2 = 'U', the orthogonal
symplectic transformation matrix

(  U21  U22  )
U2 = (            )
( -U22  U21  )

will be computed.

```
Specification
```      SUBROUTINE MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ,
\$                   H, LDH, Q1, LDQ1, Q2, LDQ2, U11, LDU11, U12,
\$                   LDU12, U21, LDU21, U22, LDU22, T, LDT, ALPHAR,
\$                   ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK,
\$                   INFO )
C     .. Scalar Arguments ..
CHARACTER          COMPQ1, COMPQ2, COMPU1, COMPU2, JOB
INTEGER            INFO, LDH, LDQ1, LDQ2, LDT, LDU11, LDU12,
\$                   LDU21, LDU22, LDWORK, LDZ, LIWORK, N
C     .. Array Arguments ..
INTEGER            IWORK( * )
DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ),
\$                   DWORK( * ), H( LDH, * ), Q1( LDQ1, * ),
\$                   Q2( LDQ2, * ), T( LDT, * ), U11( LDU11, * ),
\$                   U12( LDU12, * ), U21( LDU21, * ),
\$                   U22( LDU22, * ), Z( LDZ, * )

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; Z, T, and H will not
necessarily be put into the forms in (2); H22' is
upper Hessenberg;
= 'T': put Z, T, and H into the forms in (2), and return
the eigenvalues in ALPHAR, ALPHAI and BETA.

COMPQ1  CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q1, as follows:
= 'N': Q1 is not computed;
= 'I': the array Q1 is initialized internally to the unit
matrix, and the orthogonal matrix Q1 is returned;
= 'U': the array Q1 contains an orthogonal matrix Q01 on
entry, and the product Q01*Q1 is returned, where Q1
is the product of the orthogonal transformations
that are applied to the pencil aT*Z - bH on the
left to reduce T, Z, and H to the forms in (2),
for COMPQ1 = 'I'.

COMPQ2  CHARACTER*1
Specifies whether to compute the orthogonal transformation
matrix Q2, as follows:
= 'N': Q2 is not computed;
= 'I': the array Q2 is initialized internally to the unit
matrix, and the orthogonal matrix Q2 is returned;
= 'U': the array Q2 contains an orthogonal matrix Q02 on
entry, and the product Q02*Q2 is returned, where Q2
is the product of the orthogonal transformations
that are applied to the pencil aT*Z - bH on the
right to reduce T, Z, and H to the forms in (2),
for COMPQ2 = 'I'.

COMPU1  CHARACTER*1
Specifies whether to compute the orthogonal symplectic
transformation matrix U1, as follows:
= 'N': U1 is not computed;
= 'I': the arrays U11 and U12 are initialized internally
to the unit and zero matrices, respectively, and
the corresponding submatrices of the orthogonal
symplectic matrix U1 are returned;
= 'U': the arrays U11 and U12 contain the corresponding
submatrices of an orthogonal symplectic matrix U01
on entry, and the updated submatrices U11 and U12
of the matrix product U01*U1 are returned, where U1
is the product of the orthogonal symplectic
transformations that are applied to the pencil
aT*Z - bH to reduce T, Z, and H to the forms in (2),
for COMPU1 = 'I'.

COMPU2  CHARACTER*1
Specifies whether to compute the orthogonal symplectic
transformation matrix U2, as follows:
= 'N': U2 is not computed;
= 'I': the arrays U21 and U22 are initialized internally
to the unit and zero matrices, respectively, and
the corresponding submatrices of the orthogonal
symplectic matrix U2 are returned;
= 'U': the arrays U21 and U22 contain the corresponding
submatrices of an orthogonal symplectic matrix U02
on entry, and the updated submatrices U21 and U22
of the matrix product U02*U2 are returned, where U2
is the product of the orthogonal symplectic
transformations that are applied to the pencil
aT*Z - bH to reduce T, Z, and H to the forms in (2),
for COMPU2 = 'I'.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the pencil aS - bH.  N >= 0, even.

Z       (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the matrix Z.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the matrix Zout; otherwise, it contains the
matrix Z obtained just before the application of the
periodic QZ algorithm.
The elements of the (2,1) block, i.e., in the rows N/2+1
to N and in the columns 1 to N/2 are not set to zero, but
are unchanged on exit.

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

H       (input/output) DOUBLE PRECISION array, dimension (LDH, N)
On entry, the leading N-by-N part of this array must
contain the Hamiltonian matrix H (H22 = -H11', H12 = H12',
H21 = H21').
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the matrix Hout; otherwise, it contains the
matrix H obtained just before the application of the
periodic QZ algorithm.

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

Q1      (input/output) DOUBLE PRECISION array, dimension (LDQ1, N)
On entry, if COMPQ1 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q01, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q01 and the transformation matrix Q1
used to transform the matrices Z, T and H.
On exit, if COMPQ1 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q1.
If COMPQ1 = 'N', this array is not referenced.

LDQ1    INTEGER
The leading dimension of the array Q1.
LDQ1 >= 1,         if COMPQ1 = 'N';
LDQ1 >= MAX(1, N), if COMPQ1 = 'I' or COMPQ1 = 'U'.

Q2      (input/output) DOUBLE PRECISION array, dimension (LDQ2, N)
On entry, if COMPQ2 = 'U', then the leading N-by-N part of
this array must contain a given matrix Q02, and on exit,
the leading N-by-N part of this array contains the product
of the input matrix Q02 and the transformation matrix Q2
used to transform the matrices Z, T and H.
On exit, if COMPQ2 = 'I', then the leading N-by-N part of
this array contains the orthogonal transformation matrix
Q2.
If COMPQ2 = 'N', this array is not referenced.

LDQ2    INTEGER
The leading dimension of the array Q2.
LDQ2 >= 1,         if COMPQ2 = 'N';
LDQ2 >= MAX(1, N), if COMPQ2 = 'I' or COMPQ2 = 'U'.

U11     (input/output) DOUBLE PRECISION array, dimension
(LDU11, N/2)
On entry, if COMPU1 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper left block of a
given matrix U01, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U11 of
the product of the input matrix U01 and the transformation
matrix U1 used to transform the matrices Z, T, and H.
On exit, if COMPU1 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U11 of the
orthogonal symplectic transformation matrix U1.
If COMPU1 = 'N' this array is not referenced.

LDU11   INTEGER
The leading dimension of the array U11.
LDU11 >= 1,           if COMPU1 = 'N';
LDU11 >= MAX(1, N/2), if COMPU1 = 'I' or COMPU1 = 'U'.

U12     (input/output) DOUBLE PRECISION array, dimension
(LDU12, N/2)
On entry, if COMPU1 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper right block of a
given matrix U01, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U12
of the product of the input matrix U01 and the
transformation matrix U1 used to transform the matrices
Z, T, and H.
On exit, if COMPU1 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U12 of the
orthogonal symplectic transformation matrix U1.
If COMPU1 = 'N' this array is not referenced.

LDU12   INTEGER
The leading dimension of the array U12.
LDU12 >= 1,           if COMPU1 = 'N';
LDU12 >= MAX(1, N/2), if COMPU1 = 'I' or COMPU1 = 'U'.

U21     (input/output) DOUBLE PRECISION array, dimension
(LDU21, N/2)
On entry, if COMPU2 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper left block of a
given matrix U02, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper left block U21 of
the product of the input matrix U02 and the transformation
matrix U2 used to transform the matrices Z, T, and H.
On exit, if COMPU2 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper left block U21 of the
orthogonal symplectic transformation matrix U2.
If COMPU2 = 'N' this array is not referenced.

LDU21   INTEGER
The leading dimension of the array U21.
LDU21 >= 1,           if COMPU2 = 'N';
LDU21 >= MAX(1, N/2), if COMPU2 = 'I' or COMPU2 = 'U'.

U22     (input/output) DOUBLE PRECISION array, dimension
(LDU22, N/2)
On entry, if COMPU2 = 'U', then the leading N/2-by-N/2
part of this array must contain the upper right block of a
given matrix U02, and on exit, the leading N/2-by-N/2 part
of this array contains the updated upper right block U22
of the product of the input matrix U02 and the
transformation matrix U2 used to transform the matrices
Z, T, and H.
On exit, if COMPU2 = 'I', then the leading N/2-by-N/2 part
of this array contains the upper right block U22 of the
orthogonal symplectic transformation matrix U2.
If COMPU2 = 'N' this array is not referenced.

LDU22   INTEGER
The leading dimension of the array U22.
LDU22 >= 1,           if COMPU2 = 'N';
LDU22 >= MAX(1, N/2), if COMPU2 = 'I' or COMPU2 = 'U'.

T       (output) DOUBLE PRECISION array, dimension (LDT, N)
If JOB = 'T', the leading N-by-N part of this array
contains the matrix Tout; otherwise, it contains the
matrix T obtained just before the application of the
periodic QZ algorithm.

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

ALPHAR  (output) DOUBLE PRECISION array, dimension (N/2)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.

ALPHAI  (output) DOUBLE PRECISION array, dimension (N/2)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.

BETA    (output) DOUBLE PRECISION array, dimension (N/2)
The scalars beta defining the eigenvalues of the pencil
aS - bH.
If INFO = 0, the quantities alpha = (ALPHAR(j),ALPHAI(j)),
and beta = BETA(j) represent together the j-th eigenvalue
of the pencil aS - bH, in the form lambda = alpha/beta.
Since lambda may overflow, the ratios should not, in
general, be computed. Due to the skew-Hamiltonian/
Hamiltonian structure of the pencil, only half of the
spectrum is saved in ALPHAR, ALPHAI and BETA.
Specifically, only eigenvalues with imaginary parts
greater than or equal to zero are stored; their conjugate
eigenvalues are not stored. If imaginary parts are zero
(i.e., for real eigenvalues), only positive eigenvalues
are stored. The remaining eigenvalues have opposite signs.
If INFO = 3, one or more BETA(j) is not representable, and
the eigenvalues are returned as described below.

```
Workspace
```  IWORK   INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1), ..., IWORK(N/2) return the
scaling parameters for the eigenvalues of the pencil
aS - bH (see INFO = 3).

LIWORK  INTEGER
The dimension of the array IWORK.
LIWORK >= N+18.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and DWORK(2) returns the machine base, b.
On exit, if INFO = -31, DWORK(1) returns the minimum value
of LDWORK.

LDWORK  INTEGER
The dimension of the array DWORK.
If JOB = 'E' and COMPQ1 = 'N' and COMPQ2 = 'N' and
COMPU1 = 'N' and COMPU2 = 'N', then
LDWORK >= 3/2*N**2+MAX(N, 48)+6;
else, LDWORK >=   3*N**2+MAX(N, 48)+6.
For good performance LDWORK should generally be larger.

If LDWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the DWORK
array, returns this value as the first entry of the DWORK
array, and no error message related to LDWORK is issued by
XERBLA.

```
Error Indicator
```  INFO    INTEGER
= 0: succesful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the periodic QZ algorithm was not able to reveal
information about the eigenvalues from the 2-by-2
blocks in the SLICOT Library routine MB03BD;
= 2: the periodic QZ algorithm did not converge in the
SLICOT Library routine MB03BD;
= 3: the eigenvalues will under- or overflow if evaluated;
therefore, the j-th eigenvalue is represented by
the quantities alpha = (ALPHAR(j),ALPHAI(j)),
beta = BETA(j), and gamma = IWORK(j) in the form
lambda = (alpha/beta) * b**gamma, where b is the
machine base (often 2.0). This is not an error.

```
Method
```  The algorithm uses Givens rotations and Householder reflections to
annihilate elements in T, Z, and H such that T11, T22', Z11, Z22'
and H11 are upper triangular and H22' is upper Hessenberg. Finally
the periodic QZ algorithm is applied to transform H22' to upper
quasi-triangular form while T11, T22', Z11, Z22', and H11 stay in
upper triangular form.

```
References
```   Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
Eigenproblems.
Tech. Rep., Technical University Chemnitz, Germany,
Nov. 2007.

```
Numerical Aspects
```                                                            3
The algorithm is numerically backward stable and needs O(N ) real
floating point operations.

```
```  None
```
Example

Program Text

```*     MB04AD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER            NIN, NOUT
PARAMETER          ( NIN = 5, NOUT = 6 )
INTEGER            NMAX
PARAMETER          ( NMAX = 50 )
INTEGER            LDH, LDQ1, LDQ2, LDT, LDU11, LDU12, LDU21,
\$                   LDU22, LDWORK, LDZ, LIWORK
PARAMETER          ( LDH = NMAX, LDQ1  = NMAX,   LDQ2 = NMAX,
\$                     LDT = NMAX, LDU11 = NMAX/2, LDU12 = NMAX/2,
\$                     LDU21  = NMAX/2, LDU22 = NMAX/2,
\$                     LDWORK = 3*NMAX*NMAX + MAX( NMAX, 48 ) + 6,
\$                     LDZ = NMAX,  LIWORK = NMAX + 18 )
DOUBLE PRECISION  ZERO
PARAMETER         ( ZERO = 0.0D0 )
*
*     .. Local Scalars ..
CHARACTER          COMPQ1, COMPQ2, COMPU1, COMPU2, JOB
INTEGER            I, INFO, J, M, N
*
*     .. Local Arrays ..
INTEGER            IWORK( LIWORK )
DOUBLE PRECISION   ALPHAI( NMAX/2 ), ALPHAR( NMAX/2 ),
\$                   BETA( NMAX/2 ), DWORK( LDWORK ),
\$                   H( LDH, NMAX ), Q1( LDQ1, NMAX ),
\$                   Q2( LDQ2, NMAX ), T( LDT, NMAX ),
\$                   U11( LDU11, NMAX/2 ), U12( LDU12, NMAX/2 ),
\$                   U21( LDU21, NMAX/2 ), U22( LDU22, NMAX/2 ),
\$                   Z( LDZ, NMAX )
*
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*
*     .. External Subroutines ..
*
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*
*     .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( ( Z( I, J ), J = 1, N ), I = 1, N )
READ( NIN, FMT = * ) ( ( H( I, J ), J = 1, N ), I = 1, N )
*        Compute the eigenvalues of a real skew-Hamiltonian/Hamiltonian
*        pencil (factored version).
CALL MB04AD( JOB, COMPQ1, COMPQ2, COMPU1, COMPU2, N, Z, LDZ, H,
\$                LDH, Q1, LDQ1, Q2, LDQ2, U11, LDU11, U12, LDU12,
\$                U21, LDU21, U22, LDU22, T, LDT, ALPHAR, ALPHAI,
\$                BETA, IWORK, LIWORK, DWORK, LDWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
M = N/2
CALL DLASET( 'Full', M, M, ZERO, ZERO, Z( M+1, 1 ), LDZ )
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( T( I, J ), J = 1, N )
10       CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Z( I, J ), J = 1, N )
20       CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( H( I, J ), J = 1, N )
30       CONTINUE
IF( LSAME( COMPQ1, 'I' ) ) THEN
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q1( I, J ), J = 1, N )
40          CONTINUE
END IF
IF( LSAME( COMPQ2, 'I' ) ) THEN
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( Q2( I, J ), J = 1, N )
50          CONTINUE
END IF
IF( LSAME( COMPU1, 'I' ) ) THEN
WRITE( NOUT, FMT = 99990 )
DO 60 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U11( I, J ), J = 1, M )
60          CONTINUE
WRITE( NOUT, FMT = 99989 )
DO 70 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U12( I, J ), J = 1, M )
70          CONTINUE
END IF
IF( LSAME( COMPU2, 'I' ) ) THEN
WRITE( NOUT, FMT = 99988 )
DO 80 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U21( I, J ), J = 1, M )
80          CONTINUE
WRITE( NOUT, FMT = 99987 )
DO 90 I = 1, M
WRITE( NOUT, FMT = 99995 ) ( U22( I, J ), J = 1, M )
90          CONTINUE
END IF
WRITE( NOUT, FMT = 99986 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAR( I ), I = 1, M )
WRITE( NOUT, FMT = 99985 )
WRITE( NOUT, FMT = 99995 ) ( ALPHAI( I ), I = 1, M )
WRITE( NOUT, FMT = 99984 )
WRITE( NOUT, FMT = 99995 ) (   BETA( I ), I = 1, M )
END IF
END IF
STOP
*
99999 FORMAT( 'MB04AD EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB04AD = ', I2 )
99996 FORMAT( 'The matrix T on exit is ' )
99995 FORMAT( 50( 1X, F8.4 ) )
99994 FORMAT( 'The matrix Z on exit is ' )
99993 FORMAT( 'The matrix H is ' )
99992 FORMAT( 'The matrix Q1 is ' )
99991 FORMAT( 'The matrix Q2 is ' )
99990 FORMAT( 'The upper left block of the matrix U1 is ' )
99989 FORMAT( 'The upper right block of the matrix U1 is ' )
99988 FORMAT( 'The upper left block of the matrix U2 is ' )
99987 FORMAT( 'The upper right block of the matrix U2 is ' )
99986 FORMAT( 'The vector ALPHAR is ' )
99985 FORMAT( 'The vector ALPHAI is ' )
99984 FORMAT( 'The vector BETA is ' )
END
```
Program Data
```MB04AD EXAMPLE PROGRAM DATA
T   I   I   I   I   8
3.1472    4.5751   -0.7824    1.7874   -2.2308   -0.6126    2.0936    4.5974
4.0579    4.6489    4.1574    2.5774   -4.5383   -1.1844    2.5469   -1.5961
-3.7301   -3.4239    2.9221    2.4313   -4.0287    2.6552   -2.2397    0.8527
4.1338    4.7059    4.5949   -1.0777    3.2346    2.9520    1.7970   -2.7619
1.3236    4.5717    1.5574    1.5548    1.9483   -3.1313    1.5510    2.5127
-4.0246   -0.1462   -4.6429   -3.2881   -1.8290   -0.1024   -3.3739   -2.4490
-2.2150    3.0028    3.4913    2.0605    4.5022   -0.5441   -3.8100    0.0596
0.4688   -3.5811    4.3399   -4.6817   -4.6555    1.4631   -0.0164    1.9908
3.9090   -3.5071    3.1428   -3.0340   -1.4834    3.7401   -0.1715    0.4026
4.5929   -2.4249   -2.5648   -2.4892    3.7401   -2.1416    1.6251    2.6645
0.4722    3.4072    4.2926    1.1604   -0.1715    1.6251   -4.2415   -0.0602
-3.6138   -2.4572   -1.5002   -0.2671    0.4026    2.6645   -0.0602   -3.7009
0.6882   -1.8421   -4.1122    0.1317   -3.9090   -4.5929   -0.4722    3.6138
-1.8421    2.9428   -0.4340    1.3834    3.5071    2.4249   -3.4072    2.4572
-4.1122   -0.4340   -2.3703    0.5231   -3.1428    2.5648   -4.2926    1.5002
0.1317    1.3834    0.5231   -4.1618    3.0340    2.4892   -1.1604    0.2671
```
Program Results
```MB04AD EXAMPLE PROGRAM RESULTS
The matrix T on exit is
-3.9699   3.7658   5.5815  -1.7750  -0.8818  -0.0511  -4.2158   1.9054
0.0000   5.3686  -5.9166   4.9163   1.3839   0.8870   3.9458  -4.9167
0.0000   0.0000   5.9641   1.9432  -2.0680   2.4402  -1.4091   5.8512
0.0000   0.0000   0.0000   5.9983  -3.8172   4.0147  -2.0739  -1.2570
0.0000   0.0000   0.0000   0.0000   8.2005   0.0000   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   1.5732   8.0098   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   0.6017   2.4397   5.9751   0.0000
0.0000   0.0000   0.0000   0.0000  -2.5869   0.5598   0.2544   5.2129
The matrix Z on exit is
-6.4705  -2.5511  -4.0551  -1.9895  -2.7642   0.7532  -4.1047  -2.2046
0.0000   7.3589  -4.4480  -2.7491  -1.5465  -1.4345  -0.9272   1.3121
0.0000   0.0000   4.9125  -0.4968   5.3574   3.8579   5.2547  -1.7324
0.0000   0.0000   0.0000   9.0822   0.0460  -0.3382   3.9302   3.1084
0.0000   0.0000   0.0000   0.0000   6.1869   0.0000   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   5.5573   6.6549   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000   2.7456  -3.5789   4.3432   0.0000
0.0000   0.0000   0.0000   0.0000   0.1549   3.5335   3.1346   4.1062
The matrix H is
-7.4834   0.4404   2.3558   1.6724  -0.4630   1.9533   1.5724  -2.7254
0.0000  -7.3500   3.7414   3.7466   0.2837   0.6849   0.7727  -4.2140
0.0000   0.0000  -2.3493  -3.7994  -0.6872   1.1773  -2.6901  -5.1494
0.0000   0.0000   0.0000  -3.4719   5.3322   0.4182   1.9779   1.5175
0.0000   0.0000   0.0000   0.0000  -6.1880   0.0000   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000  -3.3324   9.0833   0.0000   0.0000
0.0000   0.0000   0.0000   0.0000  -1.8703   0.0799  -2.8180   0.0000
0.0000   0.0000   0.0000   0.0000  -2.3477   3.3110   0.6561   0.7281
The matrix Q1 is
-0.2489  -0.1409   0.3615   0.6458   0.0113   0.6063  -0.0470   0.0238
-0.2436   0.1294  -0.0874  -0.4103   0.3408   0.3628  -0.3267   0.6272
-0.4316  -0.2352   0.5553  -0.2811  -0.2198  -0.2880  -0.4564  -0.1773
0.1992  -0.2176  -0.5198   0.1561  -0.1523   0.1299  -0.7281  -0.2197
0.0161   0.7390   0.1125  -0.2226  -0.1003   0.3608  -0.1118  -0.4886
-0.5824   0.0984  -0.3052   0.1996   0.5889  -0.2442   0.0060  -0.3341
-0.3246   0.4661  -0.1835   0.3523  -0.5153  -0.3034  -0.0865   0.3931
-0.4559  -0.2961  -0.3790  -0.3127  -0.4356   0.3452   0.3642  -0.1467
The matrix Q2 is
0.0288  -0.1842  -0.6791  -0.2115  -0.4790   0.4212  -0.0417  -0.2253
-0.0666  -0.0787  -0.3711   0.1737  -0.0482  -0.5770  -0.6785   0.1607
0.1506   0.6328   0.0518  -0.6266   0.0652  -0.0790  -0.2854  -0.2994
-0.2900  -0.2737  -0.0076  -0.3671  -0.2017  -0.6241   0.4521  -0.2675
0.3353   0.4107   0.0326   0.1400  -0.6447  -0.2043   0.2561   0.4187
0.0905  -0.1648  -0.2363  -0.5323   0.3180   0.0286   0.1252   0.7126
-0.7246   0.0468   0.3328  -0.1794  -0.3639   0.2257  -0.2623   0.2786
0.4922  -0.5353   0.4803  -0.2501  -0.2723   0.0199  -0.3194  -0.0371
The upper left block of the matrix U1 is
0.4144   0.2249   0.6015  -0.1964
-0.0198   0.5131  -0.2823  -0.3058
-0.6620   0.1508   0.2237   0.0240
-0.0743  -0.4323  -0.0332  -0.7263
The upper right block of the matrix U1 is
-0.3474   0.1306  -0.3391  -0.3530
-0.3760   0.1550   0.6087  -0.1646
0.1707   0.6553  -0.1262  -0.1177
0.3048  -0.0773   0.0767  -0.4173
The upper left block of the matrix U2 is
0.1403  -0.6447  -0.6536  -0.3707
0.7069   0.2609  -0.0091  -0.1702
-0.1218  -0.1120   0.3766  -0.5154
0.0773   0.6349  -0.5070  -0.1810
The upper right block of the matrix U2 is
0.0000   0.0000   0.0000   0.0000
0.1182   0.1587   0.1930  -0.5716
0.6051  -0.2720   0.3364   0.1089
0.2823  -0.0386  -0.1529   0.4434
The vector ALPHAR is
0.0000   0.7122   0.0000   0.7450
The vector ALPHAI is
0.7540   0.0000   0.7465   0.0000
The vector BETA is
4.0000   4.0000   8.0000  16.0000
```