## MB04AZ

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

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

Purpose

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

H  T           (  B  F  )       (  Z11  Z12  )
S = J Z  J  Z and H = (      H ), Z =: (            ).       (1)
(  G -B  )       (  Z21  Z22  )

The structured Schur form of the embedded real skew-Hamiltonian/
H  T
skew-Hamiltonian pencil, aB_S - bB_T, with B_S = J B_Z  J  B_Z,

(  Re(Z11)  -Im(Z11)  |  Re(Z12)  -Im(Z12)  )
(                     |                     )
(  Im(Z11)   Re(Z11)  |  Im(Z12)   Re(Z12)  )
(                     |                     )
B_Z = (---------------------+---------------------) ,
(                     |                     )
(  Re(Z21)  -Im(Z21)  |  Re(Z22)  -Im(Z22)  )
(                     |                     )
(  Im(Z21)   Re(Z21)  |  Im(Z22)   Re(Z22)  )
(2)
( -Im(B)  -Re(B)  | -Im(F)  -Re(F)  )
(                 |                 )
(  Re(B)  -Im(B)  |  Re(F)  -Im(F)  )
(                 |                 )
B_T = (-----------------+-----------------) ,  T = i*H,
(                 |      T       T  )
( -Im(G)  -Re(G)  | -Im(B )  Re(B ) )
(                 |      T       T  )
(  Re(G)  -Im(G)  | -Re(B ) -Im(B ) )

is determined and used to compute the eigenvalues. Optionally,
if JOB = 'T', the pencil aB_S - bB_H is transformed by a unitary
matrix Q and a unitary symplectic matrix U to the structured Schur
H  T
form aB_Sout - bB_Hout, with B_Sout = J B_Zout  J  B_Zout,

( BA  BD  )              ( BB  BF  )
B_Zout = (         ) and B_Hout = (       H ),               (3)
(  0  BC  )              (  0 -BB  )

where BA and BB are upper triangular, BC is lower triangular,
and BF is Hermitian. The embedding doubles the multiplicities of
the eigenvalues of the pencil aS - bH.
Optionally, if COMPQ = 'C', the unitary matrix Q is computed.
Optionally, if COMPU = 'C', the unitary symplectic matrix U is
computed.

```
Specification
```      SUBROUTINE MB04AZ( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG,
\$                   LDFG, D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR,
\$                   ALPHAI, BETA, IWORK, LIWORK, DWORK, LDWORK,
\$                   ZWORK, LZWORK, BWORK, INFO )C     .. Scalar Arguments ..
CHARACTER          COMPQ, COMPU, JOB
INTEGER            INFO, LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK,
\$                   LDZ, LIWORK, LZWORK, N
C     .. Array Arguments ..
LOGICAL            BWORK( * )
INTEGER            IWORK( * )
DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16         B( LDB, * ), C( LDC, * ), D( LDD, * ),
\$                   FG( LDFG, * ), Q( LDQ, * ), U( LDU, * ),
\$                   Z( LDZ, * ), ZWORK( * )

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (3).
= 'T': put S and H into the forms in (3) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.

COMPQ   CHARACTER*1
Specifies whether to compute the unitary transformation
matrix Q, as follows:
= 'N': do not compute the unitary matrix Q;
= 'C': the array Q is initialized internally to the unit
matrix, and the unitary matrix Q is returned.

COMPU   CHARACTER*1
Specifies whether to compute the unitary symplectic
transformation matrix U, as follows:
= 'N': do not compute the unitary symplectic matrix U;
= 'C': the array U is initialized internally to the unit
matrix, and the unitary symplectic matrix U is
returned.

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

Z       (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, the leading N-by-N part of this array must
contain the non-trivial factor Z in the factorization
H  T
S = J Z  J  Z of the skew-Hamiltonian matrix S.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BA in (3)
not zeroed. The submatrix in the rows N/2+1 to N and the
first N/2 columns is unchanged, except possibly for the
entry (N/2+1,N/2), which might be set to zero.
If JOB = 'E', this array is unchanged on exit.

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

B       (input/output) COMPLEX*16 array, dimension (LDB, K), where
K = N, if JOB = 'T', and K = M, if JOB = 'E'.
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BB in (3)
The strictly lower triangular part is not zeroed.
If JOB = 'E', this array is unchanged on exit.

LDB     INTEGER
The leading dimension of the array B.
LDB >= MAX(1, M), if JOB = 'E';
LDB >= MAX(1, N), if JOB = 'T'.

FG      (input/output) COMPLEX*16 array, dimension (LDFG, P),
where P = MAX(M+1,N), if JOB = 'T', and
P = M+1,        if JOB = 'E'.
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
Hermitian matrix F.
On exit, if JOB = 'T', the leading N-by-N part of this
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts.
If JOB = 'E', this array is unchanged on exit.

LDFG    INTEGER
The leading dimension of the array FG.
LDFG >= MAX(1, M), if JOB = 'E';
LDFG >= MAX(1, N), if JOB = 'T'.

D       (output) COMPLEX*16 array, dimension (LDD, N)
If JOB = 'T', the leading N-by-N part of this array
If JOB = 'E', this array is not referenced.

LDD     INTEGER
The leading dimension of the array D.
LDD >= 1,         if JOB = 'E';
LDD >= MAX(1, N), if JOB = 'T'.

C       (output) COMPLEX*16 array, dimension (LDC, N)
If JOB = 'T', the leading N-by-N part of this array
METHOD).
If JOB = 'E', this array is not referenced.

LDC     INTEGER
The leading dimension of the array C.
LDC >= 1,         if JOB = 'E';
LDC >= MAX(1, N), if JOB = 'T'.

Q       (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C' and JOB = 'T', then the leading
2*N-by-2*N part of this array contains the unitary
transformation matrix Q.
If COMPQ = 'C' and JOB = 'E', this array contains the
orthogonal transformation which reduced B_Z and B_T
in the first step of the algorithm (see METHOD).
If COMPQ = 'N', this array is not referenced.

LDQ     INTEGER
The leading dimension of the array Q.
LDQ >= 1,           if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.

U       (output) COMPLEX*16 array, dimension (LDU, 2*N)
On exit, if COMPU = 'C' and JOB = 'T', then the leading
N-by-2*N part of this array contains the leading N-by-2*N
part of the unitary symplectic transformation matrix U.
If COMPU = 'C' and JOB = 'E', this array contains the
first N rows of the transformation U which reduced B_Z
and B_T in the first step of the algorithm (see METHOD).
If COMPU = 'N', this array is not referenced.

LDU     INTEGER
The leading dimension of the array U.
LDU >= 1,         if COMPU = 'N';
LDU >= MAX(1, N), if COMPU = 'C'.

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

ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
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)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent 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.

```
Workspace
```  IWORK   INTEGER array, dimension (LIWORK)

LIWORK  INTEGER
The dimension of the array IWORK.  LIWORK >= 2*N+9.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
On exit, if INFO = -25, DWORK(1) returns the minimum
value of LDWORK.

LDWORK  INTEGER
The dimension of the array DWORK.
LDWORK >= c*N**2 + N + MAX(2*N, 24) + 3, where
c = 18, if                 COMPU = 'C';
c = 16, if COMPQ = 'C' and COMPU = 'N';
c = 13, if COMPQ = 'N' and COMPU = 'N'.
For good performance LDWORK should be generally 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.

ZWORK   COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -27, ZWORK(1) returns the minimum
value of LZWORK.

LZWORK  INTEGER
The dimension of the array ZWORK.
LZWORK >= 8*N + 28, if JOB = 'T' and COMPQ = 'C';
LZWORK >= 6*N + 28, if JOB = 'T' and COMPQ = 'N';
LZWORK >= 1,        if JOB = 'E'.
For good performance LZWORK should be generally larger.

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

BWORK   LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if JOB = 'E';
LBWORK >= N, if JOB = 'T'.

```
Error Indicator
```  INFO    INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: the algorithm was not able to reveal information
about the eigenvalues from the 2-by-2 blocks in the
SLICOT Library routine MB03BD (called by MB04ED);
= 2: periodic QZ iteration failed in the SLICOT Library
routines MB03BD or MB03BZ when trying to
triangularize the 2-by-2 blocks.

```
Method
```  First T = i*H is set. Then, the embeddings, B_Z and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04ED is applied to compute the structured Schur
form, i.e., the factorizations

~      T         (  BZ11  BZ12  )
B_Z = U  B_Z Q = (              ) and
(    0   BZ22  )

~        T  T         (  T11  T12  )
B_T = J Q  J  B_T Q = (            ),
(   0   T11' )

where Q is real orthogonal, U is real orthogonal symplectic, BZ11,
BZ22' are upper triangular and T11 is upper quasi-triangular.
The notation M' denotes the transpose of the matrix M.
If JOB = 'T', the 2-by-2 blocks are triangularized using the
periodic QZ algorithm.

```
References
```  [1] 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
```                                                            3
The algorithm is numerically backward stable and needs O(N )
complex floating point operations.

```
```  This routine does not perform any scaling of the matrices. Scaling
might sometimes be useful, and it should be done externally.

```
Example

Program Text

```*     MB04AZ 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            LDB, LDC, LDD, LDFG, LDQ, LDU, LDWORK, LDZ,
\$                   LIWORK, LZWORK
PARAMETER          ( LDB = NMAX,  LDC =   NMAX, LDD = NMAX,
\$                     LDFG = NMAX, LDQ = 2*NMAX, LDU = NMAX,
\$                     LDWORK = 18*NMAX*NMAX + NMAX + MAX( 2*NMAX,
\$                                                         24 ) + 3,
\$                     LDZ = NMAX, LIWORK = 2*NMAX + 9,
\$                     LZWORK = 8*NMAX + 28 )
*
*     .. Local Scalars ..
CHARACTER          COMPQ, COMPU, JOB
INTEGER            I, INFO, J, M, N
*
*     .. Local Arrays ..
COMPLEX*16         B( LDB,   NMAX ),  C( LDC,   NMAX ),
\$                   D( LDD,   NMAX ), FG( LDFG,  NMAX ),
\$                   Q( LDQ, 2*NMAX ),  U( LDU, 2*NMAX ),
\$                   Z( LDZ,   NMAX ), ZWORK( LZWORK )
DOUBLE PRECISION   ALPHAI( NMAX ), ALPHAR( NMAX ),
\$                   BETA(   NMAX ), DWORK(LDWORK )
INTEGER            IWORK( LIWORK )
LOGICAL            BWORK( NMAX )
*
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*
*     .. External Subroutines ..
EXTERNAL           MB04AZ
*
*     .. Intrinsic Functions ..
INTRINSIC          MAX, MOD
*
*     .. 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, COMPQ, COMPU, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( (  Z( I, J ), J = 1, N ),     I=1, N )
READ( NIN, FMT = * ) ( (  B( I, J ), J = 1, N/2 ),   I=1, N/2 )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, N/2+1 ), I=1, N/2 )
*        Compute the eigenvalues of a complex skew-Hamiltonian/
*        Hamiltonian pencil (factored version).
CALL MB04AZ( JOB, COMPQ, COMPU, N, Z, LDZ, B, LDB, FG, LDFG,
\$                D, LDD, C, LDC, Q, LDQ, U, LDU, ALPHAR, ALPHAI,
\$                BETA, IWORK, LIWORK, DWORK, LDWORK, ZWORK, LZWORK,
\$                BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
M = N/2
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) (  Z( I, J ), J = 1, N )
10          CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) (  B( I, J ), J = 1, N )
20          CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
30          CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) (  D( I, J ), J = 1, N )
40          CONTINUE
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, N
WRITE( NOUT, FMT = 99995 ) (  C( I, J ), J = 1, N )
50          CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) ) THEN
WRITE( NOUT, FMT = 99990 )
DO 60 I = 1, 2*N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, 2*N )
60          CONTINUE
END IF
IF( LSAME( COMPU, 'C' ) ) THEN
WRITE( NOUT, FMT = 99989 )
DO 70 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( U( I, J ), J = 1, 2*N )
70          CONTINUE
END IF
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99987 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99986 )
WRITE( NOUT, FMT = 99987 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99985 )
WRITE( NOUT, FMT = 99987 ) (   BETA( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT ( 'MB04AZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04AZ = ', I2 )
99996 FORMAT (/' The transformed matrix Z is' )
99995 FORMAT (20(1X,F9.4,SP,F9.4,S,'i '))
99994 FORMAT (/' The transformed matrix B is' )
99993 FORMAT (/' The transformed matrix FG is' )
99992 FORMAT (/' The matrix D is' )
99991 FORMAT (/' The matrix C is' )
99990 FORMAT (/' The matrix Q is' )
99989 FORMAT (/' The upper part of the matrix U is' )
99988 FORMAT (/' The vector ALPHAR is ' )
99987 FORMAT ( 50( 1X, F8.4 ) )
99986 FORMAT (/' The vector ALPHAI is ' )
99985 FORMAT (/' The vector BETA is ' )
END
```
Program Data
```MB04AZ EXAMPLE PROGRAM DATA
T	C	C	4
(0.4941,0.8054)   (0.8909,0.8865)   (0.0305,0.9786)   (0.9047,0.0596)
(0.7790,0.5767)   (0.3341,0.0286)   (0.7440,0.7126)   (0.6098,0.6819)
(0.7150,0.1829)   (0.6987,0.4899)   (0.5000,0.5004)   (0.6176,0.0424)
(0.9037,0.2399)   (0.1978,0.1679)   (0.4799,0.4710)   (0.8594,0.0714)
(0.5216,0.7224)   (0.8181,0.6596)
(0.0967,0.1498)   (0.8175,0.5185)
0.9729            0.8003           (0.4323,0.8313)
(0.6489,0.1331)    0.4537            0.8253
```
Program Results
```MB04AZ EXAMPLE PROGRAM RESULTS

The transformed matrix Z is
0.4545  +0.0000i     0.7904  +0.0000i    -0.1601  +0.0000i    -0.2691  +0.0000i
0.7790  +0.5767i     0.4273  +0.0000i     0.1459  +0.0000i     0.1298  +0.0000i
0.7150  +0.1829i     0.6987  +0.4899i     0.6715  +0.0000i    -0.3001  +0.0000i
0.9037  +0.2399i     0.1978  +0.1679i     0.4799  +0.4710i     0.7924  +0.0000i

The transformed matrix B is
0.0000  -1.7219i     0.0000  +0.7762i     0.0000  +0.5342i     0.0000  +0.0845i
0.0000  +0.0000i     0.0000  +0.8862i     0.0000  +0.5186i     0.0000  -0.1429i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +1.1122i     0.0000  +0.2898i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.4889i

The transformed matrix FG is
0.0000  +0.0000i     0.0000  +0.4145i     0.0000  -0.7921i     0.0000  +0.5630i
0.6489  +0.1331i     0.0000  +0.0000i     0.0000  +1.5982i     0.0000  +0.5818i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.5819i
0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i

The matrix D is
2.2139  +0.0000i     0.0402  +0.0000i    -0.2787  +0.0000i     1.0465  +0.0000i
-0.5021  +0.0000i     0.5502  +0.0000i    -0.2771  +0.0000i    -0.4521  +0.0000i
-0.0398  +0.0000i     0.4046  +0.0000i     0.0149  +0.0000i     0.7577  +0.0000i
-0.1550  +0.0000i     2.0660  +0.0000i     1.6075  +0.0000i     0.5836  +0.0000i

The matrix C is
0.3159  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
0.7819  +0.0000i    -0.7575  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
-0.3494  +0.0000i     0.8622  +0.0000i     0.7539  +0.0000i     0.0000  +0.0000i
1.1178  +0.0000i     0.3133  +0.0000i     0.4638  +0.0000i     1.2348  +0.0000i

The matrix Q is
-0.4983  +0.0000i     0.3694  +0.0000i    -0.4754  +0.0000i     0.2791  +0.0000i     0.1950  +0.0000i     0.2416  +0.0000i     0.1869  +0.0000i     0.4242  +0.0000i
-0.1045  +0.0000i     0.3309  +0.0000i     0.5730  +0.0000i    -0.5566  +0.0000i     0.0877  +0.0000i     0.3543  +0.0000i     0.1761  +0.0000i     0.2779  +0.0000i
-0.2586  +0.0000i    -0.4457  +0.0000i    -0.2838  +0.0000i    -0.5436  +0.0000i     0.5524  +0.0000i    -0.1064  +0.0000i    -0.2040  +0.0000i     0.0205  +0.0000i
-0.0040  +0.0000i     0.3845  +0.0000i     0.2469  +0.0000i     0.2965  +0.0000i     0.5799  +0.0000i     0.0840  +0.0000i    -0.4788  +0.0000i    -0.3614  +0.0000i
0.7958  +0.0000i     0.1597  +0.0000i    -0.3420  +0.0000i    -0.1047  +0.0000i     0.3370  +0.0000i     0.1813  +0.0000i     0.2275  +0.0000i     0.1229  +0.0000i
-0.0600  +0.0000i    -0.4599  +0.0000i     0.3487  +0.0000i     0.3910  +0.0000i     0.4005  +0.0000i     0.1010  +0.0000i     0.5819  +0.0000i    -0.0345  +0.0000i
-0.0539  +0.0000i     0.3817  +0.0000i     0.0501  +0.0000i    -0.1114  +0.0000i     0.1751  +0.0000i    -0.8212  +0.0000i     0.3606  +0.0000i    -0.0380  +0.0000i
0.1846  +0.0000i    -0.1577  +0.0000i     0.2510  +0.0000i     0.2293  +0.0000i     0.0911  +0.0000i    -0.2834  +0.0000i    -0.3779  +0.0000i     0.7708  +0.0000i

The upper part of the matrix U is
-0.2544  +0.0000i    -0.1844  +0.0000i     0.7632  +0.0000i     0.5646  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i     0.0000  +0.0000i
0.4272  +0.0000i    -0.0454  +0.0000i    -0.2918  +0.0000i     0.5721  +0.0000i     0.1446  +0.0000i    -0.1611  +0.0000i     0.3609  +0.0000i    -0.4752  +0.0000i
0.6936  +0.0000i     0.3812  +0.0000i     0.2606  +0.0000i     0.0848  +0.0000i    -0.3359  +0.0000i    -0.1592  +0.0000i    -0.3242  +0.0000i     0.2348  +0.0000i
0.2320  +0.0000i    -0.4483  +0.0000i    -0.1629  +0.0000i     0.1784  +0.0000i    -0.2898  +0.0000i     0.7525  +0.0000i    -0.0511  +0.0000i     0.1841  +0.0000i

The vector ALPHAR is
0.0000   0.0000   0.0000   0.0000

The vector ALPHAI is
-1.4991  -1.3690   1.0985   0.9993

The vector BETA is
0.1250   0.5000   0.5000   2.0000
```