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<HTML>
<HEAD><TITLE>MB04AZ - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04AZ">MB04AZ</A></H2>
<H3>
Eigenvalues of a complex skew-Hamiltonian/Hamiltonian pencil in factored form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
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. B_H above is defined as B_H = -i*B_T.
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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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)
(see also METHOD). The strictly lower triangular part is
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)
(see also METHOD).
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. Accidental nonzero imaginary parts on
the main diagonals of F and G do not perturb the results.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the Hermitian matrix BF in (3) (see also
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts, since they have not been annihilated.
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
contains the matrix BD in (3) (see also METHOD).
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
contains the lower triangular matrix BC in (3) (see also
METHOD). The part over the first superdiagonal is not set.
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.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (LIWORK)
On exit, if INFO = 3, IWORK(1) contains the number of
(pairs of) possibly inaccurate eigenvalues, q <= N/2, and
IWORK(2), ..., IWORK(q+1) indicate their indices.
Specifically, a positive value is an index of a real or
purely imaginary eigenvalue, corresponding to a 1-by-1
block, while the absolute value of a negative entry in
IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. A 2-by-2 block may have two complex, two
real, two purely imaginary, or one real and one purely
imaginary eigenvalue. The blocks are those in B_T and B_S.
For i = q+2, ..., 2*q+1, IWORK(i) contains a pointer to
the starting location in DWORK of the i-th triplet of
1-by-1 blocks, if IWORK(i-q) > 0, or 2-by-2 blocks,
if IWORK(i-q) < 0, defining unreliable eigenvalues.
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
LIWORK INTEGER
The dimension of the array IWORK. LIWORK >= 2*N+9.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2), ..., DWORK(4) contain the
Frobenius norms of the factors of the formal matrix
product used by the algorithm. In addition, DWORK(5), ...,
DWORK(4+3*s) contain the s triplet values corresponding
to the 1-by-1 blocks. Their eigenvalues are real or purely
imaginary. Such an eigenvalue is obtained as -a1/a2/a3*i,
where a1, ..., a3 are the corresponding triplet values,
and i is the purely imaginary unit.
Moreover, DWORK(5+3*s), ..., DWORK(4+3*s+12*t) contain the
t groups of triplet 2-by-2 matrices corresponding to the
2-by-2 blocks. Their eigenvalue pairs are either complex,
or placed on the real and imaginary axes. Such an
eigenvalue pair is given by imag( ev ) - real( ev )*i,
where ev is the spectrum of the matrix product
A1*inv(A2)*inv(A3), and A1, ..., A3 define the
corresponding 2-by-2 matrix triplet.
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(6*N, 27), 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'.
</PRE>
<B>Error Indicator</B>
<PRE>
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;
= 3: some eigenvalues might be inaccurate. This is a
warning.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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 = ( T ),
( 0 T11 )
where Q is real orthogonal, U is real orthogonal symplectic, BZ11,
BZ22' are upper triangular and T11 is upper quasi-triangular.
If JOB = 'T', the 2-by-2 blocks are triangularized using the
periodic QZ algorithm.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3
The algorithm is numerically backward stable and needs O(N )
complex floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
This routine does not perform any scaling of the matrices. Scaling
might sometimes be useful, and it should be done externally.
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* MB04AZ EXAMPLE PROGRAM TEXT
*
* .. 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
</PRE>
<B>Program Data</B>
<PRE>
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
</PRE>
<B>Program Results</B>
<PRE>
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
</PRE>
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