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<HTML>
<HEAD><TITLE>MB03BZ - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03BZ">MB03BZ</A></H2>
<H3>
Finding eigenvalues of a complex generalized matrix product in Hessenberg-triangular 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 find the eigenvalues of the complex generalized matrix product
S(1) S(2) S(K)
A(:,:,1) * A(:,:,2) * ... * A(:,:,K) , S(1) = 1,
where A(:,:,1) is upper Hessenberg and A(:,:,i) is upper
triangular, i = 2, ..., K, using a single-shift version of the
periodic QZ method. In addition, A may be reduced to periodic
Schur form by unitary transformations: all factors A(:,:,i) become
upper triangular.
If COMPQ = 'V' or COMPQ = 'I', then the unitary factors are
computed and stored in the array Q so that for S(I) = 1,
H
Q(:,:,I)(in) A(:,:,I)(in) Q(:,:,MOD(I,K)+1)(in)
H (1)
= Q(:,:,I)(out) A(:,:,I)(out) Q(:,:,MOD(I,K)+1)(out),
and for S(I) = -1,
H
Q(:,:,MOD(I,K)+1)(in) A(:,:,I)(in) Q(:,:,I)(in)
H (2)
= Q(:,:,MOD(I,K)+1)(out) A(:,:,I)(out) Q(:,:,I)(out).
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03BZ( JOB, COMPQ, K, N, ILO, IHI, S, A, LDA1, LDA2,
$ Q, LDQ1, LDQ2, ALPHA, BETA, SCAL, DWORK,
$ LDWORK, ZWORK, LZWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER IHI, ILO, INFO, K, LDA1, LDA2, LDQ1, LDQ2,
$ LDWORK, LZWORK, N
C .. Array Arguments ..
INTEGER S(*), SCAL(*)
DOUBLE PRECISION DWORK(*)
COMPLEX*16 A(LDA1, LDA2, *), ALPHA(*), BETA(*),
$ Q(LDQ1, LDQ2, *), 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; A will not
necessarily be put into periodic Schur form;
= 'S': put A into periodic Schur form, and return the
eigenvalues in ALPHA, BETA, and SCAL.
COMPQ CHARACTER*1
Specifies whether or not the unitary transformations
should be accumulated in the array Q, as follows:
= 'N': do not modify Q;
= 'V': modify the array Q by the unitary transformations
that are applied to the matrices in the array A to
reduce them to periodic Schur form;
= 'I': like COMPQ = 'V', except that each matrix in the
array Q will be first initialized to the identity
matrix.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
K (input) INTEGER
The number of factors. K >= 1.
N (input) INTEGER
The order of each factor in the array A. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that each factor in A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
1 <= ILO <= IHI <= N, if N > 0;
ILO = 1 and IHI = 0, if N = 0.
S (input) INTEGER array, dimension (K)
The leading K elements of this array must contain the
signatures of the factors. Each entry in S must be either
1 or -1. By definition, S(1) must be set to 1.
A (input/output) COMPLEX*16 array, dimension (LDA1,LDA2,K)
On entry, the leading N-by-N-by-K part of this array
must contain the factors in upper Hessenberg-triangular
form, that is, A(:,:,1) is upper Hessenberg and the other
factors are upper triangular.
On exit, if JOB = 'S' and INFO = 0, the leading
N-by-N-by-K part of this array contains the factors of
A in periodic Schur form. All factors are reduced to
upper triangular form and, moreover, A(:,:,2), ...,
A(:,:,K) are normalized so that their diagonals contain
nonnegative real numbers.
On exit, if JOB = 'E', then the leading N-by-N-by-K part
of this array contains meaningless elements.
LDA1 INTEGER
The first leading dimension of the array A.
LDA1 >= MAX(1,N).
LDA2 INTEGER
The second leading dimension of the array A.
LDA2 >= MAX(1,N).
Q (input/output) COMPLEX*16 array, dimension (LDQ1,LDQ2,K)
On entry, if COMPQ = 'V', the leading N-by-N-by-K part
of this array must contain the initial unitary factors
as described in (1) and (2).
On exit, if COMPQ = 'V' or COMPQ = 'I', the leading
N-by-N-by-K part of this array contains the modified
unitary factors as described in (1) and (2).
This array is not referenced if COMPQ = 'N'.
LDQ1 INTEGER
The first leading dimension of the array Q. LDQ1 >= 1,
and, if COMPQ <> 'N', LDQ1 >= MAX(1,N).
LDQ2 INTEGER
The second leading dimension of the array Q. LDQ2 >= 1,
and, if COMPQ <> 'N', LDQ2 >= MAX(1,N).
ALPHA (output) COMPLEX*16 array, dimension (N)
On exit, if INFO = 0, the leading N elements of this
array contain the scaled eigenvalues of the matrix
product A. The i-th eigenvalue of A is given by
ALPHA(I) / BETA(I) * BASE**(SCAL(I)),
where ABS(ALPHA(I)) = 0.0 or 1.0 <= ABS(ALPHA(I)) < BASE,
and BASE is the machine base (normally 2.0).
BETA (output) COMPLEX*16 array, dimension (N)
On exit, if INFO = 0, the leading N elements of this
array contain indicators for infinite eigenvalues. That
is, if BETA(I) = 0.0, then the i-th eigenvalue is
infinite. Otherwise BETA(I) is set to 1.0.
SCAL (output) INTEGER array, dimension (N)
On exit, if INFO = 0, the leading N elements of this
array contain the scaling parameters for the eigenvalues
of A.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the minimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= MAX(1,N).
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the minimal value
of LZWORK.
LZWORK INTEGER
The length of the array ZWORK. LZWORK >= MAX(1,N).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0 : succesful exit;
< 0 : if INFO = -i, the i-th argument had an illegal
value;
= 1,..,N : the periodic QZ iteration did not converge.
A is not in periodic Schur form, but
ALPHA(I), BETA(I), and SCAL(I), for
I = INFO+1,...,N should be correct.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
A slightly modified version of the periodic QZ algorithm is
used. For more details, see [2].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Bojanczyk, A., Golub, G. H. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
In F.T. Luk (editor), Advanced Signal Processing Algorithms,
Architectures, and Implementations III, Proc. SPIE Conference,
vol. 1770, pp. 31-42, 1992.
[2] Kressner, D.
An efficient and reliable implementation of the periodic QZ
algorithm. IFAC Workshop on Periodic Control Systems (PSYCO
2001), Como (Italy), August 27-28 2001. Periodic Control
Systems 2001 (IFAC Proceedings Volumes), Pergamon.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented method is numerically backward stable.
3
The algorithm requires 0(K N ) floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* MB03BZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER KMAX, NMAX
PARAMETER ( KMAX = 6, NMAX = 50 )
INTEGER LDA1, LDA2, LDQ1, LDQ2, LDWORK, LZWORK
PARAMETER ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
$ LDQ2 = NMAX, LDWORK = NMAX, LZWORK = NMAX )
*
* .. Local Scalars ..
CHARACTER COMPQ, JOB
INTEGER I, IHI, ILO, INFO, J, K, L, N
*
* .. Local Arrays ..
COMPLEX*16 A( LDA1, LDA2, KMAX ), ALPHA( NMAX ),
$ BETA( NMAX ), Q( LDQ1, LDQ2, KMAX ),
$ ZWORK( LZWORK )
DOUBLE PRECISION DWORK( LDWORK)
INTEGER S( KMAX ), SCAL( NMAX )
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB03BZ
*
* .. 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, K, N, ILO, IHI
IF( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
READ( NIN, FMT = * ) ( S( I ), I = 1, K )
READ( NIN, FMT = * ) ( ( ( A( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
IF( LSAME( COMPQ, 'V' ) )
$ READ( NIN, FMT = * ) ( ( ( Q( I, J, L ), J = 1, N ),
$ I = 1, N ), L = 1, K )
* Compute the eigenvalues and the transformed matrices, if
* required.
CALL MB03BZ( JOB, COMPQ, K, N, ILO, IHI, S, A, LDA1, LDA2,
$ Q, LDQ1, LDQ2, ALPHA, BETA, SCAL, DWORK, LDWORK,
$ ZWORK, LZWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
IF( LSAME( JOB, 'S' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 20 L = 1, K
WRITE( NOUT, FMT = 99995 ) L
DO 10 I = 1, N
WRITE( NOUT, FMT = 99994 ) ( A( I, J, L ), J = 1, N
$ )
10 CONTINUE
20 CONTINUE
END IF
IF( .NOT.LSAME( COMPQ, 'N' ) ) THEN
WRITE( NOUT, FMT = 99993 )
DO 40 L = 1, K
WRITE( NOUT, FMT = 99995 ) L
DO 30 I = 1, N
WRITE( NOUT, FMT = 99994 ) ( Q( I, J, L ), J = 1, N
$ )
30 CONTINUE
40 CONTINUE
END IF
WRITE( NOUT, FMT = 99992 )
WRITE( NOUT, FMT = 99994 ) ( ALPHA( I ), I = 1, N )
WRITE( NOUT, FMT = 99991 )
WRITE( NOUT, FMT = 99994 ) ( BETA( I ), I = 1, N )
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99989 ) ( SCAL( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT( 'MB03BZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT( 'INFO on exit from MB03BZ = ', I2 )
99996 FORMAT(/'The matrix A on exit is ' )
99995 FORMAT( 'The factor ', I2, ' is ' )
99994 FORMAT( 50( 1X, F9.4, SP, F9.4, S, 'i ') )
99993 FORMAT(/'The matrix Q on exit is ' )
99992 FORMAT(/'The vector ALPHA is ' )
99991 FORMAT( 'The vector BETA is ' )
99990 FORMAT( 'The vector SCAL is ' )
99989 FORMAT( 50( 1X, I8 ) )
END
</PRE>
<B>Program Data</B>
<PRE>
MB03BZ EXAMPLE PROGRAM DATA
S I 3 4 1 4
1 -1 1
(0.8637,0.9326) (0.8819,0.4850) (0.5920,0.8826) (0.8991,0.9040)
(0.6994,0.8588) (0.9527,0.2672) (0.5087,0.0621) (0.9653,0.5715)
0 (0.1561,0.1898) (0.9514,0.9266) (0.6582,0.3102)
0 0 (0.8649,0.1265) (0.1701,0.0013)
(0.5113,0.7375) (0.6869,0.7692) (0.7812,0.1467) (0.7216,0.9498)
0 (0.1319,0.9137) (0.5879,0.0201) (0.9834,0.0549)
0 0 (0.7711,0.2422) (0.9468,0.3280)
0 0 0 (0.2219,0.3971)
(0.0158,0.4042) (0.0082,0.2033) (0.1028,0.9913) (0.6954,0.1987)
0 (0.5066,0.4587) (0.1060,0.6949) (0.5402,0.0970)
0 0 (0.4494,0.3700) (0.8492,0.4882)
0 0 0 (0.2110,0.5824)
</PRE>
<B>Program Results</B>
<PRE>
MB03BZ EXAMPLE PROGRAM RESULTS
The matrix A on exit is
The factor 1 is
0.6053 +1.0311i -1.7227 -0.5753i 1.2428 -1.2632i 0.9445 -0.4317i
0.0000 +0.0000i -0.2596 +1.0235i 0.4673 -0.2403i -0.5579 -1.1564i
0.0000 +0.0000i 0.0000 +0.0000i -0.3336 -0.3367i 0.0687 +0.0261i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.2014 +0.0057i
The factor 2 is
1.1118 +0.0000i -1.4173 +1.1607i 0.3271 -0.5800i 0.5291 -0.6341i
0.0000 +0.0000i 0.9051 +0.0000i 0.1710 +0.1014i -0.2696 -0.3549i
0.0000 +0.0000i 0.0000 +0.0000i 0.3599 +0.0000i 0.0231 -0.5865i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.8410 +0.0000i
The factor 3 is
1.1554 +0.0000i -0.7577 +0.0825i 0.1284 -0.0063i 1.1175 -0.0778i
0.0000 +0.0000i 0.5216 +0.0000i -0.5761 +0.2972i -0.3534 -0.3595i
0.0000 +0.0000i 0.0000 +0.0000i 0.2750 +0.0000i 0.2587 -0.1664i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.6015 +0.0000i
The matrix Q on exit is
The factor 1 is
0.6033 -0.4021i 0.3478 -0.1834i -0.0029 -0.4458i -0.3157 +0.1462i
0.1519 -0.3002i 0.6010 +0.1411i 0.0134 +0.5692i 0.4223 -0.0574i
0.1002 -0.5387i -0.4199 +0.3933i 0.1148 +0.2694i -0.3713 -0.3810i
-0.0599 -0.2395i -0.0968 +0.3521i 0.2342 -0.5801i 0.6247 -0.1555i
The factor 2 is
0.3325 +0.6289i 0.0930 +0.3421i -0.1849 -0.1715i -0.5470 -0.0734i
0.3247 -0.4396i 0.1766 -0.1691i 0.2307 -0.3746i -0.2617 -0.6160i
0.2180 -0.2074i -0.3853 +0.1978i -0.6852 +0.2781i 0.1492 -0.3909i
-0.0268 -0.3223i -0.0623 +0.7893i 0.0957 -0.4323i 0.1716 +0.2072i
The factor 3 is
0.6791 +0.1138i 0.0183 +0.2703i 0.3289 -0.2770i -0.2355 -0.4605i
0.5111 -0.1775i 0.3990 +0.1941i -0.0851 +0.4779i 0.4563 +0.2580i
0.2687 -0.3788i -0.7974 -0.0691i -0.2836 +0.1192i 0.1765 -0.1340i
0.1098 -0.0739i -0.2845 -0.0903i 0.6651 +0.2062i -0.2810 +0.5741i
The vector ALPHA is
0.6290 +1.0715i -0.2992 +1.1797i -1.0195 -1.0290i -1.1523 +0.0326i
The vector BETA is
1.0000 +0.0000i 1.0000 +0.0000i 1.0000 +0.0000i 1.0000 +0.0000i
The vector SCAL is
0 -1 -2 -3
</PRE>
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