## MB03WD

### Schur decomposition and eigenvalues of a product of matrices in periodic Hessenberg form

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

Purpose

```  To compute the Schur decomposition and the eigenvalues of a
product of matrices, H = H_1*H_2*...*H_p, with H_1 an upper
Hessenberg matrix and H_2, ..., H_p upper triangular matrices,
without evaluating the product. Specifically, the matrices Z_i
are computed, such that

Z_1' * H_1 * Z_2 = T_1,
Z_2' * H_2 * Z_3 = T_2,
...
Z_p' * H_p * Z_1 = T_p,

where T_1 is in real Schur form, and T_2, ..., T_p are upper
triangular.

The routine works primarily with the Hessenberg and triangular
submatrices in rows and columns ILO to IHI, but optionally applies
the transformations to all the rows and columns of the matrices
H_i, i = 1,...,p. The transformations can be optionally
accumulated.

```
Specification
```      SUBROUTINE MB03WD( JOB, COMPZ, N, P, ILO, IHI, ILOZ, IHIZ, H,
\$                   LDH1, LDH2, Z, LDZ1, LDZ2, WR, WI, DWORK,
\$                   LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         COMPZ, JOB
INTEGER           IHI, IHIZ, ILO, ILOZ, INFO, LDH1, LDH2, LDWORK,
\$                  LDZ1, LDZ2, N, P
C     .. Array Arguments ..
DOUBLE PRECISION  DWORK( * ), H( LDH1, LDH2, * ), WI( * ),
\$                  WR( * ), Z( LDZ1, LDZ2, * )

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Indicates whether the user wishes to compute the full
Schur form or the eigenvalues only, as follows:
= 'E':  Compute the eigenvalues only;
= 'S':  Compute the factors T_1, ..., T_p of the full
Schur form, T = T_1*T_2*...*T_p.

COMPZ   CHARACTER*1
Indicates whether or not the user wishes to accumulate
the matrices Z_1, ..., Z_p, as follows:
= 'N':  The matrices Z_1, ..., Z_p are not required;
= 'I':  Z_i is initialized to the unit matrix and the
orthogonal transformation matrix Z_i is returned,
i = 1, ..., p;
= 'V':  Z_i must contain an orthogonal matrix Q_i on
entry, and the product Q_i*Z_i is returned,
i = 1, ..., p.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix H.  N >= 0.

P       (input) INTEGER
The number of matrices in the product H_1*H_2*...*H_p.
P >= 1.

ILO     (input) INTEGER
IHI     (input) INTEGER
It is assumed that all matrices H_j, j = 2, ..., p, are
already upper triangular in rows and columns 1:ILO-1 and
IHI+1:N, and H_1 is upper quasi-triangular in rows and
columns 1:ILO-1 and IHI+1:N, with H_1(ILO,ILO-1) = 0
(unless ILO = 1), and H_1(IHI+1,IHI) = 0 (unless IHI = N).
The routine works primarily with the Hessenberg submatrix
in rows and columns ILO to IHI, but applies the
transformations to all the rows and columns of the
matrices H_i, i = 1,...,p, if JOB = 'S'.
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.

ILOZ    (input) INTEGER
IHIZ    (input) INTEGER
Specify the rows of Z to which the transformations must be
applied if COMPZ = 'I' or COMPZ = 'V'.
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

H       (input/output) DOUBLE PRECISION array, dimension
(LDH1,LDH2,P)
On entry, the leading N-by-N part of H(*,*,1) must contain
the upper Hessenberg matrix H_1 and the leading N-by-N
part of H(*,*,j) for j > 1 must contain the upper
triangular matrix H_j, j = 2, ..., p.
On exit, if JOB = 'S', the leading N-by-N part of H(*,*,1)
is upper quasi-triangular in rows and columns ILO:IHI,
with any 2-by-2 diagonal blocks corresponding to a pair of
complex conjugated eigenvalues, and the leading N-by-N
part of H(*,*,j) for j > 1 contains the resulting upper
triangular matrix T_j.
If JOB = 'E', the contents of H are unspecified on exit.

LDH1    INTEGER
The first leading dimension of the array H.
LDH1 >= max(1,N).

LDH2    INTEGER
The second leading dimension of the array H.
LDH2 >= max(1,N).

Z       (input/output) DOUBLE PRECISION array, dimension
(LDZ1,LDZ2,P)
On entry, if COMPZ = 'V', the leading N-by-N-by-P part of
this array must contain the current matrix Q of
transformations accumulated by SLICOT Library routine
MB03VY.
If COMPZ = 'I', Z need not be set on entry.
On exit, if COMPZ = 'V', or COMPZ = 'I', the leading
N-by-N-by-P part of this array contains the transformation
matrices which produced the Schur form; the
transformations are applied only to the submatrices
Z_j(ILOZ:IHIZ,ILO:IHI), j = 1, ..., P.
If COMPZ = 'N', Z is not referenced.

LDZ1    INTEGER
The first leading dimension of the array Z.
LDZ1 >= 1,        if COMPZ = 'N';
LDZ1 >= max(1,N), if COMPZ = 'I' or COMPZ = 'V'.

LDZ2    INTEGER
The second leading dimension of the array Z.
LDZ2 >= 1,        if COMPZ = 'N';
LDZ2 >= max(1,N), if COMPZ = 'I' or COMPZ = 'V'.

WR      (output) DOUBLE PRECISION array, dimension (N)
WI      (output) DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the
computed eigenvalues ILO to IHI are stored in the
corresponding elements of WR and WI. If two eigenvalues
are computed as a complex conjugate pair, they are stored
in consecutive elements of WR and WI, say the i-th and
(i+1)th, with WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the
eigenvalues are stored in the same order as on the
diagonal of the Schur form returned in H.

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

LDWORK  INTEGER
The length of the array DWORK.  LDWORK >= IHI-ILO+P-1.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
> 0:  if INFO = i, ILO <= i <= IHI, the QR algorithm
failed to compute all the eigenvalues ILO to IHI
in a total of 30*(IHI-ILO+1) iterations;
the elements i+1:IHI of WR and WI contain those
eigenvalues which have been successfully computed.

```
Method
```  A refined version of the QR algorithm proposed in [1] and [2] is
used. The elements of the subdiagonal, diagonal, and the first
supradiagonal of current principal submatrix of H are computed
in the process.

```
References
```  [1] Bojanczyk, A.W., Golub, G. and Van Dooren, P.
The periodic Schur decomposition: algorithms and applications.
Proc. of the SPIE Conference (F.T. Luk, Ed.), 1770, pp. 31-42,
1992.

[2] Sreedhar, J. and Van Dooren, P.
Periodic Schur form and some matrix equations.
Proc. of the Symposium on the Mathematical Theory of Networks
and Systems (MTNS'93), Regensburg, Germany (U. Helmke,
R. Mennicken and J. Saurer, Eds.), Vol. 1, pp. 339-362, 1994.

```
Numerical Aspects
```  The algorithm is numerically stable.

```
```  Note that for P = 1, the LAPACK Library routine DHSEQR could be
more efficient on some computer architectures than this routine,
because DHSEQR uses a block multishift QR algorithm.
When P is large and JOB = 'S', it could be more efficient to
compute the product matrix H, and use the LAPACK Library routines.

```
Example

Program Text

```*     MB03WD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX, PMAX
PARAMETER        ( NMAX = 20, PMAX = 20 )
INTEGER          LDA1, LDA2, LDTAU, LDZ1, LDZ2, LDZTA
PARAMETER        ( LDA1 = NMAX, LDA2 = NMAX, LDTAU = NMAX-1,
\$                   LDZ1 = NMAX, LDZ2 = NMAX, LDZTA = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MAX( NMAX, NMAX + PMAX - 2 ) )
DOUBLE PRECISION ZERO, ONE
PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
*     .. Local Scalars ..
DOUBLE PRECISION SSQ
INTEGER          I, IHI, IHIZ, ILO, ILOZ, INFO, J, K, KP1, N, P
CHARACTER        COMPZ, JOB
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA1,LDA2,PMAX), AS(LDA1,LDA2,PMAX),
\$                 DWORK(LDWORK), TAU(LDTAU,PMAX), WI(NMAX),
\$                 WR(NMAX), Z(LDZ1,LDZ2,PMAX), ZTA(LDZTA,NMAX)
*     .. External Functions ..
DOUBLE PRECISION DLANGE, DLAPY2
LOGICAL          LSAME
EXTERNAL         DLANGE, DLAPY2, LSAME
*     .. External Subroutines ..
EXTERNAL         DGEMM, DLACPY, MB03VD, MB03VY, MB03WD, MB03WX
*     .. Intrinsic Functions ..
INTRINSIC        MAX, MIN
*     .. Executable Statements ..
WRITE (NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, P, ILO, IHI, ILOZ, IHIZ, JOB, COMPZ
IF ( N.LT.0 .OR. N.GT.MIN( LDA1, LDA2 ) ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) P
ELSE
*           Read matrices A_1, ..., A_p from the input file.
DO 10 K = 1, P
READ ( NIN, FMT = * )
\$            ( ( A(I,J,K), J = 1, N ), I = 1, N )
CALL DLACPY( 'F', N, N, A(1,1,K), LDA1, AS(1,1,K), LDA1 )
10       CONTINUE
*           Reduce to the periodic Hessenberg form.
CALL MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
\$                   DWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
IF ( LSAME( COMPZ, 'V' ) ) THEN
DO 20 K = 1, P
CALL DLACPY( 'L', N, N, A(1,1,K), LDA1, Z(1,1,K),
\$                            LDZ1 )
20             CONTINUE
*                 Accumulate the transformations.
CALL MB03VY( N, P, ILO, IHI, Z, LDZ1, LDZ2, TAU,
\$                         LDTAU, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99996 ) INFO
STOP
ELSE
*                    Reduce to the periodic Schur form.
CALL MB03WD( JOB, COMPZ, N, P, ILO, IHI, ILOZ,
\$                            IHIZ, A, LDA1, LDA2, Z, LDZ1, LDZ2,
\$                            WR, WI, DWORK, LDWORK, INFO )
IF ( INFO.GT.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
WRITE ( NOUT, FMT = 99991 )
DO 30 I = MAX( ILO, INFO + 1 ), IHI
WRITE ( NOUT, FMT = 99990 ) WR(I), WI(I)
30                   CONTINUE
STOP
END IF
IF ( INFO.LT.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
*                       Store the isolated eigenvalues.
CALL MB03WX( ILO-1, P, A, LDA1, LDA2, WR, WI,
\$                               INFO )
IF ( IHI.LT.N )
\$                     CALL MB03WX( N-IHI, P, A(IHI+1,IHI+1,1),
\$                                  LDA1, LDA2, WR(IHI+1),
\$                                  WI(IHI+1), INFO )
WRITE ( NOUT, FMT = 99991 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99990 ) WR(I), WI(I)
40                   CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 60 K = 1, P
WRITE ( NOUT, FMT = 99994 ) K
DO 50 I = 1, N
WRITE ( NOUT, FMT = 99993 )
\$                              ( A(I,J,K), J = 1, N )
50                      CONTINUE
60                   CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 80 K = 1, P
WRITE ( NOUT, FMT = 99994 ) K
DO 70 I = 1, N
WRITE ( NOUT, FMT = 99993 )
\$                              ( Z(I,J,K), J = 1, N )
70                      CONTINUE
80                   CONTINUE
*                       Compute error.
SSQ = ZERO
DO 90 K = 1, P
KP1 = K+1
IF( KP1.GT.P ) KP1 = 1
*                          Compute NORM (Z' * A * Z - Aout)
CALL DGEMM( 'T', 'N', N, N, N, ONE, Z(1,1,K),
\$                                 LDZ1, AS(1,1,K), LDA1, ZERO, ZTA,
\$                                 LDZTA )
CALL DGEMM( 'N', 'N', N, N, N, ONE, ZTA,
\$                                 LDZTA, Z(1,1,KP1), LDZ1, -ONE,
\$                                 A(1,1,K), LDA1 )
SSQ = DLAPY2( SSQ,
\$                                   DLANGE( 'Frobenius', N, N,
\$                                           A(1,1,K), LDA1,
\$                                           DWORK ) )
90                   CONTINUE
WRITE ( NOUT, FMT = 99989 ) SSQ
END IF
END IF
END IF
END IF
END IF
END IF
STOP
99999 FORMAT (' MB03WD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from MB03WD = ', I2)
99997 FORMAT (' INFO on exit from MB03VD = ', I2)
99996 FORMAT (' INFO on exit from MB03VY = ', I2)
99995 FORMAT (/' Reduced matrices')
99994 FORMAT (/' K = ', I5)
99993 FORMAT (8F8.4)
99992 FORMAT (/' Transformation matrices')
99991 FORMAT ( ' Computed eigenvalues'/)
99990 FORMAT (4X,'( ', F17.6,' ,', F17.6,' )')
99989 FORMAT (/,' NORM (Z''*A*Z - Aout) = ', 1PD12.5)
99988 FORMAT (/, ' N is out of range.',/' N = ', I5)
99987 FORMAT (/, ' P is out of range.',/' P = ', I5)
END
```
Program Data
```MB03WD EXAMPLE PROGRAM DATA
4 2 1 4 1 4 S V
1.5 -.7 3.5 -.7
1.  0.  2.  3.
1.5 -.7 2.5 -.3
1.  0.  2.  1.
1.5 -.7 3.5 -.7
1.  0.  2.  3.
1.5 -.7 2.5 -.3
1.  0.  2.  1.
```
Program Results
``` MB03WD EXAMPLE PROGRAM RESULTS

Computed eigenvalues

(          6.449861 ,         7.817717 )
(          6.449861 ,        -7.817717 )
(          0.091315 ,         0.000000 )
(          0.208964 ,         0.000000 )

Reduced matrices

K =     1
2.2112  4.3718 -2.3362  0.8907
-0.9179  2.7688 -0.6570 -2.2426
0.0000  0.0000  0.3022  0.1932
0.0000  0.0000  0.0000 -0.4571

K =     2
2.9169  3.4539  2.2016  1.2367
0.0000  3.4745  1.0209 -2.0720
0.0000  0.0000  0.3022 -0.1932
0.0000  0.0000  0.0000 -0.4571

Transformation matrices

K =     1
0.3493  0.6751 -0.6490  0.0327
0.7483 -0.4863 -0.1249 -0.4336
0.2939  0.5504  0.7148 -0.3158
0.4813 -0.0700  0.2286  0.8433

K =     2
0.2372  0.7221  0.6490  0.0327
0.8163 -0.3608  0.1249 -0.4336
0.2025  0.5902 -0.7148 -0.3158
0.4863  0.0076 -0.2286  0.8433

NORM (Z'*A*Z - Aout) =  7.18432D-15
```