## MB03VD

### Periodic Hessenberg form of a product of p matrices using orthogonal similarity transformations

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

Purpose

```  To reduce a product of p real general matrices A = A_1*A_2*...*A_p
to upper Hessenberg form, H = H_1*H_2*...*H_p, where H_1 is
upper Hessenberg, and H_2, ..., H_p are upper triangular, by using
orthogonal similarity transformations on A,

Q_1' * A_1 * Q_2 = H_1,
Q_2' * A_2 * Q_3 = H_2,
...
Q_p' * A_p * Q_1 = H_p.

```
Specification
```      SUBROUTINE MB03VD( N, P, ILO, IHI, A, LDA1, LDA2, TAU, LDTAU,
\$                   DWORK, INFO )
C     .. Scalar Arguments ..
INTEGER           IHI, ILO, INFO, LDA1, LDA2, LDTAU, N, P
C     .. Array Arguments ..
DOUBLE PRECISION  A( LDA1, LDA2, * ), DWORK( * ), TAU( LDTAU, * )

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The order of the square matrices A_1, A_2, ..., A_p.
N >= 0.

P       (input) INTEGER
The number of matrices in the product A_1*A_2*...*A_p.
P >= 1.

ILO     (input) INTEGER
IHI     (input) INTEGER
It is assumed that all matrices A_j, j = 2, ..., p, are
already upper triangular in rows and columns 1:ILO-1 and
IHI+1:N, and A_1 is upper Hessenberg in rows and columns
1:ILO-1 and IHI+1:N, with A_1(ILO,ILO-1) = 0 (unless
ILO = 1), and A_1(IHI+1,IHI) = 0 (unless IHI = N).
If this is not the case, ILO and IHI should be set to 1
and N, respectively.
1 <= ILO <= max(1,N); min(ILO,N) <= IHI <= N.

A       (input/output) DOUBLE PRECISION array, dimension
(LDA1,LDA2,P)
On entry, the leading N-by-N-by-P part of this array must
contain the matrices of factors to be reduced;
specifically, A(*,*,j) must contain A_j, j = 1, ..., p.
On exit, the leading N-by-N upper triangle and the first
subdiagonal of A(*,*,1) contain the upper Hessenberg
matrix H_1, and the elements below the first subdiagonal,
with the first column of the array TAU represent the
orthogonal matrix Q_1 as a product of elementary
For j > 1, the leading N-by-N upper triangle of A(*,*,j)
contains the upper triangular matrix H_j, and the elements
below the diagonal, with the j-th column of the array TAU
represent the orthogonal matrix Q_j as a product of

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).

TAU     (output) DOUBLE PRECISION array, dimension (LDTAU,P)
The leading N-1 elements in the j-th column contain the
scalar factors of the elementary reflectors used to form
the matrix Q_j, j = 1, ..., P. See FURTHER COMMENTS.

LDTAU   INTEGER
The leading dimension of the array TAU.
LDTAU >= max(1,N-1).

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

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  The algorithm consists in ihi-ilo major steps. In each such
step i, ilo <= i <= ihi-1, the subdiagonal elements in the i-th
column of A_j are annihilated using a Householder transformation
from the left, which is also applied to A_(j-1) from the right,
for j = p:-1:2. Then, the elements below the subdiagonal of the
i-th column of A_1 are annihilated, and the Householder
transformation is also applied to A_p from the right.

```
References
```   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.

 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.

```
```  Each matrix Q_j is represented as a product of (ihi-ilo)
elementary reflectors,

Q_j = H_j(ilo) H_j(ilo+1) . . . H_j(ihi-1).

Each H_j(i), i = ilo, ..., ihi-1, has the form

H_j(i) = I - tau_j * v_j * v_j',

where tau_j is a real scalar, and v_j is a real vector with
v_j(1:i) = 0, v_j(i+1) = 1 and v_j(ihi+1:n) = 0; v_j(i+2:ihi)
is stored on exit in A_j(i+2:ihi,i), and tau_j in TAU(i,j).

The contents of A_1 are illustrated by the following example
for n = 7, ilo = 2, and ihi = 6:

on entry                         on exit

( a   a   a   a   a   a   a )    ( a   h   h   h   h   h   a )
( 0   a   a   a   a   a   a )    ( 0   h   h   h   h   h   a )
( 0   a   a   a   a   a   a )    ( 0   h   h   h   h   h   h )
( 0   a   a   a   a   a   a )    ( 0   v2  h   h   h   h   h )
( 0   a   a   a   a   a   a )    ( 0   v2  v3  h   h   h   h )
( 0   a   a   a   a   a   a )    ( 0   v2  v3  v4  h   h   h )
( 0   0   0   0   0   0   a )    ( 0   0   0   0   0   0   a )

where a denotes an element of the original matrix A_1, h denotes
a modified element of the upper Hessenberg matrix H_1, and vi
denotes an element of the vector defining H_1(i).

The contents of A_j, j > 1, are illustrated by the following
example for n = 7, ilo = 2, and ihi = 6:

on entry                         on exit

( a   a   a   a   a   a   a )    ( a   h   h   h   h   h   a )
( 0   a   a   a   a   a   a )    ( 0   h   h   h   h   h   h )
( 0   a   a   a   a   a   a )    ( 0   v2  h   h   h   h   h )
( 0   a   a   a   a   a   a )    ( 0   v2  v3  h   h   h   h )
( 0   a   a   a   a   a   a )    ( 0   v2  v3  v4  h   h   h )
( 0   a   a   a   a   a   a )    ( 0   v2  v3  v4  v5  h   h )
( 0   0   0   0   0   0   a )    ( 0   0   0   0   0   0   a )

where a denotes an element of the original matrix A_j, h denotes
a modified element of the upper triangular matrix H_j, and vi
denotes an element of the vector defining H_j(i). (The element
(1,2) in A_p is also unchanged for this example.)

Note that for P = 1, the LAPACK Library routine DGEHRD could be
more efficient on some computer architectures than this routine
(a BLAS 2 version).

```
Example

Program Text

```*     MB03VD 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, LDQ1, LDQ2, LDTAU
PARAMETER        ( LDA1 = NMAX, LDA2 = NMAX, LDQ1 = NMAX,
\$                   LDQ2 = NMAX, LDTAU = NMAX-1 )
INTEGER          LDWORK
PARAMETER        ( LDWORK = NMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
*     .. Local Scalars ..
DOUBLE PRECISION SSQ
INTEGER          I, IHI, ILO, INFO, J, K, KP1, N, P
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA1,LDA2,PMAX), AS(LDA1,LDA2,PMAX),
\$                 DWORK(LDWORK), Q(LDQ1,LDQ2,PMAX),
\$                 QTA(LDQ1,NMAX), TAU(LDTAU,PMAX)
*     .. External Functions ..
DOUBLE PRECISION DLANGE, DLAPY2
EXTERNAL         DLANGE, DLAPY2
*     .. External Subroutines ..
EXTERNAL         DGEMM, DLACPY, DLASET, MB03VD, MB03VY
*     .. Intrinsic Functions ..
INTRINSIC        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
IF ( N.LT.0 .OR. N.GT.MIN( LDA1, LDA2 ) ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) 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 = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99996 )
DO 30 K = 1, P
CALL DLACPY( 'L', N, N, A(1,1,K), LDA1, Q(1,1,K),
\$                         LDQ1 )
IF ( N.GT.1 ) THEN
IF ( N.GT.2 .AND. K.EQ.1 ) THEN
CALL DLASET( 'L', N-2, N-2, ZERO, ZERO,
\$                               A(3,1,K), LDA1 )
ELSE IF ( K.GT.1 ) THEN
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
\$                               A(2,1,K), LDA1 )
END IF
END IF
WRITE ( NOUT, FMT = 99995 ) K
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( A(I,J,K), J = 1, N )
20             CONTINUE
30          CONTINUE
*              Accumulate the transformations.
CALL MB03VY( N, P, ILO, IHI, Q, LDQ1, LDQ2, TAU, LDTAU,
\$                      DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99993 )
DO 50 K = 1, P
WRITE ( NOUT, FMT = 99995 ) K
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 )
\$                        ( Q(I,J,K), J = 1, N )
40                CONTINUE
50             CONTINUE
*                 Compute error.
SSQ = ZERO
DO 60 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, Q(1,1,K), LDQ1,
\$                           AS(1,1,K), LDA1, ZERO, QTA, LDQ1 )
CALL DGEMM( 'N', 'N', N, N, N, ONE, QTA, LDQ1,
\$                           Q(1,1,KP1), LDQ1, -ONE, A(1,1,K),
\$                           LDA1 )
SSQ = DLAPY2( SSQ,
\$                             DLANGE( 'Frobenius', N, N, A(1,1,K),
\$                                     LDA1, DWORK ) )
60             CONTINUE
WRITE ( NOUT, FMT = 99992 ) SSQ
END IF
END IF
END IF
END IF
STOP
99999 FORMAT (' MB03VD EXAMPLE PROGRAM RESULTS', /1X)
99998 FORMAT (' INFO on exit from MB03VD = ', I2)
99997 FORMAT (' INFO on exit from MB03VY = ', I2)
99996 FORMAT (' Reduced matrices')
99995 FORMAT (/' K = ', I5)
99994 FORMAT (8F8.4)
99993 FORMAT (/' Transformation matrices')
99992 FORMAT (/,' NORM (Q''*A*Q - Aout) = ', 1PD12.5)
99991 FORMAT (/, ' N is out of range.',/' N = ', I5)
99990 FORMAT (/, ' P is out of range.',/' P = ', I5)
END
```
Program Data
```MB03VD EXAMPLE PROGRAM DATA
4 2 1 4
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
``` MB03VD EXAMPLE PROGRAM RESULTS

Reduced matrices

K =     1
-2.3926  2.7042 -0.9598 -1.2335
4.1417 -1.7046  1.3001 -1.3120
0.0000 -1.6247 -0.2534  1.6453
0.0000  0.0000 -0.0169 -0.4451

K =     2
-2.5495  2.3402  4.7021  0.2329
0.0000  1.9725 -0.2483 -2.3493
0.0000  0.0000 -0.6290 -0.5975
0.0000  0.0000  0.0000 -0.4426

Transformation matrices

K =     1
1.0000  0.0000  0.0000  0.0000
0.0000 -0.7103  0.5504 -0.4388
0.0000 -0.4735 -0.8349 -0.2807
0.0000 -0.5209  0.0084  0.8536

K =     2
-0.5883  0.2947  0.7528 -0.0145
-0.3922 -0.8070  0.0009 -0.4415
-0.5883  0.4292 -0.6329 -0.2630
-0.3922 -0.2788 -0.1809  0.8577

NORM (Q'*A*Q - Aout) =  2.93760D-15
```