## MB02ID

### Solution of over- or underdetermined linear systems with a full rank block Toeplitz matrix

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

Purpose

```  To solve the overdetermined or underdetermined real linear systems
involving an M*K-by-N*L block Toeplitz matrix T that is specified
by its first block column and row. It is assumed that T has full
rank.
The following options are provided:

1. If JOB = 'O' or JOB = 'A' :  find the least squares solution of
an overdetermined system, i.e., solve the least squares problem

minimize || B - T*X ||.                           (1)

2. If JOB = 'U' or JOB = 'A' :  find the minimum norm solution of
the undetermined system
T
T * X = C.                                        (2)

```
Specification
```      SUBROUTINE MB02ID( JOB, K, L, M, N, RB, RC, TC, LDTC, TR, LDTR, B,
\$                   LDB, C, LDC, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         JOB
INTEGER           INFO, K, L, LDB, LDC, LDTC, LDTR, LDWORK, M, N,
\$                  RB, RC
C     .. Array Arguments ..
DOUBLE PRECISION  B(LDB,*), C(LDC,*), DWORK(LDWORK), TC(LDTC,*),
\$                  TR(LDTR,*)

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Specifies the problem to be solved as follows
= 'O':  solve the overdetermined system (1);
= 'U':  solve the underdetermined system (2);
= 'A':  solve (1) and (2).

```
Input/Output Parameters
```  K       (input) INTEGER
The number of rows in the blocks of T.  K >= 0.

L       (input) INTEGER
The number of columns in the blocks of T.  L >= 0.

M       (input) INTEGER
The number of blocks in the first block column of T.
M >= 0.

N       (input) INTEGER
The number of blocks in the first block row of T.
0 <= N <= M*K / L.

RB      (input) INTEGER
If JOB = 'O' or 'A', the number of columns in B.  RB >= 0.

RC      (input) INTEGER
If JOB = 'U' or 'A', the number of columns in C.  RC >= 0.

TC      (input)  DOUBLE PRECISION array, dimension (LDTC,L)
On entry, the leading M*K-by-L part of this array must
contain the first block column of T.

LDTC    INTEGER
The leading dimension of the array TC.  LDTC >= MAX(1,M*K)

TR      (input)  DOUBLE PRECISION array, dimension (LDTR,(N-1)*L)
On entry, the leading K-by-(N-1)*L part of this array must
contain the 2nd to the N-th blocks of the first block row
of T.

LDTR    INTEGER
The leading dimension of the array TR.  LDTR >= MAX(1,K).

B       (input/output)  DOUBLE PRECISION array, dimension (LDB,RB)
On entry, if JOB = 'O' or JOB = 'A', the leading M*K-by-RB
part of this array must contain the right hand side
matrix B of the overdetermined system (1).
On exit, if JOB = 'O' or JOB = 'A', the leading N*L-by-RB
part of this array contains the solution of the
overdetermined system (1).
This array is not referenced if JOB = 'U'.

LDB     INTEGER
The leading dimension of the array B.
LDB >= MAX(1,M*K),  if JOB = 'O'  or  JOB = 'A';
LDB >= 1,           if JOB = 'U'.

C       (input)  DOUBLE PRECISION array, dimension (LDC,RC)
On entry, if JOB = 'U' or JOB = 'A', the leading N*L-by-RC
part of this array must contain the right hand side
matrix C of the underdetermined system (2).
On exit, if JOB = 'U' or JOB = 'A', the leading M*K-by-RC
part of this array contains the solution of the
underdetermined system (2).
This array is not referenced if JOB = 'O'.

LDC     INTEGER
The leading dimension of the array C.
LDB >= 1,           if JOB = 'O';
LDB >= MAX(1,M*K),  if JOB = 'U'  or  JOB = 'A'.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0,  DWORK(1)  returns the optimal
value of LDWORK.
On exit, if  INFO = -17,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
Let x = MAX( 2*N*L*(L+K) + (6+N)*L,(N*L+M*K+1)*L + M*K )
and y = N*M*K*L + N*L, then
if MIN( M,N ) = 1 and JOB = 'O',
LDWORK >= MAX( y + MAX( M*K,RB ),1 );
if MIN( M,N ) = 1 and JOB = 'U',
LDWORK >= MAX( y + MAX( M*K,RC ),1 );
if MIN( M,N ) = 1 and JOB = 'A',
LDWORK >= MAX( y +MAX( M*K,MAX( RB,RC ),1 );
if MIN( M,N ) > 1 and JOB = 'O',
LDWORK >= MAX( x,N*L*RB + 1 );
if MIN( M,N ) > 1 and JOB = 'U',
LDWORK >= MAX( x,N*L*RC + 1 );
if MIN( M,N ) > 1 and JOB = 'A',
LDWORK >= MAX( x,N*L*MAX( RB,RC ) + 1 ).
For optimum performance LDWORK should be 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.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  the reduction algorithm failed. The Toeplitz matrix
associated with T is (numerically) not of full rank.

```
Method
```  Householder transformations and modified hyperbolic rotations
are used in the Schur algorithm , .

```
References
```   Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.

 Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.

```
Numerical Aspects
```  The algorithm requires O( L*L*K*(N+M)*log(N+M) + N*N*L*L*(L+K) )

if JOB = 'O' or JOB = 'A',
O( (K*L+RB*L+K*RB)*(N+M)*log(N+M) + N*N*L*L*RB );
if JOB = 'U' or JOB = 'A',
O( (K*L+RC*L+K*RC)*(N+M)*log(N+M) + N*N*L*L*RC );

floating point operations.

```
```  None
```
Example

Program Text

```*     MB02ID EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          KMAX, LMAX, MMAX, NMAX, RBMAX, RCMAX
PARAMETER        ( KMAX  = 20, LMAX  = 20, MMAX = 20, NMAX = 20,
\$                   RBMAX = 20, RCMAX = 20 )
INTEGER          LDB, LDC, LDTC, LDTR, LDWORK
PARAMETER        ( LDB  = KMAX*MMAX, LDC  = KMAX*MMAX,
\$                   LDTC = MMAX*KMAX, LDTR = KMAX,
\$                   LDWORK = 2*NMAX*LMAX*( LMAX + KMAX ) +
\$                            ( 6 + NMAX )*LMAX +
\$                            MMAX*KMAX*( LMAX + 1 ) +
\$                            RBMAX + RCMAX )
*     .. Local Scalars ..
INTEGER          I, INFO, J, K, L, M, N, RB, RC
CHARACTER        JOB
DOUBLE PRECISION B(LDB,RBMAX),  C(LDC,RCMAX), DWORK(LDWORK),
\$                 TC(LDTC,LMAX), TR(LDTR,NMAX*LMAX)
*     .. External Functions ..
LOGICAL          LSAME
EXTERNAL         LSAME
*     .. External Subroutines ..
EXTERNAL         MB02ID
*
*     .. Executable Statements ..
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * )  K, L, M, N, RB, RC, JOB
IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) K
ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) L
ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) N
ELSE IF ( ( LSAME( JOB, 'O' ) .OR. LSAME( JOB, 'A' ) )
\$          .AND. ( ( RB.LE.0 ) .OR. ( RB.GT.RBMAX ) ) ) THEN
WRITE ( NOUT, FMT = 99990 ) RB
ELSE IF ( ( LSAME( JOB, 'U' ) .OR. LSAME( JOB, 'A' ) )
\$          .AND. ( ( RC.LE.0 ) .OR. ( RC.GT.RCMAX ) ) ) THEN
WRITE ( NOUT, FMT = 99989 ) RC
ELSE
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,M*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,(N-1)*L ), I = 1,K )
IF ( LSAME( JOB, 'O' ) .OR. LSAME( JOB, 'A' ) ) THEN
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,RB ), I = 1,M*K )
END IF
IF ( LSAME( JOB, 'U' ) .OR. LSAME( JOB, 'A' ) ) THEN
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,RC ), I = 1,N*L )
END IF
CALL MB02ID( JOB, K, L, M, N, RB, RC, TC, LDTC, TR, LDTR, B,
\$                LDB, C, LDC, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
IF ( LSAME( JOB, 'O' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10  I = 1, N*L
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, RB )
10          CONTINUE
END IF
IF ( LSAME( JOB, 'U' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 20  I = 1, M*K
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, RC )
20          CONTINUE
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB02ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02ID = ',I2)
99997 FORMAT (' The least squares solution of T * X = B is ')
99996 FORMAT (' The minimum norm solution of T^T * X = C is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' K is out of range.',/' K = ',I5)
99993 FORMAT (/' L is out of range.',/' L = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' N is out of range.',/' N = ',I5)
99990 FORMAT (/' RB is out of range.',/' RB = ',I5)
99989 FORMAT (/' RC is out of range.',/' RC = ',I5)
END
```
Program Data
```MB02ID EXAMPLE PROGRAM DATA
3   2   4   3   1   1   A
5.0     2.0
1.0     2.0
4.0     3.0
4.0     0.0
2.0     2.0
3.0     3.0
5.0     1.0
3.0     3.0
1.0     1.0
2.0     3.0
1.0     3.0
2.0     2.0
1.0     4.0     2.0     3.0
2.0     2.0     2.0     4.0
3.0     1.0     0.0     1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
```
Program Results
``` MB02ID EXAMPLE PROGRAM RESULTS

The least squares solution of T * X = B is
0.0379
0.1677
0.0485
-0.0038
0.0429
0.1365
The minimum norm solution of T^T * X = C is
0.0509
0.0547
0.0218
0.0008
0.0436
0.0404
0.0031
0.0451
0.0421
0.0243
0.0556
0.0472
```