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 [1], [2].References
[1] Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.
SIAM Publications, Philadelphia, 1999.
[2] 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) )
and additionally
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.
Further Comments
NoneExample
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