**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)

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,*)

**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).

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'.

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.

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.

Householder transformations and modified hyperbolic rotations are used in the Schur algorithm [1], [2].

[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.

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.

None

**Program Text**

* MB02ID EXAMPLE PROGRAM TEXT * RELEASE 5.0, SLICOT COPYRIGHT 2001. * * .. 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

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

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