**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. 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.

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 * 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

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