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 [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
  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

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