## MB02ED

### Solution of T X = B or X T = B with a positive definite block Toeplitz matrix T

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

Purpose

      To solve a system of linear equations  T*X = B  or  X*T = B  with
      a symmetric positive definite (s.p.d.) block Toeplitz matrix T.
      T is defined either by its first block row or its first block
      column, depending on the parameter TYPET.

Specification

          SUBROUTINE MB02ED( TYPET, K, N, NRHS, T, LDT, B, LDB, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         TYPET
          INTEGER           INFO, K, LDB, LDT, LDWORK, N, NRHS
    C     .. Array Arguments ..
          DOUBLE PRECISION  B(LDB,*), DWORK(*), T(LDT,*)

Arguments

Mode Parameters

      TYPET   CHARACTER*1
              Specifies the type of T, as follows:
              = 'R':  T contains the first block row of an s.p.d. block
                      Toeplitz matrix, and the system X*T = B is solved;
              = 'C':  T contains the first block column of an s.p.d.
                      block Toeplitz matrix, and the system T*X = B is
                      solved.
              Note:   in the sequel, the notation x / y means that
                      x corresponds to TYPET = 'R' and y corresponds to
                      TYPET = 'C'.

Input/Output Parameters

      K       (input)  INTEGER
              The number of rows / columns in T, which should be equal
              to the blocksize.  K >= 0.

      N       (input)  INTEGER
              The number of blocks in T.  N >= 0.

      NRHS    (input)  INTEGER
              The number of right hand sides.  NRHS >= 0.

      T       (input/output)  DOUBLE PRECISION array, dimension
              (LDT,N*K) / (LDT,K)
              On entry, the leading K-by-N*K / N*K-by-K part of this
              array must contain the first block row / column of an
              s.p.d. block Toeplitz matrix.
              On exit, if  INFO = 0  and  NRHS > 0,  then the leading
              K-by-N*K / N*K-by-K part of this array contains the last
              row / column of the Cholesky factor of inv(T).

      LDT     INTEGER
              The leading dimension of the array T.
              LDT >= MAX(1,K),    if TYPET = 'R';
              LDT >= MAX(1,N*K),  if TYPET = 'C'.

      B       (input/output) DOUBLE PRECISION array, dimension
              (LDB,N*K) / (LDB,NRHS)
              On entry, the leading NRHS-by-N*K / N*K-by-NRHS part of
              this array must contain the right hand side matrix B.
              On exit, the leading NRHS-by-N*K / N*K-by-NRHS part of
              this array contains the solution matrix X.

      LDB     INTEGER
              The leading dimension of the array B.
              LDB >= MAX(1,NRHS),  if TYPET = 'R';
              LDB >= MAX(1,N*K),   if TYPET = 'C'.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0,  DWORK(1)  returns the optimal
              value of LDWORK.
              On exit, if  INFO = -10,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX(1,N*K*K+(N+2)*K).
              For optimum performance LDWORK should be larger.

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 not (numerically) positive
                    definite.

Method

      Householder transformations, modified hyperbolic rotations and
      block Gaussian eliminations 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 implemented method is numerically equivalent with forming
      the Cholesky factor R and the inverse Cholesky factor of T, using
      the generalized Schur algorithm, and solving the systems of
      equations  R*X = L*B  or  X*R = B*L by a blocked backward
      substitution algorithm.
                                3 2    2 2
      The algorithm requires 0(K N  + K N NRHS) floating point
      operations.

Further Comments

      None

Example

Program Text

    *     MB02ED EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          KMAX, NMAX
          PARAMETER        ( KMAX = 20, NMAX = 20 )
          INTEGER          LDB, LDT, LDWORK
          PARAMETER        ( LDB = KMAX*NMAX, LDT = KMAX*NMAX,
         $                   LDWORK = NMAX*KMAX*KMAX + ( NMAX+2 )*KMAX )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, K, M, N, NRHS
          CHARACTER        TYPET
    *     .. Local Arrays ..
    *     The arrays B and T are dimensioned for both TYPET = 'R' and
    *     TYPET = 'C'.
    *     NRHS is assumed to be not larger than KMAX*NMAX.
          DOUBLE PRECISION B(LDB, KMAX*NMAX), DWORK(LDWORK),
         $                 T(LDT, KMAX*NMAX)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         MB02ED
    *
    *     .. Executable Statements ..
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, K, NRHS, TYPET
          M = N*K
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE
             IF ( K.LE.0 .OR. K.GT.KMAX ) THEN
                WRITE ( NOUT, FMT = 99993 ) K
             ELSE
                IF ( NRHS.LE.0 .OR. NRHS.GT.KMAX*NMAX ) THEN
                   WRITE ( NOUT, FMT = 99992 ) NRHS
                ELSE
                   IF ( LSAME( TYPET, 'R' ) ) THEN
                      READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,M ), I = 1,K )
                   ELSE
                      READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,K ), I = 1,M )
                   END IF
                   IF ( LSAME( TYPET, 'R' ) ) THEN
                      READ (NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,
         $                                   NRHS )
                   ELSE
                      READ (NIN, FMT = * ) ( ( B(I,J), J = 1,NRHS ), I = 1,
         $                                   M )
                   END IF
    *              Compute the solution of X T = B or T X = B.
                   CALL MB02ED( TYPET, K, N, NRHS, T, LDT, B, LDB, DWORK,
         $                      LDWORK, INFO )
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      IF ( LSAME( TYPET, 'R' ) ) THEN
                         WRITE ( NOUT, FMT = 99997 )
                         DO 10  I = 1, NRHS
                            WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
       10                CONTINUE
                      ELSE
                         WRITE ( NOUT, FMT = 99996 )
                         DO 20  I = 1, M
                            WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,
         $                                                NRHS )
       20                CONTINUE
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB02ED EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB02ED = ',I2)
    99997 FORMAT (' The solution of X*T = B is ')
    99996 FORMAT (' The solution of T*X = B is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
    99993 FORMAT (/' K is out of range.',/' K = ',I5)
    99992 FORMAT (/' NRHS is out of range.',/' NRHS = ',I5)
          END

Program Data

    MB02ED EXAMPLE PROGRAM DATA
      3    3    2     C
        3.0000    1.0000    0.2000
        1.0000    4.0000    0.4000
        0.2000    0.4000    5.0000
        0.1000    0.1000    0.2000
        0.2000    0.0400    0.0300
        0.0500    0.2000    0.1000
        0.1000    0.0300    0.1000
        0.0400    0.0200    0.2000
        0.0100    0.0300    0.0200
        1.0000    2.0000
        1.0000    2.0000
        1.0000    2.0000
        1.0000    2.0000
        1.0000    2.0000
        1.0000    2.0000
        1.0000    2.0000
        1.0000    2.0000
        1.0000    2.0000

Program Results

     MB02ED EXAMPLE PROGRAM RESULTS

     The solution of T*X = B is
       0.2408   0.4816
       0.1558   0.3116
       0.1534   0.3068
       0.2302   0.4603
       0.1467   0.2934
       0.1537   0.3075
       0.2349   0.4698
       0.1498   0.2995
       0.1653   0.3307

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