## MB02DD

### Updating Cholesky factorization of a positive definite block Toeplitz matrix

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

Purpose

      To update the Cholesky factor and the generator and/or the
      Cholesky factor of the inverse of a symmetric positive definite
      (s.p.d.) block Toeplitz matrix T, given the information from
      a previous factorization and additional blocks in TA of its first
      block row, or its first block column, depending on the routine
      parameter TYPET. Transformation information is stored.

Specification

          SUBROUTINE MB02DD( JOB, TYPET, K, M, N, TA, LDTA, T, LDT, G,
         $                   LDG, R, LDR, L, LDL, CS, LCS, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JOB, TYPET
          INTEGER           INFO, K, LCS, LDG, LDL, LDR, LDT, LDTA, LDWORK,
         $                  M, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  CS(*), DWORK(*), G(LDG, *), L(LDL,*), R(LDR,*),
         $                  T(LDT,*), TA(LDTA,*)

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Specifies the output of the routine, as follows:
              = 'R':  updates the generator G of the inverse and
                      computes the new columns / rows for the Cholesky
                      factor R of T;
              = 'A':  updates the generator G, computes the new
                      columns / rows for the Cholesky factor R of T and
                      the new rows / columns for the Cholesky factor L
                      of the inverse;
              = 'O':  only computes the new columns / rows for the
                      Cholesky factor R of T.

      TYPET   CHARACTER*1
              Specifies the type of T, as follows:
              = 'R':  the first block row of an s.p.d. block Toeplitz
                      matrix was/is defined; if demanded, the Cholesky
                      factors R and L are upper and lower triangular,
                      respectively, and G contains the transposed
                      generator of the inverse;
              = 'C':  the first block column of an s.p.d. block Toeplitz
                      matrix was/is defined; if demanded, the Cholesky
                      factors R and L are lower and upper triangular,
                      respectively, and G contains the generator of the
                      inverse. This choice results in a column oriented
                      algorithm which is usually faster.
              Note:   in this routine, 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.

      M       (input)  INTEGER
              The number of blocks in TA.  M >= 0.

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

      TA      (input/output)  DOUBLE PRECISION array, dimension
              (LDTA,M*K) / (LDTA,K)
              On entry, the leading K-by-M*K / M*K-by-K part of this
              array must contain the (N+1)-th to (N+M)-th blocks in the
              first block row / column of an s.p.d. block Toeplitz
              matrix.
              On exit, if INFO = 0, the leading K-by-M*K / M*K-by-K part
              of this array contains information on the Householder
              transformations used, such that the array

                         [ T  TA ]    /    [ T  ]
                                           [ TA ]

              serves as the new transformation matrix T for further
              applications of this routine.

      LDTA    INTEGER
              The leading dimension of the array TA.
              LDTA >= MAX(1,K),   if TYPET = 'R';
              LDTA >= MAX(1,M*K), if TYPET = 'C'.

      T       (input)  DOUBLE PRECISION array, dimension (LDT,N*K) /
              (LDT,K)
              The leading K-by-N*K / N*K-by-K part of this array must
              contain transformation information generated by the SLICOT
              Library routine MB02CD, i.e., in the first K-by-K block,
              the upper / lower Cholesky factor of T(1:K,1:K), and in
              the remaining part, the Householder transformations
              applied during the initial factorization process.

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

      G       (input/output)  DOUBLE PRECISION array, dimension
              (LDG,( N + M )*K) / (LDG,2*K)
              On entry, if JOB = 'R', or 'A', then the leading
              2*K-by-N*K / N*K-by-2*K part of this array must contain,
              in the first K-by-K block of the second block row /
              column, the lower right block of the Cholesky factor of
              the inverse of T, and in the remaining part, the generator
              of the inverse of T.
              On exit, if INFO = 0 and JOB = 'R', or 'A', then the
              leading 2*K-by-( N + M )*K / ( N + M )*K-by-2*K part of
              this array contains the same information as on entry, now
              for the updated Toeplitz matrix. Actually, to obtain a
              generator of the inverse one has to set
                G(K+1:2*K, 1:K) = 0,    if TYPET = 'R';
                G(1:K, K+1:2*K) = 0,    if TYPET = 'C'.

      LDG     INTEGER
              The leading dimension of the array G.
              LDG >= MAX(1,2*K),  if TYPET = 'R' and JOB = 'R', or 'A';
              LDG >= MAX(1,( N + M )*K),
                                  if TYPET = 'C' and JOB = 'R', or 'A';
              LDG >= 1,           if JOB = 'O'.

      R       (input/output)  DOUBLE PRECISION array, dimension
              (LDR,M*K) / (LDR,( N + M )*K)
              On input, the leading N*K-by-K part of R(K+1,1) /
              K-by-N*K part of R(1,K+1) contains the last block column /
              row of the previous Cholesky factor R.
              On exit, if INFO = 0, then the leading
              ( N + M )*K-by-M*K / M*K-by-( N + M )*K part of this
              array contains the last M*K columns / rows of the upper /
              lower Cholesky factor of T. The elements in the strictly
              lower / upper triangular part are not referenced.

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

      L       (output)  DOUBLE PRECISION array, dimension
              (LDL,( N + M )*K) / (LDL,M*K)
              If INFO = 0 and JOB = 'A', then the leading
              M*K-by-( N + M )*K / ( N + M )*K-by-M*K part of this
              array contains the last M*K rows / columns of the lower /
              upper Cholesky factor of the inverse of T. The elements
              in the strictly upper / lower triangular part are not
              referenced.

      LDL     INTEGER
              The leading dimension of the array L.
              LDL >= MAX(1, M*K),         if TYPET = 'R' and JOB = 'A';
              LDL >= MAX(1, ( N + M )*K), if TYPET = 'C' and JOB = 'A';
              LDL >= 1,                   if JOB = 'R', or 'O'.

      CS      (input/output)  DOUBLE PRECISION array, dimension (LCS)
              On input, the leading 3*(N-1)*K part of this array must
              contain the necessary information about the hyperbolic
              rotations and Householder transformations applied
              previously by SLICOT Library routine MB02CD.
              On exit, if INFO = 0, then the leading 3*(N+M-1)*K part of
              this array contains information about all the hyperbolic
              rotations and Householder transformations applied during
              the whole process.

      LCS     INTEGER
              The length of the array CS.  LCS >= 3*(N+M-1)*K.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0,  DWORK(1)  returns the optimal
              value of LDWORK.
              On exit, if  INFO = -19,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX(1,(N+M-1)*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 block Toeplitz
                    matrix associated with [ T  TA ] / [ T'  TA' ]' is
                    not (numerically) positive definite.

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 implemented method is numerically stable.
                                3         2
      The algorithm requires 0(K ( N M + M ) ) floating point
      operations.

Further Comments

      For min(K,N,M) = 0, the routine sets DWORK(1) = 1 and returns.
      Although the calculations could still be performed when N = 0,
      but min(K,M) > 0, this case is not considered as an "update".
      SLICOT Library routine MB02CD should be called with the argument
      M instead of N.

Example

Program Text

    *     MB02DD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          KMAX, MMAX, NMAX
          PARAMETER        ( KMAX = 20, MMAX = 20, NMAX = 20 )
          INTEGER          LCS, LDG, LDL, LDR, LDT, LDWORK
          PARAMETER        ( LDG = KMAX*( MMAX + NMAX ),
         $                   LDL = KMAX*( MMAX + NMAX ),
         $                   LDR = KMAX*( MMAX + NMAX ),
         $                   LDT = KMAX*( MMAX + NMAX ),
         $                   LDWORK = ( MMAX + NMAX - 1 )*KMAX )
          PARAMETER        ( LCS = 3*LDWORK )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, K, M, N, S
          CHARACTER        JOB, TYPET
    *     .. Local Arrays ..
    *     The arrays are dimensioned for both TYPET = 'R' and TYPET = 'C'.
    *     Arrays G and T could be smaller.
    *     For array G, it is assumed that MMAX + NMAX >= 2.
    *     The matrix TA is also stored in the array T.
          DOUBLE PRECISION CS(LCS), DWORK(LDWORK),
         $                 G(LDG, KMAX*( MMAX + NMAX )),
         $                 L(LDL, KMAX*( MMAX + NMAX )),
         $                 R(LDR, KMAX*( MMAX + NMAX )),
         $                 T(LDT, KMAX*( MMAX + NMAX ))
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         DLACPY, MB02CD, MB02DD
    *
    *     .. 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, M, JOB, TYPET
          S = ( N + M )*K
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99989 ) N
          ELSE
             IF ( K.LE.0 .OR. K.GT.KMAX ) THEN
                WRITE ( NOUT, FMT = 99988 ) K
             ELSE
                IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
                   WRITE ( NOUT, FMT = 99987 ) M
                ELSE
                   IF ( LSAME( TYPET, 'R' ) ) THEN
                      READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,S ), I = 1,K )
                   ELSE
                      READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,K ), I = 1,S )
                   END IF
    *              Compute the Cholesky factors.
                   CALL MB02CD( JOB, TYPET, K, N, T, LDT, G, LDG, R, LDR, L,
         $                      LDL, CS, LCS, DWORK, LDWORK, INFO )
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99996 )
                      DO 10  I = 1, N*K
                         WRITE ( NOUT, FMT = 99990 ) ( R(I,J), J = 1, N*K )
       10             CONTINUE
                      IF ( LSAME( JOB, 'R' ) .OR. LSAME( JOB, 'A' ) ) THEN
                         WRITE ( NOUT, FMT = 99995 )
                         IF ( LSAME( TYPET, 'R' ) ) THEN
                            DO 20  I = 1, 2*K
                               WRITE ( NOUT, FMT = 99990 )
         $                           ( G(I,J), J = 1, N*K )
       20                   CONTINUE
                         ELSE
                            DO 30  I = 1, N*K
                               WRITE ( NOUT, FMT = 99990 )
         $                           ( G(I,J), J = 1, 2*K )
       30                   CONTINUE
                         END IF
                      END IF
                      IF ( LSAME( JOB, 'A' ) ) THEN
                         WRITE ( NOUT, FMT = 99994 )
                         DO 40  I = 1, N*K
                            WRITE ( NOUT, FMT = 99990 )
         $                        ( L(I,J), J = 1, N*K )
       40                CONTINUE
                      END IF
    *                 Update the Cholesky factors.
                      IF ( LSAME( TYPET, 'R' ) ) THEN
    *                    Copy the last block column of R.
                         CALL DLACPY( 'All', N*K, K, R(1,(N-1)*K+1), LDR,
         $                            R(K+1,N*K+1), LDR )
                         CALL MB02DD( JOB, TYPET, K, M, N, T(1,N*K+1), LDT,
         $                            T, LDT, G, LDG, R(1,N*K+1), LDR,
         $                            L(N*K+1,1), LDL, CS, LCS, DWORK,
         $                            LDWORK, INFO )
                      ELSE
    *                    Copy the last block row of R.
                         CALL DLACPY( 'All', K, N*K, R((N-1)*K+1,1), LDR,
         $                            R(N*K+1,K+1), LDR )
                         CALL MB02DD( JOB, TYPET, K, M, N, T(N*K+1,1), LDT,
         $                            T, LDT, G, LDG, R(N*K+1,1), LDR,
         $                            L(1,N*K+1), LDL, CS, LCS, DWORK,
         $                            LDWORK, INFO )
                      END IF
                      IF ( INFO.NE.0 ) THEN
                         WRITE ( NOUT, FMT = 99997 ) INFO
                      ELSE
                         WRITE ( NOUT, FMT = 99993 )
                         DO 50  I = 1, S
                            WRITE ( NOUT, FMT = 99990 ) ( R(I,J), J = 1, S )
       50                CONTINUE
                         IF ( LSAME( JOB, 'R' ) .OR. LSAME( JOB, 'A' ) )
         $                       THEN
                            WRITE ( NOUT, FMT = 99992 )
                            IF ( LSAME( TYPET, 'R' ) ) THEN
                               DO 60  I = 1, 2*K
                                  WRITE ( NOUT, FMT = 99990 )
         $                              ( G(I,J), J = 1, S )
       60                      CONTINUE
                            ELSE
                               DO 70  I = 1, S
                                  WRITE ( NOUT, FMT = 99990 )
         $                              ( G(I,J), J = 1, 2*K )
       70                      CONTINUE
                            END IF
                         END IF
                         IF ( LSAME( JOB, 'A' ) ) THEN
                            WRITE ( NOUT, FMT = 99991 )
                            DO 80  I = 1, S
                               WRITE ( NOUT, FMT = 99990 )
         $                           ( L(I,J), J = 1, S )
       80                   CONTINUE
                         END IF
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT ( ' MB02DD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT ( ' INFO on exit from MB02CD = ',I2)
    99997 FORMAT ( ' INFO on exit from MB02DD = ',I2)
    99996 FORMAT ( ' The Cholesky factor is ')
    99995 FORMAT (/' The inverse generator is ')
    99994 FORMAT (/' The inverse Cholesky factor is ')
    99993 FORMAT (/' The updated Cholesky factor is ')
    99992 FORMAT (/' The updated inverse generator is ')
    99991 FORMAT (/' The updated inverse Cholesky factor is ')
    99990 FORMAT (20(1X,F8.4))
    99989 FORMAT (/' N is out of range.',/' N = ',I5)
    99988 FORMAT (/' K is out of range.',/' K = ',I5)
    99987 FORMAT (/' M is out of range.',/' M = ',I5)
          END

Program Data

    MB02DD EXAMPLE PROGRAM DATA
            3       2       2       A       R
        3.0000    1.0000    0.1000    0.1000    0.2000    0.0500    0.1000    0.0400    0.01   0.02
        1.0000    4.0000    0.4000    0.1000    0.0400    0.2000    0.0300    0.0200    0.03   0.01

Program Results

     MB02DD EXAMPLE PROGRAM RESULTS

     The Cholesky factor is
       1.7321   0.5774   0.0577   0.0577   0.1155   0.0289
       0.0000   1.9149   0.1915   0.0348  -0.0139   0.0957
       0.0000   0.0000   1.7205   0.5754   0.0558   0.0465
       0.0000   0.0000   0.0000   1.9142   0.1890   0.0357
       0.0000   0.0000   0.0000   0.0000   1.7169   0.5759
       0.0000   0.0000   0.0000   0.0000   0.0000   1.9118

     The inverse generator is
      -0.2355   0.5231  -0.0642   0.0077   0.0187  -0.0265
      -0.5568  -0.0568   0.0229   0.0060   0.0363   0.0000
       0.5825   0.0000  -0.0387   0.0052   0.0003  -0.0575
      -0.1754   0.5231   0.0119  -0.0265  -0.0110   0.0076

     The inverse Cholesky factor is
       0.5774   0.0000   0.0000   0.0000   0.0000   0.0000
      -0.1741   0.5222   0.0000   0.0000   0.0000   0.0000
       0.0000  -0.0581   0.5812   0.0000   0.0000   0.0000
      -0.0142   0.0080  -0.1747   0.5224   0.0000   0.0000
      -0.0387   0.0052   0.0003  -0.0575   0.5825   0.0000
       0.0119  -0.0265  -0.0110   0.0076  -0.1754   0.5231

     The updated Cholesky factor is
       1.7321   0.5774   0.0577   0.0577   0.1155   0.0289   0.0577   0.0231   0.0058   0.0115
       0.0000   1.9149   0.1915   0.0348  -0.0139   0.0957  -0.0017   0.0035   0.0139   0.0017
       0.0000   0.0000   1.7205   0.5754   0.0558   0.0465   0.1145   0.0279   0.0564   0.0227
       0.0000   0.0000   0.0000   1.9142   0.1890   0.0357  -0.0152   0.0953  -0.0017   0.0033
       0.0000   0.0000   0.0000   0.0000   1.7169   0.5759   0.0523   0.0453   0.1146   0.0273
       0.0000   0.0000   0.0000   0.0000   0.0000   1.9118   0.1902   0.0357  -0.0157   0.0955
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.7159   0.5757   0.0526   0.0450
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.9118   0.1901   0.0357
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.7159   0.5757
       0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   1.9117

     The updated inverse generator is
      -0.5599   0.3310  -0.0305   0.0098   0.0392  -0.0209   0.0191  -0.0010  -0.0045   0.0035
      -0.2289  -0.4091   0.0612  -0.0012   0.0125   0.0182   0.0042   0.0017   0.0014   0.0000
       0.5828   0.0000   0.0027  -0.0029  -0.0195   0.0072  -0.0393   0.0057   0.0016  -0.0580
      -0.1755   0.5231  -0.0037   0.0022   0.0005  -0.0022   0.0125  -0.0266  -0.0109   0.0077

     The updated inverse Cholesky factor is
       0.5774   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
      -0.1741   0.5222   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
       0.0000  -0.0581   0.5812   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
      -0.0142   0.0080  -0.1747   0.5224   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
      -0.0387   0.0052   0.0003  -0.0575   0.5825   0.0000   0.0000   0.0000   0.0000   0.0000
       0.0119  -0.0265  -0.0110   0.0076  -0.1754   0.5231   0.0000   0.0000   0.0000   0.0000
      -0.0199   0.0073  -0.0391   0.0056   0.0017  -0.0580   0.5828   0.0000   0.0000   0.0000
       0.0007  -0.0023   0.0122  -0.0265  -0.0110   0.0077  -0.1755   0.5231   0.0000   0.0000
       0.0027  -0.0029  -0.0195   0.0072  -0.0393   0.0057   0.0016  -0.0580   0.5828   0.0000
      -0.0037   0.0022   0.0005  -0.0022   0.0125  -0.0266  -0.0109   0.0077  -0.1755   0.5231

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