## MB02HD

### Cholesky factorization of the matrix T' T, with T a banded block Toeplitz matrix of full rank

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

Purpose

      To compute, for a banded K*M-by-L*N block Toeplitz matrix T with
      block size (K,L), specified by the nonzero blocks of its first
      block column TC and row TR, a LOWER triangular matrix R (in band
      storage scheme) such that
                           T          T
                          T  T  =  R R .                             (1)

      It is assumed that the first MIN(M*K, N*L) columns of T are
      linearly independent.

      By subsequent calls of this routine, the matrix R can be computed
      block column by block column.

Specification

          SUBROUTINE MB02HD( TRIU, K, L, M, ML, N, NU, P, S, TC, LDTC, TR,
         $                   LDTR, RB, LDRB, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         TRIU
          INTEGER           INFO, K, L, LDRB, LDTC, LDTR, LDWORK, M, ML, N,
         $                  NU, P, S
    C     .. Array Arguments ..
          DOUBLE PRECISION  DWORK(LDWORK), RB(LDRB,*), TC(LDTC,*),
         $                  TR(LDTR,*)

Arguments

Mode Parameters

      TRIU    CHARACTER*1
              Specifies the structure, if any, of the last blocks in TC
              and TR, as follows:
              = 'N':  TC and TR have no special structure;
              = 'T':  TC and TR are upper and lower triangular,
                      respectively. Depending on the block sizes, two
                      different shapes of the last blocks in TC and TR
                      are possible, as illustrated below:

                      1)    TC       TR     2)   TC         TR

                           x x x    x 0 0      x x x x    x 0 0 0
                           0 x x    x x 0      0 x x x    x x 0 0
                           0 0 x    x x x      0 0 x x    x x x 0
                           0 0 0    x x x

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 >= 1.

      ML      (input) INTEGER
              The lower block bandwidth, i.e., ML + 1 is the number of
              nonzero blocks in the first block column of T.
              0 <= ML < M and (ML + 1)*K >= L and
              if ( M*K <= N*L ),  ML >= M - INT( ( M*K - 1 )/L ) - 1;
                                  ML >= M - INT( M*K/L ) or
                                  MOD( M*K, L ) >= K;
              if ( M*K >= N*L ),  ML*K >= N*( L - K ).

      N       (input) INTEGER
              The number of blocks in the first block row of T.
              N >= 1.

      NU      (input) INTEGER
              The upper block bandwidth, i.e., NU + 1 is the number of
              nonzero blocks in the first block row of T.
              If TRIU = 'N',   0 <= NU < N and
                               (M + NU)*L >= MIN( M*K, N*L );
              if TRIU = 'T',   MAX(1-ML,0) <= NU < N and
                               (M + NU)*L >= MIN( M*K, N*L ).

      P       (input)  INTEGER
              The number of previously computed block columns of R.
              P*L < MIN( M*K,N*L ) + L and P >= 0.

      S       (input)  INTEGER
              The number of block columns of R to compute.
              (P+S)*L < MIN( M*K,N*L ) + L and S >= 0.

      TC      (input)  DOUBLE PRECISION array, dimension (LDTC,L)
              On entry, if P = 0, the leading (ML+1)*K-by-L part of this
              array must contain the nonzero blocks in the first block
              column of T.

      LDTC    INTEGER
              The leading dimension of the array TC.
              LDTC >= MAX(1,(ML+1)*K),  if P = 0.

      TR      (input)  DOUBLE PRECISION array, dimension (LDTR,NU*L)
              On entry, if P = 0, the leading K-by-NU*L part of this
              array must contain the 2nd to the (NU+1)-st blocks of
              the first block row of T.

      LDTR    INTEGER
              The leading dimension of the array TR.
              LDTR >= MAX(1,K),  if P = 0.

      RB      (output)  DOUBLE PRECISION array, dimension
              (LDRB,MIN( S*L,MIN( M*K,N*L )-P*L ))
              On exit, if INFO = 0 and TRIU = 'N', the leading
              MIN( ML+NU+1,N )*L-by-MIN( S*L,MIN( M*K,N*L )-P*L ) part
              of this array contains the (P+1)-th to (P+S)-th block
              column of the lower R factor (1) in band storage format.
              On exit, if INFO = 0 and TRIU = 'T', the leading
              MIN( (ML+NU)*L+1,N*L )-by-MIN( S*L,MIN( M*K,N*L )-P*L )
              part of this array contains the (P+1)-th to (P+S)-th block
              column of the lower R factor (1) in band storage format.
              For further details regarding the band storage scheme see
              the documentation of the LAPACK routine DPBTF2.

      LDRB    INTEGER
              The leading dimension of the array RB.
              LDRB >= MAX( MIN( ML+NU+1,N )*L,1 ),      if TRIU = 'N';
              LDRB >= MAX( MIN( (ML+NU)*L+1,N*L ),1 ),  if TRIU = 'T'.

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.
              The first 1 + 2*MIN( ML+NU+1,N )*L*(K+L) elements of DWORK
              should be preserved during successive calls of the routine.

      LDWORK  INTEGER
              The length of the array DWORK.
              Let x = MIN( ML+NU+1,N ), then
              LDWORK >= 1 + MAX( x*L*L + (2*NU+1)*L*K,
                                 2*x*L*(K+L) + (6+x)*L ),  if P = 0;
              LDWORK >= 1 + 2*x*L*(K+L) + (6+x)*L,         if P > 0.
              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 full rank condition for the first MIN(M*K, N*L)
                    columns of T is (numerically) violated.

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 yields a factor R which has comparable
      accuracy with the Cholesky factor of T^T * T.
      The algorithm requires
                2                                  2
            O( L *K*N*( ML + NU ) + N*( ML + NU )*L *( L + K ) )

      floating point operations.

Further Comments

      None

Example

Program Text

    *     MB02HD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          KMAX, LMAX, MMAX, MLMAX, NMAX, NUMAX
          PARAMETER        ( KMAX = 20, LMAX  = 20, MMAX = 20, MLMAX = 10,
         $                   NMAX = 20, NUMAX = 10 )
          INTEGER          LDRB, LDTC, LDTR, LDWORK
          PARAMETER        ( LDRB = ( MLMAX + NUMAX + 1 )*LMAX,
         $                   LDTC = ( MLMAX + 1 )*KMAX, LDTR = KMAX )
          PARAMETER        ( LDWORK = LDRB*LMAX + ( 2*NUMAX + 1 )*LMAX*KMAX
         $                            + 2*LDRB*( KMAX + LMAX ) + LDRB
         $                            + 6*LMAX )
    *     .. Local Scalars ..
          INTEGER          I, INFO, J, K, L, LENR, M, ML, N, NU, S
          CHARACTER        TRIU
    *     .. Local Arrays ..
          DOUBLE PRECISION DWORK(LDWORK), RB(LDRB,NMAX*LMAX),
         $                 TC(LDTC,LMAX), TR(LDTR,NMAX*LMAX)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         MB02HD
    *     .. Intrinsic Functions ..
          INTRINSIC        MIN
    *
    *     .. 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, ML, N, NU, TRIU
          IF( K.LT.0 .OR. K.GT.KMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) K
          ELSE IF( L.LT.0 .OR. L.GT.LMAX ) THEN
             WRITE ( NOUT, FMT = 99991 ) L
          ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99992 ) M
          ELSE IF( ML.LT.0 .OR. ML.GT.MLMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) ML
          ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE IF( NU.LT.0 .OR. NU.GT.NUMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) NU
          ELSE
             READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,(ML+1)*K )
             READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,NU*L ), I = 1,K )
             S = ( MIN( M*K, N*L ) + L - 1 ) / L
    *        Compute the banded R factor.
             CALL MB02HD( TRIU, K, L, M, ML, N, NU, 0, S, TC, LDTC, TR,
         $                LDTR, RB, LDRB, DWORK, LDWORK, INFO )
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 )
                LENR = ( ML + NU + 1 )*L
                IF ( LSAME( TRIU, 'T' ) )  LENR = ( ML + NU )*L + 1
                LENR = MIN( LENR, N*L )
                DO 10  I = 1, LENR
                   WRITE ( NOUT, FMT = 99996 ) ( RB(I,J), J = 1,
         $                                       MIN( N*L, M*K ) )
       10       CONTINUE
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB02HD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB02HD = ',I2)
    99997 FORMAT (/' The lower triangular factor R in banded storage ')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' NU is out of range.',/' NU = ',I5)
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
    99993 FORMAT (/' ML is out of range.',/' ML = ',I5)
    99992 FORMAT (/' M is out of range.',/' M = ',I5)
    99991 FORMAT (/' L is out of range.',/' L = ',I5)
    99990 FORMAT (/' K is out of range.',/' K = ',I5)
          END

Program Data

    MB02HD EXAMPLE PROGRAM DATA
       2  2  6  2  5   1  N
         4.0     4.0
         1.0     3.0
         2.0     1.0
         2.0     2.0
         4.0     4.0
         3.0     4.0
         1.0     3.0
         2.0     1.0

Program Results

     MB02HD EXAMPLE PROGRAM RESULTS

     The lower triangular factor R in banded storage
      -7.0711  -2.4125   6.0822   2.9967   5.9732   2.8593   5.8497   2.7914   2.7298   1.9557
      -7.4953  -0.0829   5.8986  -0.5571   5.5329   0.2059   5.6797   0.3414   0.9565   0.0000
      -4.2426   0.9202   2.4747  -1.6425   2.9472  -1.0052   2.4396  -0.7785   0.0000   0.0000
      -5.2326   0.6218   2.8391  -0.0820   3.2670   0.6327   2.7067   0.0000   0.0000   0.0000
      -3.5355   0.8207   3.1160  -0.4451   3.5758   0.5701   0.0000   0.0000   0.0000   0.0000
      -4.6669  -0.5803   3.9454   0.7682   4.5481   0.0000   0.0000   0.0000   0.0000   0.0000
      -1.4142  -0.0415   1.6441   0.4848   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
      -2.1213   0.0000   2.4662   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000

Return to index