## MB02GD

### Cholesky factorization of a banded symmetric positive definite block Toeplitz matrix

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

Purpose

      To compute the Cholesky factor of a banded symmetric positive
      definite (s.p.d.) block Toeplitz matrix, defined by either its
      first block row, or its first block column, depending on the
      routine parameter TYPET.

      By subsequent calls of this routine the Cholesky factor can be
      computed block column by block column.

Specification

          SUBROUTINE MB02GD( TYPET, TRIU, K, N, NL, P, S, T, LDT, RB, LDRB,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         TRIU, TYPET
          INTEGER           INFO, K, LDRB, LDT, LDWORK, N, NL, P, S
    C     .. Array Arguments ..
          DOUBLE PRECISION  DWORK(LDWORK), RB(LDRB,*), 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; the Cholesky factor is upper
                      triangular;
              = 'C':  T contains the first block column of an s.p.d.
                      block Toeplitz matrix; the Cholesky factor is
                      lower triangular. This choice results in a column
                      oriented algorithm which is usually faster.
              Note:   in the sequel, the notation x / y means that
                      x corresponds to TYPET = 'R' and y corresponds to
                      TYPET = 'C'.

      TRIU    CHARACTER*1
              Specifies the structure of the last block in T, as
              follows:
              = 'N':  the last block has no special structure;
              = 'T':  the last block is lower / upper triangular.

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 >= 1.
              If TRIU = 'N',   N >= 1;
              if TRIU = 'T',   N >= 2.

      NL      (input)  INTEGER
              The lower block bandwidth, i.e., NL + 1 is the number of
              nonzero blocks in the first block column of the block
              Toeplitz matrix.
              If TRIU = 'N',   0 <= NL < N;
              if TRIU = 'T',   1 <= NL < N.

      P       (input)  INTEGER
              The number of previously computed block rows / columns of
              the Cholesky factor.  0 <= P <= N.

      S       (input)  INTEGER
              The number of block rows / columns of the Cholesky factor
              to compute.  0 <= S <= N - P.

      T       (input/output)  DOUBLE PRECISION array, dimension
              (LDT,(NL+1)*K) / (LDT,K)
              On entry, if P = 0, the leading K-by-(NL+1)*K /
              (NL+1)*K-by-K part of this array must contain the first
              block row / column of an s.p.d. block Toeplitz matrix.
              On entry, if P > 0, the leading K-by-(NL+1)*K /
              (NL+1)*K-by-K part of this array must contain the P-th
              block row / column of the Cholesky factor.
              On exit, if INFO = 0, then the leading K-by-(NL+1)*K /
              (NL+1)*K-by-K part of this array contains the (P+S)-th
              block row / column of the Cholesky factor.

      LDT     INTEGER
              The leading dimension of the array T.
              LDT >= MAX(1,K) / MAX(1,(NL+1)*K).

      RB      (input/output)  DOUBLE PRECISION array, dimension
              (LDRB,MIN(P+NL+S,N)*K) / (LDRB,MIN(P+S,N)*K)
              On entry, if TYPET = 'R'  and  TRIU = 'N'  and  P > 0,
              the leading (NL+1)*K-by-MIN(NL,N-P)*K part of this array
              must contain the (P*K+1)-st to ((P+NL)*K)-th columns
              of the upper Cholesky factor in banded format from a
              previous call of this routine.
              On entry, if TYPET = 'R'  and  TRIU = 'T'  and  P > 0,
              the leading (NL*K+1)-by-MIN(NL,N-P)*K part of this array
              must contain the (P*K+1)-st to (MIN(P+NL,N)*K)-th columns
              of the upper Cholesky factor in banded format from a
              previous call of this routine.
              On exit, if TYPET = 'R'  and  TRIU = 'N', the leading
              (NL+1)*K-by-MIN(NL+S,N-P)*K part of this array contains
              the (P*K+1)-st to (MIN(P+NL+S,N)*K)-th columns of the
              upper Cholesky factor in banded format.
              On exit, if TYPET = 'R'  and  TRIU = 'T', the leading
              (NL*K+1)-by-MIN(NL+S,N-P)*K part of this array contains
              the (P*K+1)-st to (MIN(P+NL+S,N)*K)-th columns of the
              upper Cholesky factor in banded format.
              On exit, if TYPET = 'C'  and  TRIU = 'N', the leading
              (NL+1)*K-by-MIN(S,N-P)*K part of this array contains
              the (P*K+1)-st to (MIN(P+S,N)*K)-th columns of the lower
              Cholesky factor in banded format.
              On exit, if TYPET = 'C'  and  TRIU = 'T', the leading
              (NL*K+1)-by-MIN(S,N-P)*K part of this array contains
              the (P*K+1)-st to (MIN(P+S,N)*K)-th columns of the lower
              Cholesky factor in banded 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.
              If TRIU = 'N',   LDRB >= MAX( (NL+1)*K,1 );
              if TRIU = 'T',   LDRB >= NL*K+1.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0,  DWORK(1)  returns the optimal
              value of LDWORK.
              On exit, if  INFO = -13,  DWORK(1)  returns the minimum
              value of LDWORK.
              The first 1 + ( NL + 1 )*K*K elements of DWORK should be
              preserved during successive calls of the routine.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= 1 + ( NL + 1 )*K*K + NL*K.
              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 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
      The algorithm requires O( K *N*NL ) floating point operations.

Further Comments

      None

Example

Program Text

    *     MB02GD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          KMAX, NMAX, NLMAX
          PARAMETER        ( KMAX = 20, NMAX = 20, NLMAX = 20 )
          INTEGER          LDRB, LDT, LDWORK
          PARAMETER        ( LDRB = ( NLMAX + 1 )*KMAX, LDT = KMAX*NMAX,
         $                   LDWORK = ( NLMAX + 1 )*KMAX*KMAX +
         $                            ( 3 + NLMAX )*KMAX )
    *     .. Local Scalars ..
          INTEGER          I, J, INFO, K, M, N, NL, SIZR
          CHARACTER        TRIU, TYPET
    *     .. Local Arrays dimensioned for TYPET = 'R' ..
          DOUBLE PRECISION DWORK(LDWORK), RB(LDRB, NMAX*KMAX),
         $                 T(LDT, NMAX*KMAX)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         MB02GD
    *
    *     .. Executable Statements ..
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) K, N, NL, TRIU
          TYPET = 'R'
          M = ( NL + 1 )*K
          IF( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) N
          ELSE IF( NL.LE.0 .OR. NL.GT.NLMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) NL
          ELSE IF( K.LE.0 .OR. K.GT.KMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) K
          ELSE
             READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,M ), I = 1,K )
    *        Compute the banded Cholesky factor.
             CALL MB02GD( TYPET, TRIU, K, N, NL, 0, N, T, LDT, RB, LDRB,
         $                DWORK, LDWORK, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 )
                IF ( LSAME( TRIU, 'T' ) ) THEN
                   SIZR = NL*K + 1
                ELSE
                   SIZR = ( NL + 1 )*K
                END IF
                DO 10  I = 1, SIZR
                   WRITE ( NOUT, FMT = 99996 ) ( RB(I,J), J = 1, N*K )
       10       CONTINUE
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB02GD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB02GD = ',I2)
    99997 FORMAT (/' The upper Cholesky factor in banded storage format ')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' N is out of range.',/' N = ',I5)
    99994 FORMAT (/' NL is out of range.',/' NL = ',I5)
    99993 FORMAT (/' K is out of range.',/' K = ',I5)
          END

Program Data

    MB02GD EXAMPLE PROGRAM DATA
      2    4    2    T
      3.0000    1.0000    0.1000    0.4000    0.2000    0.0000
      0.0000    4.0000    0.1000    0.1000    0.0500    0.2000

Program Results

     MB02GD EXAMPLE PROGRAM RESULTS

     The upper Cholesky factor in banded storage format
       0.0000   0.0000   0.0000   0.0000   0.1155   0.1044   0.1156   0.1051
       0.0000   0.0000   0.0000   0.2309  -0.0087   0.2290  -0.0084   0.2302
       0.0000   0.0000   0.0577  -0.0174   0.0541  -0.0151   0.0544  -0.0159
       0.0000   0.5774   0.0348   0.5704   0.0222   0.5725   0.0223   0.5724
       1.7321   1.9149   1.7307   1.9029   1.7272   1.8996   1.7272   1.8995

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