MB02JX

Low rank QR factorization with column pivoting of a block Toeplitz matrix

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

Purpose

  To compute a low rank QR factorization with column pivoting of a
  K*M-by-L*N block Toeplitz matrix T with blocks of size (K,L);
  specifically,
                                  T
                        T P =  Q R ,

  where R is lower trapezoidal, P is a block permutation matrix
  and Q^T Q = I. The number of columns in R is equivalent to the
  numerical rank of T with respect to the given tolerance TOL1.
  Note that the pivoting scheme is local, i.e., only columns
  belonging to the same block in T are permuted.

Specification
      SUBROUTINE MB02JX( JOB, K, L, M, N, TC, LDTC, TR, LDTR, RNK, Q,
     $                   LDQ, R, LDR, JPVT, TOL1, TOL2, DWORK, LDWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      CHARACTER         JOB
      INTEGER           INFO, K, L, LDQ, LDR, LDTC, LDTR, LDWORK, M, N,
     $                  RNK
      DOUBLE PRECISION  TOL1, TOL2
C     .. Array Arguments ..
      DOUBLE PRECISION  DWORK(LDWORK), Q(LDQ,*), R(LDR,*), TC(LDTC,*),
     $                  TR(LDTR,*)
      INTEGER           JPVT(*)

Arguments

Mode Parameters

  JOB     CHARACTER*1
          Specifies the output of the routine as follows:
          = 'Q':  computes Q and R;
          = 'R':  only computes R.

Input/Output Parameters
  K       (input)  INTEGER
          The number of rows in one block of T.  K >= 0.

  L       (input)  INTEGER
          The number of columns in one block of T.  L >= 0.

  M       (input)  INTEGER
          The number of blocks in one block column of T.  M >= 0.

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

  TC      (input) DOUBLE PRECISION array, dimension (LDTC, L)
          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)
          The leading K-by-(N-1)*L part of this array must contain
          the first block row of T without the leading K-by-L
          block.

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

  RNK     (output)  INTEGER
          The number of columns in R, which is equivalent to the
          numerical rank of T.

  Q       (output)  DOUBLE PRECISION array, dimension (LDQ,RNK)
          If JOB = 'Q', then the leading M*K-by-RNK part of this
          array contains the factor Q.
          If JOB = 'R', then this array is not referenced.

  LDQ     INTEGER
          The leading dimension of the array Q.
          LDQ >= MAX(1,M*K),  if JOB = 'Q';
          LDQ >= 1,           if JOB = 'R'.

  R       (output)  DOUBLE PRECISION array, dimension (LDR,RNK)
          The leading N*L-by-RNK part of this array contains the
          lower trapezoidal factor R.

  LDR     INTEGER
          The leading dimension of the array R.
          LDR >= MAX(1,N*L)

  JPVT    (output)  INTEGER array, dimension (MIN(M*K,N*L))
          This array records the column pivoting performed.
          If JPVT(j) = k, then the j-th column of T*P was
          the k-th column of T.

Tolerances
  TOL1    DOUBLE PRECISION
          If TOL1 >= 0.0, the user supplied diagonal tolerance;
          if TOL1 < 0.0, a default diagonal tolerance is used.

  TOL2    DOUBLE PRECISION
          If TOL2 >= 0.0, the user supplied offdiagonal tolerance;
          if TOL2 < 0.0, a default offdiagonal tolerance is used.

Workspace
  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal value
          of LDWORK;  DWORK(2) and DWORK(3) return the used values
          for TOL1 and TOL2, respectively.
          On exit, if INFO = -19,  DWORK(1) returns the minimum
          value of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= MAX( 3, ( M*K + ( N - 1 )*L )*( L + 2*K ) + 9*L
                              + MAX(M*K,(N-1)*L) ),    if JOB = 'Q';
          LDWORK >= MAX( 3, ( N - 1 )*L*( L + 2*K + 1 ) + 9*L,
                              M*K*( L + 1 ) + L ),     if JOB = 'R'.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  due to perturbations induced by roundoff errors, or
                removal of nearly linearly dependent columns of the
                generator, the Schur algorithm encountered a
                situation where a diagonal element in the negative
                generator is larger in magnitude than the
                corresponding diagonal element in the positive
                generator (modulo TOL1);
          = 2:  due to perturbations induced by roundoff errors, or
                removal of nearly linearly dependent columns of the
                generator, the Schur algorithm encountered a
                situation where diagonal elements in the positive
                and negative generator are equal in magnitude
                (modulo TOL1), but the offdiagonal elements suggest
                that these columns are not linearly dependent
                (modulo TOL2*ABS(diagonal element)).

Method
  Householder transformations and modified hyperbolic rotations
  are used in the Schur algorithm [1], [2].
  If, during the process, the hyperbolic norm of a row in the
  leading part of the generator is found to be less than or equal
  to TOL1, then this row is not reduced. If the difference of the
  corresponding columns has a norm less than or equal to TOL2 times
  the magnitude of the leading element, then this column is removed
  from the generator, as well as from R. Otherwise, the algorithm
  breaks down. TOL1 is set to norm(TC)*sqrt(eps) and TOL2 is set
  to N*L*sqrt(eps) by default.
  If M*K > L, the columns of T are permuted so that the diagonal
  elements in one block column of R have decreasing magnitudes.

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 0(K*RNK*L*M*N) floating point operations.

Further Comments
  None
Example

Program Text

*     MB02JX EXAMPLE PROGRAM TEXT
*     RELEASE 5.0, SLICOT COPYRIGHT 2001.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          KMAX, LMAX, MMAX, NMAX
      PARAMETER        ( KMAX = 20, LMAX = 20, MMAX = 20, NMAX = 20 )
      INTEGER          LDR, LDQ, LDTC, LDTR, LDWORK
      PARAMETER        ( LDR  = NMAX*LMAX, LDQ  = MMAX*KMAX,
     $                   LDTC = MMAX*KMAX, LDTR = KMAX,
     $                   LDWORK = ( MMAX*KMAX + NMAX*LMAX )
     $                            *( LMAX + 2*KMAX ) + 5*LMAX
     $                            + MMAX*KMAX + NMAX*LMAX )
*     .. Local Scalars .. 
      CHARACTER        JOB
      INTEGER          I, INFO, J, K, L, M, N, RNK
      DOUBLE PRECISION TOL1, TOL2
*     .. Local Arrays ..
      INTEGER          JPVT(NMAX*LMAX)
      DOUBLE PRECISION DWORK(LDWORK), Q(LDQ,NMAX*LMAX),
     $                 R(LDR,NMAX*LMAX), TC(LDTC,LMAX),
     $                 TR(LDTR,NMAX*LMAX)
*     .. External Functions ..
      LOGICAL          LSAME 
      EXTERNAL         LSAME
*     .. External Subroutines ..
      EXTERNAL         MB02JX
*     .. 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, N, TOL1, TOL2, JOB
      IF( K.LE.0 .OR. K.GT.KMAX ) THEN
         WRITE ( NOUT, FMT = 99991 ) K
      ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
         WRITE ( NOUT, FMT = 99990 ) L
      ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
         WRITE ( NOUT, FMT = 99989 ) M
      ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
         WRITE ( NOUT, FMT = 99988 ) N
      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 )
*        Compute the required part of the QR factorization.
         CALL MB02JX( JOB, K, L, M, N, TC, LDTC, TR, LDTR, RNK, Q, LDQ,
     $                R, LDR, JPVT, TOL1, TOL2, DWORK, LDWORK, INFO )
         IF ( INFO.NE.0 ) THEN
            WRITE ( NOUT, FMT = 99998 ) INFO
         ELSE
            WRITE ( NOUT, FMT = 99994 ) RNK
            IF ( LSAME( JOB, 'Q' ) ) THEN
               WRITE ( NOUT, FMT = 99997 )
               DO 10  I = 1, M*K
                  WRITE ( NOUT, FMT = 99993 ) ( Q(I,J), J = 1, RNK )
   10          CONTINUE
            END IF
            WRITE ( NOUT, FMT = 99996 )
            DO 20  I = 1, N*L
               WRITE ( NOUT, FMT = 99993 ) ( R(I,J), J = 1, RNK )
   20       CONTINUE
            WRITE ( NOUT, FMT = 99995 )
            WRITE ( NOUT, FMT = 99992 ) ( JPVT(I), 
     $                                    I = 1, MIN( M*K, N*L ) )
         END IF
      END IF            
      STOP
*
99999 FORMAT (' MB02JX EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02JX = ',I2)
99997 FORMAT (/' The factor Q is ')
99996 FORMAT (/' The factor R is ')
99995 FORMAT (/' The column permutation is ')
99994 FORMAT (/' Numerical rank ',/' RNK = ',I5)
99993 FORMAT (20(1X,F8.4))
99992 FORMAT (20(1X,I4))
99991 FORMAT (/' K is out of range.',/' K = ',I5)
99990 FORMAT (/' L is out of range.',/' L = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
      END
Program Data
MB02JX EXAMPLE PROGRAM DATA
   3   3   4   4  -1.0D0  -1.0D0   Q
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     2.0     3.0
     1.0     0.0     1.0
     1.0     1.0     0.0
     2.0     2.0     0.0
     1.0     2.0     3.0     1.0     2.0     3.0     0.0     1.0     1.0
     1.0     2.0     3.0     1.0     2.0     3.0     1.0     2.0     1.0
     1.0     2.0     3.0     1.0     2.0     3.0     1.0     1.0     1.0
     1.0     2.0     3.0     1.0     2.0     3.0     0.0     1.0     0.0
Program Results
 MB02JX EXAMPLE PROGRAM RESULTS


 Numerical rank 
 RNK =     7

 The factor Q is 
  -0.3313  -0.0105  -0.0353   0.0000  -0.4714  -0.8165   0.0000
  -0.3313  -0.0105  -0.0353   0.0000  -0.4714   0.4082   0.7071
  -0.3313  -0.0105  -0.0353   0.0000  -0.4714   0.4082  -0.7071
  -0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
  -0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
  -0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
  -0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
  -0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
  -0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
  -0.1104   0.2824   0.9529   0.0000   0.0000   0.0000   0.0000
   0.0000   0.4288  -0.1271   0.8944   0.0000   0.0000   0.0000
   0.0000   0.8576  -0.2541  -0.4472   0.0000   0.0000   0.0000

 The factor R is 
  -9.0554   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
  -3.0921   2.3322   0.0000   0.0000   0.0000   0.0000   0.0000
  -5.9633   1.9557  -1.2706   0.0000   0.0000   0.0000   0.0000
  -9.2762   4.4238   0.7623   1.3416   0.0000   0.0000   0.0000
  -6.1842   2.9492   0.5082   0.8944   0.0000   0.0000   0.0000
  -3.0921   1.4746   0.2541   0.4472   0.0000   0.0000   0.0000
  -9.2762   4.4238   0.7623   1.3416   0.0000   0.0000   0.0000
  -6.1842   2.9492   0.5082   0.8944   0.0000   0.0000   0.0000
  -3.0921   1.4746   0.2541   0.4472   0.0000   0.0000   0.0000
  -7.2885   4.4866   0.9741   1.3416   2.8284   0.0000   0.0000
  -2.7608   1.4851   0.2894   0.4472   0.4714   0.8165   0.0000
  -5.5216   2.9701   0.5788   0.8944   0.9428   0.4082   0.7071

 The column permutation is 
    3    1    2    6    5    4    9    8    7   12   10   11

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