## MB03PD

### Matrix rank determination by incremental condition estimation (row pivoting)

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

Purpose

      To compute (optionally) a rank-revealing RQ factorization of a
      real general M-by-N matrix  A,  which may be rank-deficient,
      and estimate its effective rank using incremental condition
      estimation.

      The routine uses an RQ factorization with row pivoting:
         P * A = R * Q,  where  R = [ R11 R12 ],
                                    [  0  R22 ]
      with R22 defined as the largest trailing submatrix whose estimated
      condition number is less than 1/RCOND.  The order of R22, RANK,
      is the effective rank of A.

      MB03PD  does not perform any scaling of the matrix A.

Specification

          SUBROUTINE MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
         $                   RANK, SVAL, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          JOBRQ
          INTEGER            INFO, LDA, M, N, RANK
          DOUBLE PRECISION   RCOND, SVLMAX
    C     .. Array Arguments ..
          INTEGER            JPVT( * )
          DOUBLE PRECISION   A( LDA, * ), SVAL( 3 ), TAU( * ), DWORK( * )

Arguments

Mode Parameters

      JOBRQ   CHARACTER*1
              = 'R':  Perform an RQ factorization with row pivoting;
              = 'N':  Do not perform the RQ factorization (but assume
                      that it has been done outside).

Input/Output Parameters

      M       (input) INTEGER
              The number of rows of the matrix A.  M >= 0.

      N       (input) INTEGER
              The number of columns of the matrix A.  N >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension
              ( LDA, N )
              On entry with JOBRQ = 'R', the leading M-by-N part of this
              array must contain the given matrix A.
              On exit with JOBRQ = 'R',
              if M <= N, the upper triangle of the subarray
              A(1:M,N-M+1:N) contains the M-by-M upper triangular
              matrix R;
              if M >= N, the elements on and above the (M-N)-th
              subdiagonal contain the M-by-N upper trapezoidal matrix R;
              the remaining elements, with the array TAU, represent the
              orthogonal matrix Q as a product of min(M,N) elementary
              reflectors (see METHOD).
              On entry and on exit with JOBRQ = 'N',
              if M <= N, the upper triangle of the subarray
              A(1:M,N-M+1:N) must contain the M-by-M upper triangular
              matrix R;
              if M >= N, the elements on and above the (M-N)-th
              subdiagonal must contain the M-by-N upper trapezoidal
              matrix R;
              the remaining elements are not referenced.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= max(1,M).

      JPVT    (input/output) INTEGER array, dimension ( M )
              On entry with JOBRQ = 'R', if JPVT(i) <> 0, the i-th row
              of A is a final row, otherwise it is a free row. Before
              the RQ factorization of A, all final rows are permuted
              to the trailing positions; only the remaining free rows
              are moved as a result of row pivoting during the
              factorization.  For rank determination it is preferable
              that all rows be free.
              On exit with JOBRQ = 'R', if JPVT(i) = k, then the i-th
              row of P*A was the k-th row of A.
              Array JPVT is not referenced when JOBRQ = 'N'.

      RCOND   (input) DOUBLE PRECISION
              RCOND is used to determine the effective rank of A, which
              is defined as the order of the largest trailing triangular
              submatrix R22 in the RQ factorization with pivoting of A,
              whose estimated condition number is less than 1/RCOND.
              RCOND >= 0.
              NOTE that when SVLMAX > 0, the estimated rank could be
              less than that defined above (see SVLMAX).

      SVLMAX  (input) DOUBLE PRECISION
              If A is a submatrix of another matrix B, and the rank
              decision should be related to that matrix, then SVLMAX
              should be an estimate of the largest singular value of B
              (for instance, the Frobenius norm of B).  If this is not
              the case, the input value SVLMAX = 0 should work.
              SVLMAX >= 0.

      TAU     (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
              On exit with JOBRQ = 'R', the leading min(M,N) elements of
              TAU contain the scalar factors of the elementary
              reflectors.
              Array TAU is not referenced when JOBRQ = 'N'.

      RANK    (output) INTEGER
              The effective (estimated) rank of A, i.e. the order of
              the submatrix R22.

      SVAL    (output) DOUBLE PRECISION array, dimension ( 3 )
              The estimates of some of the singular values of the
              triangular factor R:
              SVAL(1): largest singular value of
                       R(M-RANK+1:M,N-RANK+1:N);
              SVAL(2): smallest singular value of
                       R(M-RANK+1:M,N-RANK+1:N);
              SVAL(3): smallest singular value of R(M-RANK:M,N-RANK:N),
                       if RANK < MIN( M, N ), or of
                       R(M-RANK+1:M,N-RANK+1:N), otherwise.
              If the triangular factorization is a rank-revealing one
              (which will be the case if the trailing rows were well-
              conditioned), then SVAL(1) will also be an estimate for
              the largest singular value of A, and SVAL(2) and SVAL(3)
              will be estimates for the RANK-th and (RANK+1)-st singular
              values of A, respectively.
              By examining these values, one can confirm that the rank
              is well defined with respect to the chosen value of RCOND.
              The ratio SVAL(1)/SVAL(2) is an estimate of the condition
              number of R(M-RANK+1:M,N-RANK+1:N).

Workspace

      DWORK   DOUBLE PRECISION array, dimension ( LDWORK )
              where LDWORK = max( 1, 3*M ),           if JOBRQ = 'R';
                    LDWORK = max( 1, 3*min( M, N ) ), if JOBRQ = 'N'.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      The routine computes or uses an RQ factorization with row
      pivoting of A,  P * A = R * Q,  with  R  defined above, and then
      finds the largest trailing submatrix whose estimated condition
      number is less than 1/RCOND, taking the possible positive value of
      SVLMAX into account.  This is performed using an adaptation of the
      LAPACK incremental condition estimation scheme and a slightly
      modified rank decision test.

      The matrix Q is represented as a product of elementary reflectors

         Q = H(1) H(2) . . . H(k), where k = min(m,n).

      Each H(i) has the form

         H = I - tau * v * v'

      where tau is a real scalar, and v is a real vector with
      v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit
      in A(m-k+i,1:n-k+i-1), and tau in TAU(i).

      The matrix P is represented in jpvt as follows: If
         jpvt(j) = i
      then the jth row of P is the ith canonical unit vector.

References

      [1] Bischof, C.H. and P. Tang.
          Generalizing Incremental Condition Estimation.
          LAPACK Working Notes 32, Mathematics and Computer Science
          Division, Argonne National Laboratory, UT, CS-91-132,
          May 1991.

      [2] Bischof, C.H. and P. Tang.
          Robust Incremental Condition Estimation.
          LAPACK Working Notes 33, Mathematics and Computer Science
          Division, Argonne National Laboratory, UT, CS-91-133,
          May 1991.

Numerical Aspects

      The algorithm is backward stable.

Further Comments

      None

Example

Program Text

    *     MB03PD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          DOUBLE PRECISION ZERO, ONE
          PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX
          PARAMETER        ( NMAX = 10, MMAX = 10 )
          INTEGER          LDA
          PARAMETER        ( LDA = NMAX )
          INTEGER          LDTAU
          PARAMETER        ( LDTAU = MIN(MMAX,NMAX) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 3*MMAX )
    *     .. Local Scalars ..
          CHARACTER*1      JOBRQ
          INTEGER          I, INFO, J, M, N, RANK
          DOUBLE PRECISION RCOND, SVAL(3), SVLMAX
    *     ..
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU)
          INTEGER          JPVT(MMAX)
    *     .. External Subroutines ..
          EXTERNAL         MB03PD
    *     .. 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 = * ) M, N, JOBRQ, RCOND, SVLMAX
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99972 ) N
          ELSE
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99971 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
    *           RQ with row pivoting.
                DO 10 I = 1, M
                   JPVT(I) = 0
       10       CONTINUE
                CALL MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
         $                   RANK, SVAL, DWORK, INFO )
    *
                IF ( INFO.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99998 ) INFO
                ELSE
                   WRITE ( NOUT, FMT = 99995 ) RANK
                   WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,M )
                   WRITE ( NOUT, FMT = 99993 ) ( SVAL(I), I = 1,3 )
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' MB03PD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB03PD = ',I2)
    99995 FORMAT (' The rank is ',I5)
    99994 FORMAT (' Row permutations are ',/(20(I3,2X)))
    99993 FORMAT (' SVAL vector is ',/(20(1X,F10.4)))
    99972 FORMAT (/' N is out of range.',/' N = ',I5)
    99971 FORMAT (/' M is out of range.',/' M = ',I5)
          END

Program Data

     MB03PD EXAMPLE PROGRAM DATA
       6     5     R  5.D-16     0.0
       1.    2.    6.    3.    5.
      -2.   -1.   -1.    0.   -2.
       5.    5.    1.    5.    1.
      -2.   -1.   -1.    0.   -2.
       4.    8.    4.   20.    4.
      -2.   -1.   -1.    0.   -2.

Program Results

     MB03PD EXAMPLE PROGRAM RESULTS

     The rank is     4
     Row permutations are
      2    4    6    3    1    5
     SVAL vector is
        24.5744     0.9580     0.0000

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