## MB03PY

### Matrix rank determination by incremental condition estimation, during the pivoted RQ factorization process (row pivoting)

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

Purpose

      To compute 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 a truncated RQ factorization with row pivoting:
                                    [ R11 R12 ]
         P * A = R * Q,  where  R = [         ],
                                    [  0  R22 ]
      with R22 defined as the largest trailing upper triangular
      submatrix whose estimated condition number is less than 1/RCOND.
      The order of R22, RANK, is the effective rank of A.  Condition
      estimation is performed during the RQ factorization process.
      Matrix R11 is full (but of small norm), or empty.

      MB03PY  does not perform any scaling of the matrix A.

Specification

          SUBROUTINE MB03PY( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
         $                   TAU, DWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, M, N, RANK
          DOUBLE PRECISION   RCOND, SVLMAX
    C     .. Array Arguments ..
          INTEGER            JPVT( * )
          DOUBLE PRECISION   A( LDA, * ), DWORK( * ), SVAL( 3 ), TAU( * )

Arguments

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, the leading M-by-N part of this array must
              contain the given matrix A.
              On exit, the upper triangle of the subarray
              A(M-RANK+1:M,N-RANK+1:N) contains the RANK-by-RANK upper
              triangular matrix R22;  the remaining elements in the last
              RANK  rows, with the array TAU, represent the orthogonal
              matrix Q as a product of  RANK  elementary reflectors
              (see METHOD).  The first  M-RANK  rows contain the result
              of the RQ factorization process used.

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

      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.
              0 <= RCOND <= 1.
              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.

      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).

      JPVT    (output) INTEGER array, dimension ( M )
              If JPVT(i) = k, then the i-th row of P*A was the k-th row
              of A.

      TAU     (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
              The trailing  RANK  elements of TAU contain the scalar
              factors of the elementary reflectors.

Workspace

      DWORK   DOUBLE PRECISION array, dimension ( 3*M-1 )

Error Indicator

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

Method

      The routine computes a truncated RQ factorization with row
      pivoting of A,  P * A = R * Q,  with  R  defined above, and,
      during this process, 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
      factorization process stops when  RANK  has been determined.

      The matrix Q is represented as a product of elementary reflectors

         Q = H(k-rank+1) H(k-rank+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

      None

Program Data

      None

Program Results

      None

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