MB03PY

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

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.

      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( * )

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.

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

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

  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.

  [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.

  The algorithm is backward stable.

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