MB03OY

Matrix rank determination by incremental condition estimation, during the pivoted QR factorization process

Purpose

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

  MB03OY  does not perform any scaling of the matrix A.

      SUBROUTINE MB03OY( 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 leading RANK-by-RANK upper triangular part
          of A contains the triangular factor R11, and the elements
          below the diagonal in the first  RANK  columns, with the
          array TAU, represent the orthogonal matrix Q as a product
          of  RANK  elementary reflectors.
          The remaining  N-RANK  columns contain the result of the
          QR 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 leading triangular
          submatrix R11 in the QR 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 R11.

  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(1:RANK,1:RANK);
          SVAL(2): smallest singular value of R(1:RANK,1:RANK);
          SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
                   if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
                   otherwise.
          If the triangular factorization is a rank-revealing one
          (which will be the case if the leading columns 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(1:RANK,1:RANK).

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

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

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

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

  The routine computes a truncated QR factorization with column
  pivoting of A,  A * P = Q * R,  with  R  defined above, and,
  during this process, finds the largest leading submatrix whose
  estimated condition number is less than 1/RCOND, taking the
  possible positive value of SVLMAX into account.  This is performed
  using 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(1) H(2) . . . H(k), where k = rank <= 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(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
  A(i+1:m,i), and tau in TAU(i).

  The matrix P is represented in jpvt as follows: If
     jpvt(j) = i
  then the jth column 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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Program Text

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