## MB3OYZ

### Matrix rank determination by incremental condition estimation, during the pivoted QR factorization process (complex case)

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

Purpose

      To compute a rank-revealing QR factorization of a complex 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.

      MB3OYZ  does not perform any scaling of the matrix A.

Specification

          SUBROUTINE MB3OYZ( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
         $                   TAU, DWORK, ZWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, M, N, RANK
          DOUBLE PRECISION   RCOND, SVLMAX
    C     .. Array Arguments ..
          INTEGER            JPVT( * )
          COMPLEX*16         A( LDA, * ), TAU( * ), ZWORK( * )
          DOUBLE PRECISION   DWORK( * ), SVAL( 3 )

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) COMPLEX*16 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 unitary 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) COMPLEX*16 array, dimension ( MIN( M, N ) )
              The leading  RANK  elements of TAU contain the scalar
              factors of the elementary reflectors.

Workspace

      DWORK   DOUBLE PRECISION array, dimension ( 2*N )

      ZWORK   COMPLEX*16 array, dimension ( 3*N-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 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 complex scalar, and v is a complex 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.

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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