## MB03OD

### Matrix rank determination by incremental condition estimation

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

Purpose

      To compute (optionally) 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 QR factorization with column pivoting:
         A * P = Q * R,  where  R = [ R11 R12 ],
                                    [  0  R22 ]
      with R11 defined as the largest leading submatrix whose estimated
      condition number is less than 1/RCOND.  The order of R11, RANK,
      is the effective rank of A.

      MB03OD  does not perform any scaling of the matrix A.

Specification

          SUBROUTINE MB03OD( JOBQR, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
         $                   RANK, SVAL, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          JOBQR
          INTEGER            INFO, LDA, LDWORK, M, N, RANK
          DOUBLE PRECISION   RCOND, SVLMAX
    C     .. Array Arguments ..
          INTEGER            JPVT( * )
          DOUBLE PRECISION   A( LDA, * ), SVAL( 3 ), TAU( * ), DWORK( * )

Arguments

Mode Parameters

      JOBQR   CHARACTER*1
              = 'Q':  Perform a QR factorization with column pivoting;
              = 'N':  Do not perform the QR 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 JOBQR = 'Q', the leading M by N part of this
              array must contain the given matrix A.
              On exit with JOBQR = 'Q', the leading min(M,N) by N upper
              triangular part of A contains the triangular factor R,
              and the elements below the diagonal, with the array TAU,
              represent the orthogonal matrix Q as a product of
              min(M,N) elementary reflectors.
              On entry and on exit with JOBQR = 'N', the leading
              min(M,N) by N upper triangular part of A contains the
              triangular factor R, as determined by the QR factorization
              with pivoting.  The elements below the diagonal of A are
              not referenced.

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

      JPVT    (input/output) INTEGER array, dimension ( N )
              On entry with JOBQR = 'Q', if JPVT(i) <> 0, the i-th
              column of A is an initial column, otherwise it is a free
              column. Before the QR factorization of A, all initial
              columns are permuted to the leading positions; only the
              remaining free columns are moved as a result of column
              pivoting during the factorization.  For rank determination
              it is preferable that all columns be free.
              On exit with JOBQR = 'Q', if JPVT(i) = k, then the i-th
              column of A*P was the k-th column of A.
              Array JPVT is not referenced when JOBQR = 'N'.

      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.
              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 JOBQR = 'Q', the leading min(M,N) elements of
              TAU contain the scalar factors of the elementary
              reflectors.
              Array TAU is not referenced when JOBQR = 'N'.

      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).

Workspace

      DWORK   DOUBLE PRECISION array, dimension ( LDWORK )
              On exit, if  INFO = 0,  DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= 3*N + 1,                 if JOBQR = 'Q';
              LDWORK >= max( 1, 2*min( M, N ) ), if JOBQR = 'N'.
              For good performance when JOBQR = 'Q', LDWORK should be
              larger. Specifically, LDWORK >= 2*N + ( N + 1 )*NB, where
              NB is the optimal block size for the LAPACK Library
              routine DGEQP3.

              If LDWORK = -1, then a workspace query is assumed;
              the routine only calculates the optimal size of the
              DWORK array, returns this value as the first entry of
              the DWORK array, and no error message related to LDWORK
              is issued by XERBLA.

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 a QR factorization with column
      pivoting of A,  A * P = Q * R,  with  R  defined above, and then
      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.

Further Comments

      None

Example

Program Text

    *     MB03OD 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*NMAX + 1 )
    *     .. Local Scalars ..
          CHARACTER*1      JOBQR
          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(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         MB03OD
    *     .. 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, JOBQR, 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 )
    *           QR with column pivoting.
                DO 10 I = 1, N
                   JPVT(I) = 0
       10       CONTINUE
                CALL MB03OD( JOBQR, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
         $                   RANK, SVAL, DWORK, LDWORK, 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,N )
                   WRITE ( NOUT, FMT = 99993 ) ( SVAL(I), I = 1,3 )
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' MB03OD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB03OD = ',I2)
    99995 FORMAT (' The rank is ',I5)
    99994 FORMAT (' Column 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

     MB03OD EXAMPLE PROGRAM DATA
       6     5     Q  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

     MB03OD EXAMPLE PROGRAM RESULTS

     The rank is     4
     Column permutations are
      4    3    1    5    2
     SVAL vector is
        22.7257     1.4330     0.0000

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