## MB03MD

### Upper bound for L singular values of a bidiagonal matrix

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

Purpose

      To compute an upper bound THETA using a bisection method such that
      the bidiagonal matrix

               |q(1) e(1)  0    ...   0   |
               | 0   q(2) e(2)        .   |
           J = | .                    .   |
               | .                  e(N-1)|
               | 0   ...        ...  q(N) |

      has precisely L singular values less than or equal to THETA plus
      a given tolerance TOL.

      This routine is mainly intended to be called only by other SLICOT
      routines.

Specification

          SUBROUTINE MB03MD( N, L, THETA, Q, E, Q2, E2, PIVMIN, TOL, RELTOL,
         $                   IWARN, INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, IWARN, L, N
          DOUBLE PRECISION  PIVMIN, RELTOL, THETA, TOL
    C     .. Array Arguments ..
          DOUBLE PRECISION  E(*), E2(*), Q(*), Q2(*)

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the bidiagonal matrix J.  N >= 0.

      L       (input/output) INTEGER
              On entry, L must contain the number of singular values
              of J which must be less than or equal to the upper bound
              computed by the routine.  0 <= L <= N.
              On exit, L may be increased if the L-th smallest singular
              value of J has multiplicity greater than 1. In this case,
              L is increased by the number of singular values of J which
              are larger than its L-th smallest one and approach the
              L-th smallest singular value of J within a distance less
              than TOL.
              If L has been increased, then the routine returns with
              IWARN set to 1.

      THETA   (input/output) DOUBLE PRECISION
              On entry, THETA must contain an initial estimate for the
              upper bound to be computed. If THETA < 0.0 on entry, then
              one of the following default values is used.
              If L = 0, THETA is set to 0.0 irrespective of the input
              value of THETA; if L = 1, then THETA is taken as
              MIN(ABS(Q(i))), for i = 1,2,...,N; otherwise, THETA is
              taken as ABS(Q(N-L+1)).
              On exit, THETA contains the computed upper bound such that
              the bidiagonal matrix J has precisely L singular values
              less than or equal to THETA + TOL.

      Q       (input) DOUBLE PRECISION array, dimension (N)
              This array must contain the diagonal elements q(1),
              q(2),...,q(N) of the bidiagonal matrix J. That is,
              Q(i) = J(i,i) for i = 1,2,...,N.

      E       (input) DOUBLE PRECISION array, dimension (N-1)
              This array must contain the superdiagonal elements
              e(1),e(2),...,e(N-1) of the bidiagonal matrix J. That is,
              E(k) = J(k,k+1) for k = 1,2,...,N-1.

      Q2      (input) DOUBLE PRECISION array, dimension (N)
              This array must contain the squares of the diagonal
              elements q(1),q(2),...,q(N) of the bidiagonal matrix J.
              That is, Q2(i) = J(i,i)**2 for i = 1,2,...,N.

      E2      (input) DOUBLE PRECISION array, dimension (N-1)
              This array must contain the squares of the superdiagonal
              elements e(1),e(2),...,e(N-1) of the bidiagonal matrix J.
              That is, E2(k) = J(k,k+1)**2 for k = 1,2,...,N-1.

      PIVMIN  (input) DOUBLE PRECISION
              The minimum absolute value of a "pivot" in the Sturm
              sequence loop.
              PIVMIN >= max( max( |q(i)|, |e(k)| )**2*sf_min, sf_min ),
              where i = 1,2,...,N, k = 1,2,...,N-1, and sf_min is at
              least the smallest number that can divide one without
              overflow (see LAPACK Library routine DLAMCH).
              Note that this condition is not checked by the routine.

Tolerances

      TOL     DOUBLE PRECISION
              This parameter defines the multiplicity of singular values
              by considering all singular values within an interval of
              length TOL as coinciding. TOL is used in checking how many
              singular values are less than or equal to THETA. Also in
              computing an appropriate upper bound THETA by a bisection
              method, TOL is used as a stopping criterion defining the
              minimum (absolute) subinterval width.  TOL >= 0.

      RELTOL  DOUBLE PRECISION
              This parameter specifies the minimum relative width of an
              interval. When an interval is narrower than TOL, or than
              RELTOL times the larger (in magnitude) endpoint, then it
              is considered to be sufficiently small and bisection has
              converged.
              RELTOL >= BASE * EPS, where BASE is machine radix and EPS
              is machine precision (see LAPACK Library routine DLAMCH).

Warning Indicator

      IWARN   INTEGER
              = 0:  no warnings;
              = 1:  if the value of L has been increased as the L-th
                    smallest singular value of J coincides with the
                    (L+1)-th smallest one.

Error Indicator

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

Method

      Let s(i), i = 1,2,...,N, be the N non-negative singular values of
      the bidiagonal matrix J arranged so that s(1) >= ... >= s(N) >= 0.
      The routine then computes an upper bound T such that s(N-L) > T >=
      s(N-L+1) as follows (see [2]).
      First, if the initial estimate of THETA is not specified by the
      user then the routine initialises THETA to be an estimate which
      is close to the requested value of THETA if s(N-L) >> s(N-L+1).
      Second, a bisection method (see [1, 8.5]) is used which generates
      a sequence of shrinking intervals [Y,Z] such that either THETA in
      [Y,Z] was found (so that J has L singular values less than or
      equal to THETA), or

         (number of s(i) <= Y) < L < (number of s(i) <= Z).

      This bisection method is applied to an associated 2N-by-2N
      symmetric tridiagonal matrix T" whose eigenvalues (see [1]) are
      given by s(1),s(2),...,s(N),-s(1),-s(2),...,-s(N). One of the
      starting values for the bisection method is the initial value of
      THETA. If this value is an upper bound, then the initial lower
      bound is set to zero, else the initial upper bound is computed
      from the Gershgorin Circle Theorem [1, Theorem 7.2-1], applied to
      T". The computation of the "number of s(i) <= Y (or Z)" is
      achieved by calling SLICOT Library routine MB03ND, which applies
      Sylvester's Law of Inertia or equivalently Sturm sequences
      [1, 8.5] to the associated matrix T". If

         Z - Y <= MAX( TOL, PIVMIN, RELTOL*MAX( ABS( Y ), ABS( Z ) ) )

      at some stage of the bisection method, then at least two singular
      values of J lie in the interval [Y,Z] within a distance less than
      TOL from each other. In this case, s(N-L) and s(N-L+1) are assumed
      to coincide, the upper bound T is set to the value of Z, the value
      of L is increased and IWARN is set to 1.

References

      [1] Golub, G.H. and Van Loan, C.F.
          Matrix Computations.
          The Johns Hopkins University Press, Baltimore, Maryland, 1983.

      [2] Van Huffel, S. and Vandewalle, J.
          The Partial Total Least Squares Algorithm.
          J. Comput. and Appl. Math., 21, pp. 333-341, 1988.

Numerical Aspects

      None.

Further Comments

      None

Example

Program Text

    *     MB03MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          DOUBLE PRECISION ZERO
          PARAMETER        ( ZERO = 0.0D0 )
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 20 )
    *     .. Local Scalars ..
          DOUBLE PRECISION PIVMIN, RELTOL, SAFMIN, THETA, TOL
          INTEGER          I, INFO, IWARN, L, N
    *     .. Local Arrays ..
          DOUBLE PRECISION E(NMAX-1), E2(NMAX-1), Q(NMAX), Q2(NMAX)
    *     .. External Functions ..
          DOUBLE PRECISION DLAMCH
          EXTERNAL         DLAMCH
    *     .. External Subroutines ..
          EXTERNAL         MB03MD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, THETA, L, TOL, RELTOL
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99991 ) N
          ELSE IF ( L.LT.0 .OR. L.GT.N ) THEN
             WRITE ( NOUT, FMT = 99990 ) L
          ELSE
             READ ( NIN, FMT = * ) ( Q(I), I = 1,N )
             READ ( NIN, FMT = * ) ( E(I), I = 1,N-1 )
    *        Print out the bidiagonal matrix J.
             WRITE ( NOUT, FMT = 99997 )
             DO 20 I = 1, N - 1
                WRITE ( NOUT, FMT = 99996 ) I, I, Q(I), I, (I+1), E(I)
       20    CONTINUE
             WRITE ( NOUT, FMT = 99995 ) N, N, Q(N)
    *        Compute Q**2, E**2, and PIVMIN.
             Q2(N) = Q(N)**2
             PIVMIN = Q2(N)
             DO 40 I = 1, N - 1
                Q2(I) = Q(I)**2
                E2(I) = E(I)**2
                PIVMIN = MAX( PIVMIN, Q2(I), E2(I) )
       40    CONTINUE
             SAFMIN = DLAMCH( 'Safe minimum' )
             PIVMIN = MAX( PIVMIN*SAFMIN, SAFMIN )
             TOL = MAX( TOL, ZERO )
             IF ( RELTOL.LE.ZERO )
         $      RELTOL = DLAMCH( 'Base' )*DLAMCH( 'Epsilon' )
    *        Compute an upper bound THETA such that J has 3 singular values
    *        < =  THETA.
             CALL MB03MD( N, L, THETA, Q, E, Q2, E2, PIVMIN, TOL, RELTOL,
         $                IWARN, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                IF ( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99994 ) IWARN
                WRITE ( NOUT, FMT = 99993 ) THETA
                WRITE ( NOUT, FMT = 99992 ) L
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' MB03MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB03MD = ',I2)
    99997 FORMAT (' The Bidiagonal Matrix J is',/)
    99996 FORMAT (2(' (',I1,',',I1,') = ',F7.4,2X))
    99995 FORMAT (' (',I1,',',I1,') = ',F7.4)
    99994 FORMAT (' IWARN on exit from MB03MD = ',I2,/)
    99993 FORMAT (/' The computed value of THETA is ',F7.4)
    99992 FORMAT (/' J has ',I2,' singular values < =  THETA')
    99991 FORMAT (/' N is out of range.',/' N = ',I5)
    99990 FORMAT (/' L is out of range.',/' L = ',I5)
          END

Program Data

     MB03MD EXAMPLE PROGRAM DATA
       5     -3.0     3     0.0     0.0
       1.0  2.0  3.0  4.0  5.0
       2.0  3.0  4.0  5.0

Program Results

     MB03MD EXAMPLE PROGRAM RESULTS

     The Bidiagonal Matrix J is

     (1,1) =  1.0000   (1,2) =  2.0000
     (2,2) =  2.0000   (2,3) =  3.0000
     (3,3) =  3.0000   (3,4) =  4.0000
     (4,4) =  4.0000   (4,5) =  5.0000
     (5,5) =  5.0000

     The computed value of THETA is  4.7500

     J has  3 singular values < =  THETA

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