## SB08MD

### Spectral factorization of polynomials (continuous-time case)

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

Purpose

      To compute a real polynomial E(s) such that

         (a)  E(-s) * E(s) = A(-s) * A(s) and
         (b)  E(s) is stable - that is, all the zeros of E(s) have
              non-positive real parts,

      which corresponds to computing the spectral factorization of the
      real polynomial A(s) arising from continuous optimality problems.

      The input polynomial may be supplied either in the form

         A(s) = a(0) + a(1) * s + ... + a(DA) * s**DA

      or as

         B(s) = A(-s) * A(s)
              = b(0) + b(1) * s**2  + ... + b(DA) * s**(2*DA)        (1)

Specification

          SUBROUTINE SB08MD( ACONA, DA, A, RES, E, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         ACONA
          INTEGER           DA, INFO, LDWORK
          DOUBLE PRECISION  RES
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(*), DWORK(*), E(*)

Arguments

Mode Parameters

      ACONA   CHARACTER*1
              Indicates whether the coefficients of A(s) or B(s) =
              A(-s) * A(s) are to be supplied as follows:
              = 'A':  The coefficients of A(s) are to be supplied;
              = 'B':  The coefficients of B(s) are to be supplied.

Input/Output Parameters

      DA      (input) INTEGER
              The degree of the polynomials A(s) and E(s).  DA >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (DA+1)
              On entry, this array must contain either the coefficients
              of the polynomial A(s) in increasing powers of s if
              ACONA = 'A', or the coefficients of the polynomial B(s) in
              increasing powers of s**2 (see equation (1)) if ACONA =
              'B'.
              On exit, this array contains the coefficients of the
              polynomial B(s) in increasing powers of s**2.

      RES     (output) DOUBLE PRECISION
              An estimate of the accuracy with which the coefficients of
              the polynomial E(s) have been computed (see also METHOD
              and NUMERICAL ASPECTS).

      E       (output) DOUBLE PRECISION array, dimension (DA+1)
              The coefficients of the spectral factor E(s) in increasing
              powers of s.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)

      LDWORK  INTEGER
              The length of the array DWORK.  LDWORK >= 5*DA+5.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  if on entry, A(I) = 0.0, for I = 1,2,...,DA+1.
              = 2:  if on entry, ACONA = 'B' but the supplied
                    coefficients of the polynomial B(s) are not the
                    coefficients of A(-s) * A(s) for some real A(s);
                    in this case, RES and E are unassigned;
              = 3:  if the iterative process (see METHOD) has failed to
                    converge in 30 iterations;
              = 4:  if the last computed iterate (see METHOD) is
                    unstable. If ACONA = 'B', then the supplied
                    coefficients of the polynomial B(s) may not be the
                    coefficients of A(-s) * A(s) for some real A(s).

Method

          _                                               _
      Let A(s) be the conjugate polynomial of A(s), i.e., A(s) = A(-s).

      The method used by the routine is based on applying the
      Newton-Raphson iteration to the function
                _       _
         F(e) = A * A - e * e,

      which leads to the iteration formulae (see [1]):

         _(i)   (i)  _(i)   (i)     _      )
         q   * x   + x   * q    = 2 A * A  )
                                           )   for i = 0, 1, 2,...
          (i+1)    (i)   (i)               )
         q     = (q   + x   )/2            )

                     (0)         DA
      Starting from q   = (1 + s)   (which has no zeros in the closed
                                                   (1)   (2)   (3)
      right half-plane), the sequence of iterates q   , q   , q   ,...
      converges to a solution of F(e) = 0 which has no zeros in the
      open right half-plane.

      The iterates satisfy the following conditions:

               (i)
         (a)  q   is a stable polynomial (no zeros in the closed right
              half-plane) and

               (i)        (i-1)
         (b)  q   (1) <= q     (1).

                                        (i-1)                       (i)
      The iterative process stops with q     , (where i <= 30)  if q
      violates either (a) or (b), or if the condition
                        _(i) (i)  _
         (c)  RES  = ||(q   q   - A A)|| < tol,

      is satisfied, where || . || denotes the largest coefficient of
                      _(i) (i)  _
      the polynomial (q   q   - A A) and tol is an estimate of the
                                                     _(i)  (i)
      rounding error in the computed coefficients of q    q   . If there
      is no convergence after 30 iterations then the routine returns
      with the Error Indicator (INFO) set to 3, and the value of RES may
      indicate whether or not the last computed iterate is close to the
      solution.

      If ACONA = 'B', then it is possible that the equation e(-s) *
      e(s) = B(s) has no real solution, which will be the case if A(1)
      < 0 or if ( -1)**DA * A(DA+1) < 0.

References

      [1] Vostry, Z.
          New Algorithm for Polynomial Spectral Factorization with
          Quadratic Convergence II.
          Kybernetika, 12, pp. 248-259, 1976.

Numerical Aspects

      The conditioning of the problem depends upon the distance of the
      zeros of A(s) from the imaginary axis and on their multiplicity.
      For a well-conditioned problem the accuracy of the computed
      coefficients of E(s) is of the order of RES. However, for problems
      with zeros near the imaginary axis or with multiple zeros, the
      value of RES may be an overestimate of the true accuracy.

Further Comments

      In order for the problem e(-s) * e(s) = B(s) to have a real
      solution e(s), it is necessary and sufficient that B(j*omega)
      >= 0 for any purely imaginary argument j*omega (see [1]).

Example

Program Text

    *     SB08MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          DAMAX
          PARAMETER        ( DAMAX = 10 )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 5*DAMAX+5 )
    *     .. Local Scalars ..
          DOUBLE PRECISION RES
          INTEGER          DA, I, INFO
          CHARACTER*1      ACONA
    *     .. Local Arrays ..
          DOUBLE PRECISION A(DAMAX+1), DWORK(LDWORK), E(DAMAX+1)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         SB08MD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
          READ ( NIN, FMT = '()' )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = * ) DA, ACONA
          IF ( DA.LE.-1 .OR. DA.GT.DAMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) DA
          ELSE
             READ ( NIN, FMT = * ) ( A(I), I = 1,DA+1 )
    *        Compute the spectral factorization of the given polynomial.
             CALL SB08MD( ACONA, DA, A, RES, E, DWORK, LDWORK, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                IF ( LSAME( ACONA, 'A' ) ) THEN
                   WRITE ( NOUT, FMT = 99997 )
                   DO 20 I = 0, DA
                      WRITE ( NOUT, FMT = 99995 ) 2*I, A(I+1)
       20          CONTINUE
                   WRITE ( NOUT, FMT = * )
                END IF
                WRITE ( NOUT, FMT = 99996 )
                DO 40 I = 0, DA
                   WRITE ( NOUT, FMT = 99995 ) I, E(I+1)
       40       CONTINUE
                WRITE ( NOUT, FMT = 99994 ) RES
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' SB08MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from SB08MD = ',I2)
    99997 FORMAT (' The coefficients of the polynomial B(s) are ',//' powe',
         $       'r of s     coefficient ')
    99996 FORMAT (' The coefficients of the spectral factor E(s) are ',
         $       //' power of s     coefficient ')
    99995 FORMAT (2X,I5,9X,F9.4)
    99994 FORMAT (/' RES = ',1P,E8.1)
    99993 FORMAT (/' DA is out of range.',/' DA = ',I5)
          END

Program Data

     SB08MD EXAMPLE PROGRAM DATA
       3     A
       8.0  -6.0  -3.0  1.0

Program Results

     SB08MD EXAMPLE PROGRAM RESULTS

     The coefficients of the polynomial B(s) are

     power of s     coefficient
          0           64.0000
          2          -84.0000
          4           21.0000
          6           -1.0000

     The coefficients of the spectral factor E(s) are

     power of s     coefficient
          0            8.0000
          1           14.0000
          2            7.0000
          3            1.0000

     RES =  2.7E-15

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