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
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. 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 1.0Program 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