## 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
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 ):

_(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
```   Vostry, Z.
New Algorithm for Polynomial Spectral Factorization with
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.

```
```  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 ).

```
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  1.0
```
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
```