A complete discussion of the mathematical theory behind the approach taken in this book can be found for example in []. The following abridged version should give enough information to use the given routine reliably. This routine is available from the Maple share library.
A polynomial 
is a
Hurwitz polynomial if all its roots are in the
left half plane.
The paraconjugate 
of p is defined as
the polynomial whose roots
are the roots of p reflected across the imaginary axis.
Algebraically, 
.
So if 
, then
.
The auxiliary Maple subroutine
`Hurwitz/data` computes the sequence of partial
quotients in the Stieltjes continued
fraction of the rational function
to determine if 
for 
, which will happen
if and only if p is Hurwitz.
The subroutine returns a list with two elements --- the
first element of the list is another list,
of the partial quotients in the Stieltjes continued
fraction, and the second element of the first list
is the gcd of p and 
.
If this gcd is nontrivial, then there are pairs of roots symmetric on
reflection through the imaginary axis, or else purely imaginary
roots, and in either case p is not Hurwitz.
We then compute the partial quotients of the Stieltjes continued
fraction for 
essentially by the Euclidean algorithm.
We put 
where q is the quotient on polynomial division
of 
by 
, and put 
and
repeat the process with the resulting numerator and denominator.
We terminate the process when we encounter a zero remainder.
For example, consider 
.
We see 
and since the
coefficients of p are real,
and so
Then the first step of the algorithm gives 
and
.
Replacing h by 
and starting again, we get
(which is our first partial quotient)
and 
, so
. Repeating the process
twice more gives quotients 
and 
. Since not all
of these are positive, the original polynomial is not Hurwitz.
The Maple routine `Hurwitz/data` does all this calculation
automatically and returns a list containing the GCD of p and
and a sub-list of the partial quotients.
The first element of the returned list of partial quotients
is special. If it is
of higher degree than 1 in 
, p is not Hurwitz. If it
is of the form 
, where 
or b < 0, then p
is not Hurwitz. If each subsequent polynomial in the
returned sequence is of the form 
, where 
and
b > 0, then p is Hurwitz, and not otherwise.
See [] for details.
The Maple routine Hurwitz
will return true if it can use these rules to
decide if p is Hurwitz, false if it can decide that p is not
Hurwitz, and FAIL otherwise, in which case it is up to the user to
decide when p is Hurwitz, given the partial quotients. Of course,
if the polynomial has symbolic coefficients then there will usually
be no way that this routine can tell automatically when the
polynomial is Hurwitz, and thus the described mechanism for returning
the partial quotients to the user for inspection is required.
Consider the matrix
where c = a + ib. The characteristic polynomial of this matrix is
We can easily tell directly when the roots of this polynomial are in
the left half plane as it is linear in c and hence a and b.
Putting 
gives us the parametric equations
and 
describing the location
in parameter space where
roots of p cross the imaginary axis. See Figure 5.1.
Figure 1: Values of a and b where the
roots of 
have purely imaginary roots.
This parametric curve is symmetric and has a loop. Inspection shows that all roots are in the left half plane precisely when c = a+ib has values inside the loop.
We now show how our
implementation of the Hurwitz criterion is used to determine the same
information.
We load in the Hurwitz package from the Maple share library
and call the
routine according to the instructions in the help file. The
immediate result is the answer FAIL --- the Maple routine cannot
decide by itself
if the polynomial is Hurwitz or not, as this depends on the
value of the parameter c, which Maple knows nothing about. This is
the expected behaviour, and we then examine the gcd of p and
, which turns out to be 1, and the sequence of
partial quotients computed, which turns out to be
Now we see that the first partial quotient has positive real part
(1/6) for
the coefficient of 
and
zero real part for the constant term (in fact zero imaginary part,
also). The second partial quotient will have positive real part if
and only if a > -56. The third
partial quotient will have positive real part if and only if
(one sees this by isolating 
and factoring the right-hand side).
Plotting of this region gives exactly the interior of the
loop shown earlier.
The Maple calculation is as below.
> p := lambda^3 + 6*lambda^2 + 10*lambda + 4 - a - I*b;
3 2
p := lambda + 6 lambda + 10 lambda + 4 - a - I b
> with(share):
> readlib(Hurwitz):
> Hurwitz(p,lambda);
FAIL
> Stieltjes_continued_fraction_and_GCD := `Hurwitz/data`(p,lambda);
Stieltjes_continued_fraction_and_GCD := [
I b lambda
[1/6 lambda, 216 --------- + 36 ------,
2 56 + a
(56 + a)
2
I (56 + a) b
---------------------------------------
2 3 2
- 12544 + 2688 a + 108 a + a + 216 b
3
(56 + a) lambda
- 1/6 ---------------------------------------], 1]
2 3 2
- 12544 + 2688 a + 108 a + a + 216 b
The 1 on the end is the GCD of p and its paraconjugate.
Exercises
Hurwitz?
is the truncation to degree n of the Taylor
series for 
about x=0, show that 
is Hurwitz
for k = 0, 1, 2, 3, 4, 5 but not for k = 6. In fact it is
not Hurwitz for any k > 5 [].
is
the Bessel polynomial of degree n, show that it is
stable for n = 5, 10, 15. It can be shown [] that
these polynomials are stable for all n.