QUARTERLY OF APPLIED MATHEMATICS 577
JANUARY, 1972
ON THE COMPUTATION OF THE CAUCHY INDEX*
By BRIAN D. O. ANDERSON (University of Newcastle, Australia)
Abstract. The Cauchy index of a real rational function can be computed by evaluat-
ing the signature of a certain Hankel matrix. Alternative procedures for its computation
are presented here, one of which offers greater computational simplicity.
1. Introduction. Let p{x) and q(x) be relatively prime polynomials with real coeffi-
cients. The Cauchy index over (— =°. =°) of p(x)/q(x), written I*_Z(p(x)/q(x), is defined
as the number of jumps from — co to +00 less the number of jumps from + to —
of the fraction p(x)/q{x) when x varies from — <=° to + <» [l].
Of course, it is possible to evaluate ItZ{p(x)/q(x)) by determining the real zeros
of q(x) and the associated residues of p(x)/q(x) at these real zeros. But the efficiency of
such a procedure is clearly tied to the case with which real roots of a polynomial can
be found. An alternative and more appealing procedure for the evaluation of
I12(p(x)/q(x)) is discussed in [1]; it is based on computing the signature of a Hankel
matrix constructed from the Markov parameters associated with p(x)/q(x). (Precise
definitions are given later.)
Our aim here is to give two alternative characterizations of the Cauchy index; one
is especially attractive from the computational point of view as it avoids construction
of the Markov parameters or indeed any rational functions of the coefficients of p{x)
and q(x)\ it does, however require computation of integral, as opposed to rational,
functions of the coefficients. The second characterization leads to an infinity of matrices
whose signature defines the Cauchy index of p(x')/q(x). Amongst this family is the matrix
of integral functions of the coefficients.
As a preliminary simplification, we demand that the degree of p(x) be less than that
of q(x). If this is initially not the case, we can of course write
p{x)/q{x) = r(x) + p\x)/q{x), (1.1)
where p'(x) and r(x) are polynomials, with the degree of p'(x) less than that of q(x). Then
IlZ(p(x)/q(x)) = HZ(p'(x)/q(x)). (1.2)
2. The Hankel matrix of Markov parameters. The Markov parameters s,, i = 0,
1, 2, ■ • ■ , associated with p(x)/q(x) are defined as follows, where we now assume that
the degree of p(x) is less than that of q(x):
p{x)/q{x) = s0/x Si/x2 + sjxz + ■ • • . (2.1)
* Received April 8, 1970; revised version received May 21, 1971. Work supported by Australian
Research Grants Committee, Australian-American Educational Foundation in Australia and USAF
Contract F44620-68-C-0023 of the Air Force Office of Scientific Research. Part of this work was performed
while the author was a Visiting Professor in the Information and Control Sciences Center, Institute of
Technology, Southern Methodist University, Dallas, Texas 75222, U.S.A. Sincere thanks are extended
to Professor Andrew P. Sage for his hospitality at this time.
578 BRIAN D. 0. ANDERSON
Note that the region of convergence of the right-hand side of (2.1) is irrelevant in defining
the S; . The associated infinite Hankel matrix is
H =
50
51 §2
52 •
(2.2)
L
Let Hmm denote the matrix consisting of the first m rows and columns of H.
In [1], the following is established.
Theorem 1. Suppose q(x) has degree n. Then for all m > n, rank Hmm = rank Hnn
and signature Hmm = signature Hnn = ItZ(p(x)/q(x)).
Thus to evaluate the Cauchy index, it is necessary to determine the first 2n — 1
Markov parameters associated with p(x)/q(x), then to form Hnn consisting of the first
n rows and columns of H in (2.2), and finally to determine the signature of Hnn. Of course,
the signature of Hnn can be found by examining the signs of the leading minors [1].
3. Matrix of integral functions of polynomial coefficients. Let the polynomials
p(-) and <?(•) be defined by
p(x) = baxm + biXm 1 + ■ • • + bm , (3 1)
q(x) = a0xn + a^x71'1 + •••+»„,
with n > m. From these polynomials p(-) and q(-), we define coefficients c^, by the
Bezoutian form
P(y)q(x) - q(y)p(x) = £ c^-y-, (3 2)
x y i.i-i
Clearly, c,,- = c,, . Below we shall prove the following result.
Theorem 2. Let C = (c,-,), where the entries c,,- are formed as indicated above. Then
signature C is the Cauchy index over (— , =°) o/ p(x)/q{x).
First, let us note a simple expression for the coefficients c,-, in terms of the coefficients
of the polynomials p(-) and q(-). From (3.2), it follows by multiplying on the left and
right by x — y and equating coefficients of like powers that
Gn — ibm—i Cij+1 Ci + lj •
From this identity we obtain
C,-,' = (fln-i-kbm-i+l+k Q*-j + l+l:bm-i-k) (3.3)
k> 0
where we take a: = 0 if Z < 0 or I > n and b, = 0 if I < 0 or I > m.
Proof of Theorem 2. Recalling that, formally,
p(x) = q(x)
NOTES 579
we have
x>„*-v-
= p(y)q(x) - g(y)p(x)
X — y
q(x)q(y) 1 [~ (l _ l\ (l_ _ 1^\ "1
xy (\/y) ~ (l/x)\_°\y x) Ay2 x2) "J
«2WsM[" + (1 + 1)+ A+J.+ n 1
xy L \y xl \y yx x / J
<3-4)
where the superscript prime denotes matrix transposition.
Now write q(x)/x' as q,{x) + r,(x) where qt (x) is a polynomial in x, and find r,(x)
polynomial in \/x with no constant terms. Thus, for example, qx(x) = a0x"~L + ■ • • + a„_,
and Ti{x) = ajx. Then (3.4) yields
X) cux'~y'_1 = [qi(x) + ry(x) q2(x) + r2(x) ■ • • ]ff[g,(y) + r.ft/) g2(t/) + r2(y) •••]'.
Since the left side has no terms in l/x or l/y it follows that
X Cux'-y-1 = foOc) q2(x) ■ ■ ■ qn(x) 0 0 • - -Mg^y) g2(t/) • ■ • q.(y) 0 0 • • •]
= [?i(z) q2(x) ■ ■ ■ qn(x)]Hnn[qi(y) q2(y) • • • qn(y)]'. (3.5)
It is readily verified that
[gi(x) q2(x) ■ • ■ g„(z)] =
&»-1 ®n-2 * * * &1 &0
&n-2 ^n-3 * * " &0 0
ax Q/q ' * * 0 0
l a0 0 •••00.
(3.6)
and so with
A =
&n-l a0
2 3 ' a0 0
ai a0 ■■■00
_ a0 0 • • • 0 0.
(3.7)
Eqs. (3.5), (3.6) and (3.7) imply C = A'HnnA. The rank and signature of C are the same
as the rank and signature of Hnn in view of the obvious nonsingularity of A. Application
of Theorem 1 then yields the desired result.
4. A class of matrices yielding the Cauchy index. We shall make use of the langu-
age of linear system theory [2], As earlier, we assume p and q relatively prime, p of degree
m and q of degree n > m. Let F be an n X n matrix and let g and h be n-vectors with the
580 BRIAN D. O. ANDERSON
entries of F, g and h all real. Then the triple {F, g, h) is a minimal realization of p(x)/q(x)
if
p(x)/q(x) = h'(xl — F)~1g. (4.1)
Minimal realizations always exist—in fact there is an infinity of minimal realizations
associated with the rational function p(x)/q(x).
Our main result is as follows.
Theorem 3. Let F, g, hbe a minimal realization of p(x)/q(x) in the above sense. Then
there exists a unique symmetric matrix P satisfying
PF = F'P, Pg = h, (4.2)
and signature P is I12(p(x)/q(x)). Further, P is computable by simple algebraic operations
which exclude polynomial factoring or its equivalent.
Proof, (i) Computation. From (4.2) we have
PFg = F'Pg = F'h, PF2g = F'2Pg = (Fr)2h,
and in fact P[g Fg ■ ■ ■ F"~'g] = [k F'h ■ ■ ■ (FT'1*] or
PV = W (4.3)
with obvious definitions of V and W. The matrix V is invertible because IF. g, h] is
minimal and thus completely controllable [2], Therefore, P = WV'1.
We must also verify that (4.3) implies (4.2). The second equation of (4.2) follows by
equating the first column on each side of (4.3). To check the first equation of (4.2), we
proceed as follows. By the Cayley-Hamilton theorem (4.3) implies PeF'g = eF''h.
Differentiate to obtain PFeF'g = F'eF''h = F'PeFtg. So (PF — F'P)eFtg — 0, and an
application of complete controllability [2] yields that PF = F'P.
(ii) Symmetry. Observe that V'PV = V'W is symmetric, because the i — j term
of V'W is g^F'-y^'y-'h = h'Fi+i~2g.
(iii) Uniqueness. Eqs. (4.2) and (4.3) are equivalent and (4.3) has a unique solution.
(iv) Signature property. Since V is nonsingular, signature P = signature V'PV =
signature V'W by (4.3). The i — j term of V'W is h'F'+'~2g which can be verified, using
(3.1), to be S;+,_2. Hence V'W -is precisely Hnn, the first n rows and columns of the infinite
Hankel matrix defined by the Markov parameters of p(x)/q(x). Theorem 1 yields the
desired result.
We shall show now that one particular minimal realization of p(x)/q(x) yields a
matrix P which is, to within a positive constant multiplier, precisely the matrix C of
Sec. 2.
It may be verified by direct calculation, and it is shown in [2], that with p(x) and q(x)
defined as in (3.1), a minimal realization of p(x)/q(x) is provided by
To 1 0 ••• 0
0 0 1- 0
F =
flw-1
L G0
h =
K
(X o
a0
(4.4)
NOTES 581
Theorem 4. Let p(x) and q(x) be defined as in (3.1), and let a minimal realization
of p(x)/q(x) be as given in (4.4). Then the solution P of Eqs. (4.2) is P = a^2C, where
C = (cu) with Cn defined by (3.3).
To prove the theorem, we make use of the following lemma, the proof of which is
immediate, by direct verification.
Lemma. Let F, g and h be as in (4.4), and let t(x) = [Ix x2 ■ ■ ■ xn~1]'. Then Fir(x) =
xir(x) mod q(x) and tiir(x) = p{x)/aa .
Proof of Theorem 4. From (3.2) and the definitions of C and ir(x) we have
- pWgfa) _ j.
x y >. >■ -1
and so
p(y)q(x) - p(x)q(y) = xir'(x)CT(y) - Tr'(x)Cir(y)y. (4.5)
Now reduce both sides modulo q(x) and then modulo q(y). The left side becomes zero,
while the right side may be evaluated using the lemma to yield v'(x)[F'C — CF]ir(y) — 0.
Since this holds for all x and y,
CF = F'C. (4.6)
Xext, observe that in Eq. (4.5) the coefficient of y" on the left-hand side is —a0p(x) =
—alh'ir(x) by the lemma. The coefficient on the right side is evidently —ir'(x)Cg. Conse-
quently ir'(x)[Cg — alh] = 0, whence
Cg = api. (4.7)
Xow the uniqueness of P and comparison of (4.2), (4.G) and (4.7) yield the result C = a\P.
5. Concluding remarks. Various special choices of p(x) and q(x) lead to the Cauchy
index of p(x)/q(x) being the number of real roots of a polynomial, or the number of right
half plane zeros of a polynomial. In the latter case, the matrix C of Sec. 3 becomes
identical with the Hermite matrix [3], [4] whose relation to the matrix Hnn of Sec. 2 has
been explored in [5]. A theorem similar to Theorem 4 has been derived in [6], where special
choices of p(-) and q( - ) enable counting of the zeros of a polynomial inside the unit circle.
We note too the use of Bezoutian form of (3.2) in [7] to compute the degree of the highest
common factor of p(x) and q(x).
The Cauchy index can also be computed from the signs of the Hurwitz determinants.
Their relation to the signs of a sequence of nested determinants of the Hankel matrix
appears in [8] and [9], and to the signs of a sequence of nested determinants of the Be-
zoutian matrix C in [7].
References
[1] F. R. Gantmacher, The theory of matrices, Chelsea, New York, 1959
[2] L. A. Zadeh and C. A. Desoer, Linear system theory, McGraw-Hill, New York, 1963
[3] C. Hermite, Sur le nombre des racines d'une equation alg&rique comprise entre des limites donnces,
J. Reine Angew. Math. 52, 39-51, (1854) and Oeuvres 1, pp. 397—114
[4] P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Lyapunov,
Proc. Cambridge Philos. Soc. 58, part 4, pp. 694-702 (1962)
[5] B. D. O. Anderson, Application of the second method of Lyapunov to the proof of the Markov stability
criterion, Internat. J. Control. 5, 473—482 (1967)