The Construction of Rational Iterating Functions

MATHEMATICS OF COMPUTATION, VOLUME 32, NUMBER 143

JULY 1978, PAGES 811-827

The Construction of Rational Iterating Functions

By W. F. Smyth

Abstract. Suppose integers n > 1 and o > 2 are given, together with n distinct points

I], . . . , zn, in the complex plane. Define <t>M = *j^fto; zx.zn) to be the class

of rational functions <¡> (z) = g (z)/hg(z) (where g and 7i are polynomials of degree

p > 1 and q > 1, respectively) such that (l)p+qr + l=Ai, (2) 0 when iterated con-

verges with order a at each z¡, i = 1, . . . , n. Then if M < on, 4)^ is null; if M =

on, *j^ contains exactly on elements. For every M > on, we show how to construct

all the elements of <t>j^ by expressing, for each choice of p and q which satisfies p +

q + 1 = M, the coefficients of g„ and 7i in terms of M — on arbitrarily chosen values.

In fact, these coefficients are expressed in terms of generalized Newton sums S¿ =i k

S¿ (zj, . . . , zn), 1 <, j <, n, k > n, which we show may be calculated by recursion

from the normal Newton sums Sn' . Hence, given a polynomial fn(z) with n distinct

(unknown) zeros zx, . . . , zn, we may construct all <j>„ _(z) which converge to the z(-

with order a: in the case a = 2, the choice p = n, q = n — 1, yields the Newton-

Raphson iteration <j>n n_x e *2n ; the Schröder and König iterations are shown to be

elements of *2(2a—3)(n—1)+2 an<' *2(a—l)(n—1)+2 ' resPectiyely- We show by ex-

ample that there exist cases in which <j>n n_x has an undesirable property (attractive

cycles) not shared by other iterating functions in the same class <t>2n.

Introduction. This research has its original motivation in the work of Julia [1],

Fatou [2], and Montel [3], [4] on the global convergence properties of iterations of

meromorphic, particularly rational, functions. It appears that, for cases of practical

interest, convergence of the sequence of iterates z0, 0(zo), 02(zo), . . . , is assured for

every choice of starting point z0 in the extended complex plane, except when z0 is a

point of a certain nowhere dense perfect set E [5]. Usually, of course, we have con-

structed 0 in such a manner that the sequence of iterates will converge to a zero of a

given function f(z); that is, we have replaced the problem of finding z such that f(z)

= 0 by the problem of finding z such that 0(z) = z. (The most common form of 0 is

¥,z) = z- f(z)/f(z),

the Newton-Raphson iteration.) It may happen, however, that for choice of starting

point z0 in a certain domain, convergence takes place, not to a zero of f(z), but to a

set of r > 2 distinct points (ax, . . . ,4tr) such that

<p(ax) = cx2, . . . , <piar_x) = a,, 0(cg = a, ;

so that, in fact, for each k = I, . . . , r, z = ak is a solution of 0r(z) = z. The best-

known example of this phenomenon is due to Turing and Brooker [6], who pointed

Received March 21, 1977; revised December 7, 1977.

AMS (MOS) subject classifications (1970). Primary 30A08, 40A05.Copyright © 1978, American Mathematical Society

811

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812 W. F. SMYTH

out that the Newton-Raphson iteration applied to the seemingly innocuous function

f(z) = z4 -6z2 - 11

gives rise to the attractive cycle (+1,-1) for starting points z0 in a certain neighbor-

hood of either +1 or -1.

Indeed, on the basis of the general theory of global convergence of rational

functions, it is not surprising that such attractive cycles should sometimes exist. For

let g = gpiz) and h = h (z) be polynomials of degree p and q, respectively, with no

common factor. Then a basic result of the theory is that, in the immediate domain

of convergence of each attractive fixpoint or attractive cycle of the rational iterating

function

<t>p.q(z) - Sp(z)lhq(z),

there exists at least one critical point of the inverse iteration 0-1 [7]. Since these

critical points are just the solutions of

<j>'iz) = ig'h - gh')/h2 =0,

we see that in general they shall be p + q - 1 in number. In other words, convergence

of the sequence of iterates of 0 may take place to at most p + q - 1 distinct attrac-

tive fixpoints or attractive cycles; the attractive fixpoints will be solutions of 0(z) =

z, hence of g — zh = 0, and the attractive cycles (if they exist) will be solutions of

0r(z) = z, for certain values of r > 2. In practice, as observed above, we shall have

constructed 0(z) in such a manner that, of the mavAp, q + I) fixpoints of 0(z) = z, n

shall be attractive fixpoints and also solutions of a given equation f(z) = 0. Hence,

in general, there remains a possible total of (p + q - 1) - n attractive fixpoints or

attractive cycles, to which convergence may take place from starting points in certain

domains of the complex plane. In the case of the Newton-Raphson iteration, p = n,

q = n - 1, and we see that up to 77 - 2 such cases may occur.

An ultimate aim of our research is therefore to find a means of constructing

rational iterating functions which do not give rise to attractive cycles or superfluous

attractive fixpoints, and which, therefore, converge globally-that is, almost every-

where in the complex plane—to solutions of the given problem f(z) = 0. In this paper

we take a step toward this goal by showing how to construct all the rational iterating

functions 0 = 0 which converge with given order of convergence a to 77 given

distinct points z¡,i=l,...,n; in particular, how to construct all the 0 which

converge with order a to the distinct (but unknown) zeros z¡ of a given polynomial

fn(z). Denote by

$m = *m(°> zx,...,zn)

the class of rational functions 0 such that (1) p + q + 1 = M, (2) the sequence of

iterates of 0 converges with order a at each z¡; then our construction technique

opens up to us the new possibility of inspecting the iterating functions of each 4>M to

determine which give rise to attractive cycles, and which do not—or, indeed, to


CONSTRUCTION OF RATIONAL ITERATING FUNCTIONS 813

LZ*,

determine which may or may not have other desirable or undesirable properties. In

this paper we do not propose any techniques for inspection of the elements of 4>M ;

but we do show, by an example derived from Brooker's, that, even though the Newton-

Raphson iteration <j>n n_x G 4>2n gives rise to attractive cycles, there may nevertheless

exist other functions in the same class which have no such cycles.

Rational Iterating Functions With 77 Given Fixpoints. Consider

<Pp,q(z) = gp(z)lhq(z) = X aft! Z 0,0<;«p / 0<k<q

where p > 0,q > 0, a =£ 0, 0 ^ 0, and the polynomials g and h have no common

factor. In this section we try to choose the coefficients (a-, 0fc) to satisfy 77 given

conditions

(1) <t>P,q(zd = Zv i=l,-..,n,

where i + 1 =*■ z¡ =£ z,'. That is, we must find (a-, 0fc) such that

(2) g(z¡) - Zih(z,) = 0;

in other words, such that

(3)«0 + (<*, -ß0)z, + ■ ■ ■ + (ap - ßp_x)zf + (0 - ßp)zf+1

+ ■ - + (0-ßq)zf+1 =0

. ,p.for each z¡, where if 17 < p, we define ßk = 0, k = q + 1

Setting

(4) vO)=,v -Oj - 0,-,, / = 0, . . . m,

where 777 = max(p, q + I), and defining ß_x = 0, a- = 0 for / = p + 1,

rewrite (3) as

h *i---*r\ lw\

\i *.••■*/ U1}/

= 0.

Introducing the notation

h s+i

?r,s =

-I-+I

l*"\

r' r,s

*nl u1)

, m, we


814 W. F. SMYTH

for integers r and s, 0 < r < s, this becomes

(5) z°n'm r0>m = o,

a set of n homogeneous linear equations in 777 + 1 variables yj1^. We observe that for

7?7 + 1 = 77, det(Z^¡mx) is simply the Vandermonde determinant

Dn= n (z.-z.0*0.n>i>i>l

Accordingly, for 777 < tj, the only solution of (5) is ro>m = 0, which implies ay =

0/_1,/ = 0, . . . , 777, hence that <Pp¡q(z) = z, identically.

However, if we choose m = n, the rank of Z°'m is also 77, and, therefore, (5)

possesses a nontrivial solution in the yjl ^ which is unique but for multiplication by a

constant factor. Thus the minimum value of 777 leading to a nontrivial iterating func-

tion 0 is 777 = 77. Rewriting (5) for m = n, we get

70,n-lr . = -7n,n (1)^n l 0,n-l n 'n >

which yields the (unique) solution

7(1)=_(Z37+1,«/Dj7(l); f = 0,...,„-l,

where in general D^k is the determinant obtained by replacing the /th column of

Dn with Zk,k. In the Appendix we show (Theorem 5) that the quantities

S^ = (-l)n-'Di-nlDn, l</<77,

are the familiar Newton sums

Sn'" = zlz2 ' ' ' zn> ■ ■ ■ ' Sn = 2-, zízí'' Sn'" = ¿*> zïi¥>i' Ki<n

We write then for the case m = n :

(6) t}1 > = (- i)"-'s>+! 'ny(nl\ / = 0,..., « - 1,

and we observe that the yj1' axe always symmetric functions of the z¡.

More generally, fox m > n conditions (1),

v'' ¿n '0,11-1 n ln,m>

and we see that each element of the solution ro n_x may be expressed as a linear

combination of 777 - 77 + 1 arbitrarily chosen values 7„ , • • • , 7¿J • If corresponding

to each K = n, . . . , m, we choose these 777-77 + 1 values so that y^ =£ 0; yk1^ =

0 for k ^ K, we derive m - n + I linearly independent particular solutions

7JK)--0n+1,K/Dn)yk1)' ! = 0,...,n-l,

each dependent on a single arbitrarily chosen value y^^- We may then reconstitute a

general solution of (7) by forming a linear combination of these 777-77 + 1 particular

solutions. Accordingly, summing the yjjJ over K and making use of the generalized

Newton sums S'¿k defined in the Appendix, we write the general solution of the sys-

tem (5) as



(8) y^=(-l)n-i Z S>n+l'Ky%\ / = 0,. ..,n-l,n<K<m

where the y^ are arbitrary complex numbers, not all zero. The quantities Sn+l'K

axe by definition symmetric functions of the z¡, and may be calculated as indicated

by the Algorithm and Theorem 7 given in the Appendix. Hence we state

Theorem 1. In order that a rational function 0 (z) satisfy the n conditions

(I), it is necessary that m = maxip, q + 1) > n. 0 (z) satisfying (1) may be con-

structed by arbitrarily choosing m - n + 1 "combined coefficients" yy' as defined

in (4), and then using (8) or an analogous system to find the remaining n combined

coefficients, which are then uniquely determined as symmetric functions of the given

points z¡, i = 1, . . . , 77.

We have thus found a way to generate all rational functions which as iterating

functions have fixpoints at 77 given distinct points.

Rational Iterating Functions With 77 Given Attractive Fixpoints. The conditions

now to be satisfied may be expressed as follows :

W <¡>p,q<¿i) = z/> <t>'P,q(zi) = *,

for some 0 < lei < 1 and every i = I, . . . ,n. These are 2n independent simulta-

neous linear constraints on the p + q + 2 coefficients (a-, 0fc), which we may re-

express in the form of (2) :

(10<0 g(zi)-zih(zl) = 0,

(10b) g'(zi)-zih'(zi) = e.

From (10b) we obtain after some manipulation a set of equations analogous to (3) :

(11) Z [0' + l)a/+1 - (j + e)ßj]zi = 0, 1 = 1, . . . , 77.0</<m-l

It is convenient to define

(12) yj2) = U + l)aj+ !-(/ + e)0,, / = 0,. . . , m - 1,

analogous to (4). Then (11) simplifies to

(13) Z z/7J2) = 0, 7 = 1,..., n,0</'<m-l

a set of 77 homogeneous linear equations in 777 unknowns yj2^ which is identical in

form to (5). Proceeding as in the previous section, we observe that for 777 < n + 1

there are only trivial solutions of (13), and that for ttj = 77 + 1 there is a unique

solution

(14) yf) = (- iyHSl+ 1 .»r(2), / = 0, . . . , 77 - 1,

analogous to (6). Then for 777 > 77 + 1, we may show as before that general solutions

yj2\ j = 0, . . . , 7i - 1, of (10b) may be written


816 W. F. SMYTH

(15) 7/(2) = (-l)n-/ Z &**-#>. ; = 0,..., 77-1,n«Ä:<m-l

identical in form to (8). The solutions (8) and (15) taken together are 2« linear ho-

mogeneous equations in p + q + 2 variables (a;-, 0k) equivalent to the original

2n equations (9); they will possess a solution if p + q + 1 > 2n, a condition in

some respects stricter than m > n + 1, but which allows us to solve the equations (8)

and (15) in the important "Newton-Raphson" case, characterized by n = m = p = q

+ 1 and e = 0. We have then

Theorem 2. In order that a rational function 0 (z) satisfy the 2n conditions

(9), it is necessary that p + q + 1 > 2n. 0 (z) satisfying (9) may be constructed

by arbitrarily choosing (p + q + 2) - 2n of the coefficients (a-, 0fc), and then using

(4), (8), (12), and (15) to find the remaining 2n coefficients, which are then uniquely

determined as symmetric functions of the given points z¡, i = I, . . . , n.

We have thus found a way to generate all rational functions which as iterating

functions have attractive fixpoints at 77 given distinct points. If e = 0, the generated

iterating functions will have convergence of order 2 at each of the given points.

Rational Iterating Functions of Given Order a. Evidently, the above results

may be generalized. Given 77 distinct points zx, . . . , z„, and any integer a > 1, we

may construct rational iterating functions 0 with order of convergence a at each

of the given points by finding solutions to the 077 linear homogeneous equations

(16) *p.¿2¡> = z"

^l\z{) = 0, s = 2,..., a,

where 7=1,...,«. If we introduce the notation g^ = g, /r°) = h, we may briefly

reexpress these equations in the form of (10) :

(17) g^-l\zi)-zih^-l\zi) = 0,

for s = 1, . . . , a, i = 1, . . . , 77. If consistent with (4) and (12), we define

^■('mV-CmVfor s = 1, . . . , a, 7 = 0, . . . , 7?7 - s + 1, then we may rewrite (17) in the form

(19) Z ztfP = 0,0<j*im~s+ 1

for s = 1, . . . , a, i = 1, . . . , 77, and, as before, derive general solutions

(20) y(s) = (rly-i £ S'n+l*y%\n<K<m-s+l

s = 1, . . . , a, j = 0, . . . , 77 - 1, valid provided p + q + 1 > on. We have

Theorem 3. In order that a rational function <t>PiQ(z) satisfy the an conditions

(16), it is necessary that p + q + 1 > an. 0 (z), satisfying (16), may be constructed



by arbitrarily choosing (p + q + 2)-an of the coefficients (a-, 0fc), <277<i then using (18)

and (20) to find the remaining an coefficients, which are then uniquely determined as

symmetric functions of the given points z¡, i = I, . . . ,n.

In terms of the iteration classes $M defined in the Introduction, Theorem 3

tells us that 4>M is null whenever M < an. Moreover, for M = on, each of the an

possible choices of p and q yields a single unique iterating function 0 ; in other

words, <&on contains exactly on members. For M> on, each choice of p and q yields

an infinite subclass of iterating functions with coefficients dependent on the particular

choice of the excess (p + q + 2)-an coefficients. The case M = an is, therefore, of

particular interest, not only because it is a minimum (hence minimizing time required

for function evaluations, as well as number of possible attractive cycles), but also be-

cause <í>an is finite.

Rational Iterating Functions for Zeros of a Polynomial f„(z). As mentioned

earlier, it is shown in the Appendix that the quantities 5¿fc = S'r¡k(zx, . . . ,zn) used

to compute the coefficients of 0 may themselves be computed from the Newton

sums. Since the Newton sums may in their turn be computed from the coefficients

only of a polynomial [8, p. 453], without direct knowledge of the roots, it is natural

to seek to apply the method to construct rational iterating functions convergent to

the unknown (but distinct) zeros of a given polynomial fn(z). Such an application is

straightforward, and an example is provided in the next section.

In this section we establish a connection between our iteration classes <$>M and

one or two well-known iterating functions (or classes of iterating functions) which are

applied to the zeros of polynomials. We observe first that for a = 2 the iterating

function 0„ „_j G $2n(2; zx, . . . , zn) is exactly the Newton-Raphson iteration ap-

plied to a polynomial fn(z) with 77 distinct roots zx, . . . , zn. This result is a direct

consequence of the uniqueness of <pn n_x (Theorem 3), but may also be proved direct-

ly.

A general class of iterations was discovered by Eggers and Schröder [9] about

a century ago and has been connected in a recent paper [10] to a number of more

modern iterations. Given a polynomial fn(z) of degree 77, Schröder described a method

of finding a rational iterating function of any desired order of convergence a > 1 con-

vergent to the zeros of fn ; his iterations took the form

0l"1(z) = z - bx(z)fn(z) + ■■■ + (-l)°-lba_x(z)\fn(z)]°-\

where bx(z) = U^Xz), bs(z) = b^\(z)l[sf^\z)\, s = 2, . . . , a - 1, and turn out

to have numerator of degree (2a - 3)(t? - 1) + 1 and denominator of degree one less

(they are Newton-Raphson-like in form). In our terminology Schroder's 0^CT'(z) is

one of the infinitely many iterations

0(2o-3)(n-l)+l,(2o-3)(/t-l) 6 *2(2a-3)(n-l) + 2 >

and we see that 0'a ' actually depends upon an implicit arbitrary choice of

2(2ct - 3)(n -I)+ 2-on ox (a- 2)(3n - 4) coefficients. For a = 2, 012 ' =

Kn-i-


818 W. F. SMYTH

Householder [11, p. 103] describes another general class of iterations due to

König, which are also designed to converge with order o to the zeros of polynomial

/„. König's iteration takes the form

0l"l(z) = z + (o- l)[l//B(z)](°-2V[l//,t(*)](,r-1),

is also Newton-Raphson-like, and has numerator of degree (a - 1)(t7 - 1) + 1.

0'CT'(z)is one of the infinitely many iterations

<t>(o-l)(n-l)+l,(o-l)(n-l) € *2(a-l)(n-l) + 2

and depends upon an implicit arbitrary choice of (o - 2\n - 2) coefficients.

Application to /4(z) = z4 - 4z2 - 5. This polynomial, suggested by Brooker's

example [6], shares the property that the Newton-Raphson iteration applied to it

gives rise to an attractive cycle. It has zeros ±\JS and ±i, and the Newton sums may

be read off from the coefficients :

S¿'4=-5, S2'4 = 0, S** -4, S4'4 0.

With these values as our only input data, and making use of Theorem 3 and the Algo-

rithm and results developed in the Appendix, we generate all the rational iterating

functions 0 which converge with order a = 2 to the zeros of /4 and which belong

to the minimum class <ï>8(2; +\JS, -\/S, +i, -/). Table 1 displays the results : ob-

serve that while 04 3, for example, gives rise to the attractive cycle (+y/5¡l, -\j5/7),

the iterations 02 s, 06 x, 0X 6, and 07 0 converge to some root of /4 from each crit-

ical point.

Table 1

Iterating functions for /4(z) = z4 - 4z2 - 5 which belong to

minimum class $s(2; +\/5, -\/5, +i, -i)

m p q iterating function </>„

4 4 3 (3z4 - 4z2 + 5)/(4z3 - 8r)

5 3 4 (8z3 + 20z)/(z4 + 4z2 + 15)

5 5 2 (z5 - 12z3 - 25z)/(-8z2 - 20)

6 2 5 (92z2-40)/(3z5-20r3 + 109z)

6 6 1 (z6 - 6z4 - 33z2 + 10)/(-36z)

7 1 6 -180z/(2z6-21z4 +42z2 - 115)

7 7 0 (2z7 - 2Iz5 + 42z3 + 245z)/180

8 0 7 -900/(23z7 - 174z5 + 33z3 + 1130z)

additionalcritical points

(cps}*

±V2/3-

+ VÎ5/2.'

±(5/V6)i

± s/218/207/

±s/ïfSi

tV23/10

ty/7Í2

±V226/161

fixpoints or cyclesreached from cps

(+V5/7, -V5/7)

(+V7Í.-0Ooo

±i

V

±i

±s/5"

(+0.9285,-0.9285)

The roots ±\J5, ±i are critical points of each <¡>~lq. Thus, in each case, two other critical points remain

to be specified, and these may possibly belong to the immediate domain of convergence of an attractive

cycle, if such exist, or of a root. Note that "convergence" to °° is possible.



Table 2 displays 5 of the 12 iterating functions in 4>12(3; + \/5, -\/5> +i,~i);ihat

is, the minimal class of rational iterating functions which converge with order o = 3

to the zeros of/4.

Table 2

Iterating functions for /4(z) which belong to $X2(3; +\/5, —s/5, +7, ~i)

m p q iterating function 0

6 6 5 (49z6 - 12z4 + 675z2 - 200)/(87zs - 204z3 + 645z)

7 5 6 (129zs + 204z3 + 435z)/(8z6 + 135z4 + 12z2 + 245)

7 7 4 (8z7 - 231zs - 252z3 - 805z)/(-273z4 + 84z2 - 435)

8 4 7 (1443z4 - 8220z2 + 3225)/(49z7 - 576z5 + 3363z3 - 8900z)

8 8 3 (29z8 - 280z6 - 1050z4 + 4760z2 - 2275)/(-2016z3 + 5760z)

The determination of the coefficients (a¡, ßk) for this example involved the solution

of the simultaneous equations (20) in accordance with Theorem 3. This solution was

facilitated by the fact that these equations are sparse in the (a-, 0fc); and an estimate

of the maximum density of the matrix representing the equations may, therefore, be

of general interest. (20) represents 077 equations in p + q + 2 variables. The left-

hand side of (20) specifies at most 2ütí nonzero coefficients (a-, 0k), hence represents

at most that many matrix positions. The right-hand side specifies at most

SKK(J (777-S+I-77 + I) nonzero coefficients. Maximum nonzero entries in the

coefficient matrix are, therefore,

2o77 +2(777 - s - n + 2) = (0/2X2/1 + 2m - o + 3),

out of a total an(p + q + 2) positions. The maximum ratio of nonzero positions in

the matrix representation of the Eqs. (20) is, therefore,

(2n+2m-o + 3)/2n(p + q + 2).

If we let 777 take its maximum value on, and confine ourselves to the "minimal"

case p + q + 1 = n, this ratio becomes after some manipulation

I/77 + (2n - a + l)/2n(an + 1),

which, since a > 2, is less than 3/2n. The system (20) will thus, in general, be fairly

sparse.

Extension of Results to Multiple Fixpoints. In the preceding development it

has most obviously been the definition of S(¡k in terms of an expression with denom-

inator Dn (vanishing whenever z¡ = z¡-, i ¥= i') which has precluded consideration of

multiple fixpoints. In this section it will be found that the vanishing of Dn is not in

itself a fundamental problem, but that other difficulties, similar to those encountered

when applying the Newton-Raphson iteration to a function with multiple zeros, inhibit

full extension of the results to the multiple fixpoint case.


820 W. F. SMYTH

We begin by considering Eq. (2), which, it has been remarked [12], expresses

the condition on g(z) - zh(z), a polynomial of degree m, that it have 77 < 777 roots z¡;

that is, that

0</<m

= g(z) - zh(z)

c(z) n (z-zi)=¡c(z) £ (-íy-'s^'VlK/<» 0«i<n

where the S'n+l'n are as before simply the Newton sums, S" + 1'n = 1, and

<Xz) = c„ +cn+xz + - •• +cm_xz'm—n—l

+ C Z^ cm¿ #0,

is a polynomial with arbitrary coefficients introduced so that the bracketed polyno-

mials should be of equal degree 777. We recall, however, that some of the left-hand

coefficients-to be precise, y^\ K = 77, . . . , 777—may also be arbitrarily chosen; and

we realize then that we may either express the 7^ in terms of the cK, K = n, . . . ,

777, or vice versa, simply by equating the coefficients of like powers of z. We find

then

'y^ = (-l)"Sn'"cn

y(l) = (_l}n-ls2n,nCn+{_irsln,nCn + i

(A)'

■vO) =ln-1

£ (_x)K+lsn-K,nCn + K

0<K<min(n—1 ,m—n)

■y(l) =•n Z (-DKS"n-K+1'"Cn + K

0<K<,min(n,m—n)

(B)

LO) _ ç.n+1,/1 +/_i\çn,nf,'m-1 °n cm-n-l T v l)t3n cm-n

-,(1) _ oii+l,n1 m n rn—n'

We may imagine, first of all, solving (B) for the cK in terms of the 7^' \ K =

n, . . . ,777; this we may clearly accomplish by a process of back substitution. Then,

if we substitute the resulting expressions for the cK into (A), we shall be able to

write down solutions which express the 7J1), 7 = 0, . . . , 77 - 1, in terms of the y%\

k = n, . . . , m, and the Newton sums, just as in the preceding sections, but this time

avoiding the use of Vandermonde determinants. We do not need actually to carry

out this complex calculation, however, since we already know, from our previous



work, the result of it. For distinct z¡, we shall certainly find, as before, that

(8) 7(D = (-!)"-/ £ Sf„+1*y%\ 7 = 0,..., 77-1,n<K<m

but with the important difference now that the values Sn+ l ,K axe no longer defined

in terms of Vandermonde determinants, but may now be regarded as defined by the

relation (30), found in the Appendix, which states that for k > n > 1 and 1 </' < 77

s'nk= Z (-iysnn-k-i'-'sn-i''n,

0</'</-l

with the particular case

s"n'k= Z (-lys^-i'-'s"^'-",0</'<n-l

which allows us, using the Algorithm given in the Appendix, to recursively determine

the Sjj,k, hence the S¡¡k, from the Newton sums and S"'n~l = 1. Based on this sec-

ond definition of the SJ,+1 ̂ , these calculations may now equally be carried out for

the case of multiple z¡; and they will, by the continuity of 7J1^(z1, . . . , zn), yield

the same result (8). We state then

Theorem l'. Theorem 1 holds also in the case that the n given fixpoints z¡

are not necessarily distinct.

Difficulties arise, however, when we try to extend Theorems 2 and 3 to the

case of multiple fixpoints. For suppose z¡ is a fixpoint of exact multiplicity a > 1.

Then

g(z) - zh(z) = (z - zjfA(z),

for some polynomial ,4(z) such that A(z¡) ¥= 0. Since

g'(z) - zh'iz) = aiz - ajf-lA(z) + (z - z,)aA'(z) + h(z)

and

0'(X>= \g'(z)-<Kz)h'(z)]ih(z),

we conclude that if (z - z,)0-1 is 77or a factor of /t(z),

0'(zf) = lim [aiz - zAa~xAiz) + (z - zAaA\z) + A(z) - (0(z) - z)h'(z)]/h(z)

Z-+Zj

= 1,

so that z¡ cannot be an attractive, but is rather an indifferent [13], fixpoint of 0. In

this case, then, Eqs. (10) and (17) are both internally inconsistent, and we cannot ex-

pect a meaningful solution.

Let us suppose then that (z - zjf~l is a factor of h(z), and that a > 1. Then

h(z) = (z - zi)a~1B(z), for some polynomial B(z), and

0(z) = (z - z,.)«"1 [zB(z) + (z - zAA(z)]l(z - zjf-lB(z),


822 W. F. SMYTH

which, since our construction process makes no provision for cancellation of identical

factors in numerator and denominator, is of the form 0/0 at z = z,-. Furthermore,

0'(z) is of the same form 0/0 at z = z¡, and we conclude that in this case also multiple

fixpoints introduce internal inconsistencies into Eqs. (10) and (17).

It becomes clear then that for multiple fixpoints we require an improved con-

struction process which will somehow cancel the terms (z - z¡f~l to yield

Hz) = z + (z- zAA(z)lB(z) and 0'(z)lz=z. = 1 + A(zi)/B(zi),

effectively reducing the problem to the case of distinct fixpoints, and enabling us to

solve (10) and (17) as before.

It is nevertheless interesting to see what happens when the equations (10) or

(17) are applied to a particular example of multiple fixpoints. The results of such an

application are displayed in Table 3. Provided both g„(z) and h (z) axe of degree suf-

ficiently large to contain a factor (z - zjf~l, it seems that the iterating function

0p may be used to solve the given problem, but with a lower order of convergence

at multiple fixpoints z¡. The case in which (z - z,)"-1 does not divide h(z) apparent-

ly does not occur. The repeated result 0'(1) = 16. suggests a possible generalization of

the well-known fact [14, p. 53] that for the Newton-Raphson iteration, at a fixpoint

z¡ of multiplicity a,

0'(z,.) = 1 - I/o.

This typical reduction of the order of convergence at multiple fixpoints is a problem

not only for iterations of the form treated in this paper-that is, z^+1) = 0(z^)—

but also for more general forms of iteration: an analysis of the local convergence

properties of one such iterative form, using some techniques similar to those em-

ployed here and including the case of multiple roots, is presented in [15].

Remark. The method described here provides a means of constructing and

classifying rational iterating functions of specified order of convergence. It would be

valuable to extend these results so as to be able to construct efficiently an iterating

function for a given problem which in addition possesses desirable convergence char-

acteristics.

Acknowledgement. I should like to thank James L. Howland, University of

Ottawa, for introducing me to this topic and assisting me in the early stages of this

research.

Appendix

Given integers n> I, j, and k, and distinct (real or complex) values z¡, i = 1,

. . . , n, we define quantities S'¿k = SÍ¡k(zx, . . . , zn) as follows:

S'¿k = (- l)n-'D>¿k/Dn for 1 < / < n and k > 0;

= 0 otherwise,

where Dn = Dn(zx,. . . , z„) is the Vandermonde determinant of order n,


CONSTRUCTION OF RATIONAL ITERATING FUNCTIONS

o o o o o o

+1 +1 +1 +1 +1 +1

^ rN rN rN (N rN fN

+1

1

+

+w-1CO<H

(N

II

•S S

o ud. g

+

M

+

I

2+

+

I

IN

+

(N

N'•£>

n

I

5?¡t.

•â

•s.

I

++ +

i c.

+

i <N « N I

CL

5

*3> rs ü-i — se

r^i in (N \0 —

KT> *s~i \0 \C T-*-


824 W. F. SMYTH

1 zx

1 z-,

• • zn-1

• • Zn-1

1 Z„ ' • ' Zn-1

and iy^k is the determinant obtained by replacing the /th column of Dn, which we

denote (z{--1), by the new column (zf). Then, in particular, Dx = 1 and S},fc =

D\'k = z\. For 77 > 2, recall that

(21) 2)„ = rL,£>„_i,

where fl„ = ni<I<„_1(z„ - zj), from which it follows that

(22)

Note also that when k <n -

(23)

Dn= n (zn-zt).Ki<i'<n

D{¡k=Dn foxj = k+l;

= 0 otherwise.

Theorem 4. For k > n > 2,

sn*= Z &'z'n+ Z s£*-f-izi.Kj'<k-n + l 0</'<fc-«+l

Proof The result is trivially true for / < 1 and for / > n. For / = 77,

jyik

1 zr--z"xn-2 ,fe

1 zn • ■ ■ z'nn-2 zk

Subtracting the 77th row of this determinant from each of the others, and then

extracting the common factor Xln, we find that D",k is equal to (-l)"_1(-l)"_1n„

times an 77 - 1 by 77 - 1 determinant whose hth column, 1 < h < n - 2, may be rep-

resented

z zjrï-'V1<7'<7i

and whose (77 - l)th column is

Z zk-'z{^Kj'<k

Then, subtracting z„ " times the (n - 2)th column from the (77 - l)th column, and



afterwards successively subtracting zn times the (h - l)th column from the hih col-

umn h = n - 2, . . . , 2, we find

D»k = n„

Tn-21

j'k-j'-lZ'Z

0<7 Kk-n-Yln'l

'«-1 7n-2zn-l

J'k-j'-l

0<j'^k-n+l

= n„ x Dn^-'-'zi;,0</ <fc-n + l

from which, in view of (21) and the definition of 5¿,k, the result follows immediately.

Similar manipulations establish the theorem in the case 1 < / < 77.

We note some special cases of Theorem 4. For/ = 1,

(24)

and for 7 = 77,

(25)

For k = n,

(26)

whence

J„ - ¿- J„_i zn,

Kj'<k-n+l

Snnk -

0<7 «fc-n+1

vn-l,k-j-lJ°M—1 Z".

°n ¿n-l zn ^ ¿n-l

(27)

(28)

çl,n _ cl,n-1'n J«-l zn '

c",n _ nn-l,n—1 ¡¿n - ¿n-1 +

and the following theorem may easily be proved by induction.

Theorem 5. For 1 < / < 77 the quantities S'¿n(zx, . . . , z„) are the Newton

sums for a polynomial of degree n with distinct zeros z¡, i = 1, . . . , 77; that is, 5„,n

is the sum of all distinct products of the z¡ taken n - 7 + 1 at a time.*

Theorem 5 establishes then the important fact that given any polynomial f„(z)

of degree 77, we may determine corresponding quantities 5¿'" computed only from the

coefficients of fn. We go on now to show that in fact every S(¡k may be computed

from the S\¡", hence also from the coefficients of fn.

Theorem 6. For k > n > 1 and 1 < 7 < 77,

snk +Sn~Uk~l =S"n'k-iS>;n.

* Using Theorem 4, it may also be shown more generally that for 1 < /' < n and k > n,

S1' is the sum of all distinct products z" z* . . . , where max(a', b',...) = k - n + 1 and a' +a o

b + = k -j + 1.


826 W. F. SMYTH

Proof. Using the definition of S'¿k, we may easily verify this result for tí = 1.

Suppose we know that

(29) S'-_\'k + S'nZ\k-1 = S£¡ '"-'S'-1,"-1

for all k > n - 1 > 1 and 1 < / < 77. Then

Si,k+Sj-i,k-i= z s£?4+ £ s£x>k-i'-'zn'Kj'<k-n + l 0<7'<fc-n+l

+ Z ¿rVfcW'-lz„' + Z C?i^'-2z„'Kj'<k-n 0<j'<k-n

= Z is£Z + s>r-\'k-,'-lHK/'<fc-n + l

+ Z {sn^k-'-1+sn-^k-'-2}zn-0<j'<k-n

5/-1,n-2_k-n+l 1 o/-l ,n-2,k-n+ 1„_!' Z„ + ¿n-l zn

= V çn-l,k-j'-lctj,n-l f¿* ¿n-1 ¿n-1 zn

+ y cn-l,k-j'-2d-l,n-l j'+ ¿- ¿n-1 ¿n-1 zn>

0</'<fc-n

by the inductive hypothesis, if 7 =£ 77; then by (25) and (26),

s,,k + 5¿_1,k_1 = z s^i^-MWi-r1 + Sitv*"1 >0<;'<k-n

aj-aj.*-!.

if/ = n,

5n,fc+5n-l,fc-l= £ SS:1^'-V„ Z ¿rl'*-/'-2Z„'Ky'<k-n + l 0<j'<k-n

Zen— l,k-j'-l r _ 1 on—l,n—l-i _ çyi.k—1 on,n£„_,' {Z„ +S„_,' } -Sn- Sn ,

0<j'<k-n

by application of (25) and (28). This completes the proof.

An immediate consequence of Theorem 6 is

Theorem 7. For k > n > 1 and 1 < 7" < 77,

(30) s¡¡k = Z (-ly'^'*-'"-1^-'"-".0</'</-l

Theorem 7 shows that the determination of the S¡¡k may be reduced to the

determination of the Newton sums Sn'",... , S¡¡" together with the quantities

S"*fc_1, . . . , S",n + 1. But these latter quantities also may be calculated from the

Newton sums, since by rewriting (30) for the special case 7 = 77,



snnk= Z (-lys^-i'-'s^-i''",0</'<n-l

we see that we may successively set k = n + 1,77 + 2, . . . , and recursively deter-

mine all the S"'k in terms of the Newton sums and S"'n~l = 1. This is done by the

following

Algorithm. Given Newton sums S„'", . . . , S"'" stored in SN(l), . . . , SN(n),

respectively, to calculate the values S"n'"~l = 1.0, S£'M = SN(n), S"„-n+l, . . . ,

S"-n + K, and store them in SK(l), . . . , SK(K + 2).

1. SK(l) «— 1.0, SK(2) <— 5/V(t7), k «— 77 + 1.

2. /' 4- 0, SIGN *— 1.0, a <— it - n + 2, SK(a) «— 0.0.

3. 5/T(a) +- 52C(a) + SIGN*S/V(t7 -f')*SK(a -/' - 1).

4. Ifjfc -/'#/! and/'¥=77- 1, SIGN <-SIGN,/'*-/' + 1, go to 3.

5. If fc=É2í, * <— A: + 1, go to 2.

6. Exit.

84 Dupont Street

Toronto, Ontario M5R 1V2, Canada

1. GASTON JULIA, "Mémoires sur l'itération des fonctions rationnelles," /. Math. Pures

Appl., V. 1, 1918, pp. 47-245.

2. P. FATOU, "Sur les équations fonctionnelles," Bull. Soc. Math. France, v. 47, 1919, pp.

161-271.

3. PAUL MONTEL, Leçons sur les Familles Normales de Fonctions Analytiques et Leurs

Applications, Gauthier-Villars, Paris, 1927.

4. PAUL MONTEL, "L'itération," Univ. Nac. La Plata Publ. Fac. Ci. Fisicomat. Serie

Segunda Rev., v. 3, 1940, pp. 201-211.

5. H. RADSTRÖM, "On the iteration of analytic functions," Math. Scand., v. 1, 1953, pp.

85-92.

6. R. A. BROOKER, "The solution of algebraic equations on the EDSAC," Proc. Cam-

bridge Philos. Soc, v. 48, 1952, pp. 255-270.

7. HUBERT CREMER, "Über die Iteration rationaler Funktionen,"Jber. Deutsch. Math.-

Verein., v. 9, 1952, pp. 185-210.

8. H. S. HALL & S. R. KNIGHT, Higher Algebra, Macmillan, New York, 1950.

9. ERNST SCHRÖDER, "Über unendlich viele Algorithmen zur auflösung der Gleichungen,

Math. Ann., v. 23, 1870, pp. 317-365.

10. A. S. HOUSEHOLDER, "Schröder and Trudi: a historical excursion," SIAM Rev.,

v. 16, 1974, pp. 344-348.

11. A. S. HOUSEHOLDER, Principles of Numerical Analysis, McGraw-Hill, New York,

1953.

12. JAMES L. HOWLAND, Private communication, 1977.

13. L. LEAU, "Étude sur les équations fonctionnelles," Ann. Fac. Sei. Univ. Toulouse, v.

11, 1897, pp. 22, 29-35.

14. E. DURAND, Solutions Numériques des Equations Algébriques. I, Masson, Paris, 1960.

15. P. JARRATT & D. NUDDS, "The use of rational functions in the iterative solution

of equations on a digital computer," Comput. J., v. 8, 1965, pp. 62-65.


The Construction of Rational Iterating Functions

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