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A rational iteration function for solving equationsBy P. Jarratt*

A new rational iteration function is given which is shown to enjoy high-order convergenceTheoretical comparisons of computational efficiency are made with three other methods, andsome rules to assist in the selection of an iteration function are derived.

1. IntroductionThe application of rational approximations to theproblem of computing solutions of equations of theformhas been investigated by a number of authors includingOstrowski (1960), Tornheim (1964), Traub (1964) andJar ratt and Nudd s (1965). In particular, Ostrowskiproposed a functional iteration in which a linear fraction

x abx + c (1.2)is fitted to f(x) at three points, two of which are co-inciden t. Thus a step in the iteration consists ofmatching / and y at the points xn_, and xn and / ' andy' at xn only, the next approximation being given by thezero of (1.2), i.e. by xn+l a. Ostrowski showed thatthe order of the process so obtained is 2-414 and aseach step requires the evaluation o f / a n d / ', it thereforecompares favourably with the second-order Newton-Rap hso n meth od. Howev er, since the derivative f'n_ iis always available once an iteration has been started,it seems natural to try to obtain more rapid convergence

= x xn, we construct the rational functioni> ay = (2.1

such that y and / and y' and / ' respectively coincide athe points 0 and 8 /

The next approximation is obtained from the zero o(2.1) at a. and hence, solving (2.2) for a, we l(/n ~ /n - l ) + 4>n- \jnf'n- l]

^ + ' ~ n-which gives us

fnM'n) + VJn- fn- .)

2 /n /n _ ,(/ - / _ , ) - (X n -X n_ {){f2nf'n- , + / - \f'n) (2.3

by approximating/^) by a rational function of the formx abx2 +cx (1.3)

where the parameters a, b, c an d d are determined bythe conditions that / and y, f an d y' coincide at both* _ i an d xn. In this paper we first examine propertiesof the iteration based on (1.3), and the usefulness of themethod is then assessed by some theoretical comparisonswith other iteration functions.2. The basic methodIn deriving the actual formula for the iteration (1.3),we first shift our origin to the point xn so as to preservenum erical accuracy. Thu s, with new co-ordinates* Bradford Institute of Technology, Bradford, Yorks.

The advantage of this incremental form lies in the facthat the second term on the right-hand side of (2.3) canbe regarded as a correction term to xn. Thus, althoughas we approach the root this term will be computed ona digital computer with relatively few significant figuresnevertheless there will generally be sufficient accuracy tocalculate the root to nearly full precision.3. Convergence of the method

In order to examine the convergence of the iterativprocess defined by (2.3), we revert to the original coordinates and consider the set of equations

2*,/,) + + CX7+7*/i +])+4a

r; xn+ \= * / >

I =\n,

(3 .n

1

304

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Rational iteration function(3.1) is a set of five equations in the four unknownsa, b, c, d and hence for consistency the determinantalcondition

X/i+1xnXn-i11

1

00

0 0r

0= 0 (3.2)

must be satisfied, where F is the arrayx2f x f

xn-lfn-l *n-l /n- lX2 f 4 - 2X f X f A- fVlJa-lTiVl/n-l *n-l/n-

/--IJn

Next, by assuming a root of (1.1) at x = 6 and definingthe error en in the nth approximation by xn = en + 6, wecan substitute in (3.2) and use the elementary propertiesof determinants to obtain

1100

(3.3)

where A is the array-?/-1f 2 n /nL e n _?/ ;_ 1 +2 e n _ ,/ n _ 1

en-lfn-lenf'n +fn

fn/-I

(3.4)(3.3) can be further developed by using the Taylor

CO

expansions of / and / ' about 9. We have / = crer,f = rce'-1 where c r = fM(d)/r\ and c0 = f(ff) = 0,r = land substituting in (3.4) results in

A =

Zi -Penr = looZJ c r e n 1r = l

r = l

r = l

r = l

r = l

r = l(r

(3-5)Using (3.5) we may now develop the numerator anddenominator of (3.3) as infinite series of determinants.Thus, assuming that the root at 6 is simple so thatc, =^ 0an d writing ?,_, = max flej.le,,-!!} we obtain,

by expanding the non-zero determinants of lowest o rder,the relations

. . - . A1

and10

Combining (3.6) and (3.7) we have

where K, the asymptotic error constant, is

2c,c2c3 + ci)^e2_,(en - en_,) 4X [ l + O , _, ) ] (3.6)

(3.7)

(3.8)

From (3.8) it can be seen that mono tonic convergencewill be guaranteed provided the initial estimates, x0 andJCI, are chosen sufficiently close to the ro ot. The limitingdifference equation obtained by neglecting O(e nr n_! )can be linearized by taking logarithms of both sides andits solution shown to be

log = .4(1 + V3)n + B(l - V3)" - (3.9)where A and B are constants depending on the initialvalues x0 and x{. The second term on the right-handside of (3.9) tends to zero for large n and we can henceshow that asymptotically en+i = Kuvie ln+V 3. Theorder of the process defined by (2.3) is therefore1 + V 3 = 2-732.

For the analysis of the convergence in the presence ofa double root we may set c, = 0 and, with c 2 = 0expand (3.3) as before. In this case we obtain the results. . - . A

1100

(3.10)

+ _ ,)(en - en_ ,) 4X [l +O fo ., , . ,) ] , (3.11)and from (3.10) and (3.11) we have

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Rational iteration functionone extra significant figure approximately every threeiterations.

The behaviour of (2.3) in the presence of higher-orderroots has been investigated but no analytic results werefound. However, numerical evidence suggests that weagain obtain geometric convergence in which th easymptotic error constant tends to 1 as the order ofthe root increases.The method described may be readily generalized toone of fitting a rational function t o / a n d / ' at the latest

m points, and in this case we should havex a

boxm b2m_m - 2By proceeding as before it can readily be shown thatfor simple roots, the limiting difference equation des-cribing the behaviour of the errors is of the formen+1 = A

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Table 1Best methods for 0 < dx < 1

Rational iteration function(0

R A N G E

0-237