It is well known that the Steffensen and Aitken-Steffensen type methods are obtained from the chord method, using controlled nodes. The chord method is an interpolatory method, with two distinct nodes. Using this remark, the Steffensen and Aitken-Steffensen methods have been generalized using interpolatory methods obtained from the inverse interpolation polynomial of Lagrange or Hermite type. In this paper we study the convergence and efficiency of some Steffensen type methods which are obtained from the inverse interpolatory polynomial of Hermite type with two controlled nodes

Ion Păvăloiu

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 35 (2006) no. 1, pp. 87–94ictp.acad.ro/jnaatON A STEFFENSEN-HERMITE TYPE METHODFOR APPROXIMATING THE SOLUTIONS OF NONLINEAR EQUATIONS∗ION PĂVĂLOIU†Abstract. It is well known that the Steffensen and Aitken-Steffensen type methodsare obtained from the chord method, using controlled nodes. The chord method is aninterpolatory method, with two distinct nodes. Using this remark, the Steffensen andAitken-Steffensen methods have been generalized using interpolatory methods obtainedfrom the inverse interpolation polynomial of Lagrange or Hermite type. In this paperwe study the convergence and efficiency of some Steffensen type methods which areobtained from the inverse interpolatory polynomial of Hermite type with two controllednodes.MSC 2000. 65H05.Keywords. Nonlinear equations in R, Steffensen and Aitken-Steffensen methods, in-verse interpolatory polynomial of Hermite type.1. INTRODUCTIONThe most well-known methods for approximating the solutions of nonlinear equa-tions are of interpolatory type.The Newton type methods and, more generally, the Chebyshev type methods areobtained from the inverse interpolatory polynomial of Taylor type, with one node.The chord method and its generalizations are obtained from the inverse interpolationpolynomial of Lagrange, or more generally, from the inverse interpolation polynomialof Hermite [4], [7], [10]. The Steffensen and Aitken-Steffensen methods are also ofinterpolatory type, but their nodes are controlled at each step [6]–[9]. For the methodsof interpolatory type it is necessary to compute the values of the derivative of theinverse function at different points in R.Let f : [c, d] −→ R be a given function, where c, d ∈ R, c < d, and denote byF = f([c, d]) the set of the values of f on [c, d]. Assume that f is one-to-one andtherefore there exists f−1 : F −→ [c, d].Consider the equation(1) f(x) = 0.If equation (1) has a solution x̄ ∈ [c, d] then obviouslyx̄ = f−1(0).∗This work has been supported by the Romanian Academy under grant GAR 14/2005.†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail:pavaloiu@ictp.acad.ro, web: www.ictp.acad.ro/ p̃avaloiu.88 Ion Păvăloiu 2It is therefore natural to seek methods of approximation of the values of f−1 aty = 0, in order to determine different approximations to x̄.Several methods of approximation for f−1(0) have been studied in papers such as[2]–[5], [7], [8], [10].In the following we shall consider a method of Hermite type with two interpolationnodes. Such a method has been studied in [2], [3], [8], etc.Let p, q ∈ N, p, q ≥ 1, and x1, x2 ∈ [c, d]. Denote by(2) H(y) = H(y1, p; y2, q; f−1 | y)the Hermite polynomial associated to the inverse function f−1 with the multiplenodes: y1 of order p, and y2 of order q, where yi = f(xi), i = 1, 2. In order thatpolynomial (2) exists, it suffices that function f−1 to be differentiable up to the orderp+ q in F.In this sense the following theorem holds [4], [11].Theorem 1. If f satisfies the following conditions:i1. f : [c, d] −→ F is one-to-one;ii1. f is differentiable up to the order n ∈ N at each point x ∈ [c, d];iii1. f ′(x) 6= 0 for all x ∈ [c, d],then the inverse function f−1 is differentiable up to the order n at each point y ∈ F ,and the following relations hold:(3) [f−1(y)](k) =∑ (2k−2−i1)(−1)k−1+i1i2!i3!...ik![f ′(x)]2k−1(f ′(x)1!)i1.(f ′′(x)2!)i2. . .(f(k)(x)k!)ik,where y = f(x), and the above sum extends over all integer solutions of the system(4) i2 + 2i3 + · · ·+ (k − 1)ik = k − 1i1 + i2 + i3 + · · ·+ ik = k − 1.Taking into account the above relations, under the hypothesis that f admits deriva-tives up to the order p+ q on [c, d], f ′(x) 6= 0,∀x ∈ [c, d], and using the remainder ofthe Hermite interpolation, it follows:(5) f−1(y) = H(y) + [(f−1(η)](p+q)(p+q)! (y − y1)p(y − y2)qfor some η ∈ int(F ).Setting y = 0 in (5), then we obtain for x̄ the following relation:(6) x̄ = f−1(0) = H(0) + [f−1(ξ)](p+q)(p+q)! (−1)p+qyp1yq2.Denote by x3 the next approximation for x̄, obtained from (6)(7) x3 = H(y1, p; y2, q; f−1|0).If we assume that the derivative of order p+ q of the inverse function is bounded,i.e., there exists M ∈ R,M > 0, such that(8)∣∣∣[f−1(y)](p+q)∣∣∣ ≤M, for all y ∈ F,then by (2), (6) and (7) we deduce:|x̄− x3| ≤ M(p+q)! |f(x1)|p |f(x2)|q .3 On a Steffensen-Hermite type method 89In general, if xn, xn+1 ∈ [c, d] are two approximations for x̄, then(9) xn+2 = H(yn, p; yn+1, q; f−1 | 0), n = 1, 2, . . . ,where yn = f(xn) while yn+1 = f(xn+1) is a new approximation for x̄, and obviously(10) |x̄− xn+2| ≤ M(p+q)! |f(xn)|p |f(xn+1|q , n = 1, 2, . . . .Taking into account all the above relations one can generate the sequence of ap-proximations (xn)n≥1, under the assumption that at each iteration step, the newapproximation xn+2 obtained by (9) from xn and xn+1, lies in [c, d].It can be easily seen that the convergence order of method (9) is given by thepositive root of equation [1]–[4], [8], [10]:t2 − qt− p = 0i.e.,(11) ω = q+√q2+4p2 .If we assume now that the Fréchet derivative f ′ of f satisfies(12) supx∈[c,d]|f ′(x)| ≤ a, a ∈ R, a > 0,then (10) becomes(13) |x̄− xn+2| ≤ Map+q(p+q)! |x̄− xn+1|q |x̄− xn|p , n = 1, 2, . . . .In the following we shall denote ρi, i = 1, 2, . . . , the expressions(14) ρi = a[Ma(p+q)!] 1p+q−1 |x̄− xi| .By (13) and (14) we obtainρn+2 ≤ ρqn+1ρpn, n = 1, 2, . . . .If we assume now that ρ1 obeys(15) ρ1 < 1and, moreover, ρ2 ≤ ρω1 , with ω given by (11), we easily deduce that for all n ∈ N, n ≥2 we haveρn ≤ ρωn−11 , n = 2, 3, . . .andlim ρn = 0which implieslimn→∞xn = x̄.Theorem 2. If the following conditions hold:i2. function f obeys the assumptions of Theorem 1 for n = p+ q;ii2. equation (1) has a root x̄ ∈]c, d[;iii2. [f−1(y)](p+q) verifies (8);iv2. f ′ (x) verifies (12);v2. ρ1 given by (14) verifies (15) and ρ2≤ρω1 , where ω is given by (11);vi2. the sequence (xn)n≥1 generated by (9) remains in [c, d],then the following relations hold:90 Ion Păvăloiu 4j2. |x̄− xn| ≤ a−1[ Ma(p+q)! ]− 1p+q−1 ρωn−11 , n = 1, 2, . . . ;jj2. lim xn = x̄.Remark 1.1. The uniqueness of the root x̄ of equation (1) is ensured by hypothesisiii1 from Theorem 1. We consider in the following, besides (1), an equivalent equation, of the form(16) x− ϕ(x) = 0whereϕ : [c, d] −→ [c, d], ϕ(x̄) = x̄.We shall show in the following section that if instead of method (9) we considerthe iterative method(17) xn+1 = H(f(xn), p; f(ϕ(xn)), q; f−1 | 0), n = 1, 2, . . . , x1 ∈ [c, d],then the r-convergence order of this resulted method, which we call of Steffensen-Hermite type, is at least p+ q, i.e., substantially higher than ω, given by (11).We shall also determine the values of p and q for which the methods is the class(17) are optimal with respect to the efficiency index [4].In section 3 we shall study the convergence of the optimal methods determined inSection 2.2. THE CONVERGENCE AND THE EFFICIENCY OF THE STEFFENSEN-HERMITETYPE METHODSWe shall assume that the function ϕ from (16) is Lipschitz, i.e., there exists b ∈ R,b > 0, such that(18) |ϕ(x)− ϕ(y)| ≤ b |x− y| , x, y ∈ [c, d].For the approximation of the solution x̄ of (1), consider the sequence (xm)m≥1,generated by (17).Concerning the convergence of this method the following result holds.Theorem 3. If the functions f and ϕ obey the following conditions:i3. function f obeys conditions i2–iv2 from Theorem 2;ii3. the approximation x1 ∈ [c, d] verifies(19) δ1 = a[Mabq(p+q)!] 1p+q−1 |x̄− x1| < 1,where b is the Lipschitz constant from relation (18) and 0 < b < 1;iii3. the elements of the sequence (xn)n≥1 generated by (17) remain in [c, d],then the following relations hold:j3. |x̄− xn| ≤ a−1[ Mabq(p+q)! ]− 1p+q−1 δ(p+q)(n−1)1 ;jj3. lim xn = x̄.Proof. The proof can be done along the same lines as in Theorem 2. From the inequality given in j3 one can see that the r-convergence order of method(16) is p+ q.5 On a Steffensen-Hermite type method 91The following function evaluations are required by method (17) at each iterationstep n in order to determine xn+1, under the hypothesis that function ϕ from (16) isexpressed with the aid of function f and its derivatives up to the order r = min{p, q}:f(xn), f ′(xn), . . . , f (p−1)(xn),f(ϕ(xn), f ′(ϕ(xn)), . . . , f (q−1)(ϕ(xn)),i.e., p+ q function evaluations.The values of the successive derivatives of f−1 at the points yn = f(xn) andȳn = f(ϕ(xn)) respectively, are determined by (3).We shall admit that the number of function evaluations which must be performedfrom iteration step n to n+ 1 is proportional to p+ q, with a factor α ∈ R, α > 0. Inthis case, the efficiency index (see, e.g., [4], [10]) of method (17) is given by relationI(p, q) = (p+ q)1α(p+q) .An elementary study on the function g : (0,+∞) −→ (0,+∞) given byg(t) = t 1αt ,show that this function attains its maximum at t̄ = e, and also that g is increasingon (0, e) and decreasing on (e,+∞), i.e. t̄ is a unique stationary point.From here it follows thatI3(p, q) = [ 3√3]1αwhile for p+ q = 2I2(p, q) = [ 2√2] 1α .It is clear that I3(p, q) > I2(p, q) and therefore I(p, q) has the maximum value forp+ q = 3.We shall point out and study the optimal methods in the following section.3. THE CONVERGENCE OF THE OPTIMAL METHODSWe shall determine in the beginning the methods which correspond to the twocases which may be considered for p+ q = 3, i.e., p = 1, q = 2 or p = 2, q = 1.In the first case (p = 1, q = 2) we want to determine a polynomial H of the form:H(y) = a0 + a1y + a2y2with the conditions:H(y1) = x1, x1 ∈ [c, d],(20)H(y2) = ϕ(x1),H ′(y2) = 1f ′(ϕ(x1)) = [f−1(y2)]′,where y1 = f(x1) and y2 = f(ϕ(x1)).Conditions (20) lead to a system of 3 equations with the unknowns a0, a1, a2. Forour purpose, of approximating the solution of (1), we are interested only in the valueH(0) = a0. Denoting by [ϕ(x1), ϕ(x1); f ] = f ′(ϕ(x1)), then for a0 we obtain thefollowing two equivalent expressions:a0 = ϕ(x1)− f(ϕ(x1))[ϕ(x1),ϕ(x2);f ] −[x1,ϕ(x1),ϕ(x1);f ][ϕ(x1),ϕ(x1);f ][x1,ϕ(x1);f ]2 f2(ϕ(x1))= x1 − f(x1)[x1,ϕ(x1);f ] −[x1,ϕ(x1),ϕ(x1);f ][ϕ(x1),ϕ(x1);f ][x1,ϕ(x1);f ]2 f(x1)f(ϕ(x1)).92 Ion Păvăloiu 6The above expressions for a0 lead us to the sequences (xn)n≥1 and (ϕ(xn))n≥1,generated by(21) xn+1 = xn− f(xn)[xn,ϕ(xn);f ]−[xn,ϕ(xn),ϕ(xn);f ][ϕ(xn),ϕ(xn);f ][xn,ϕ(xn);f ]2 f(xn)f (ϕ(xn)) , n = 1, 2, . . .or equivalently(22)xn+1 = ϕ(xn)− f(ϕ(xn))[ϕ(xn),ϕ(xn);f ] −[xn,ϕ(xn),ϕ(xn);f ][ϕ(xn),ϕ(xn);f ][xn,ϕ(xn);f ]2 f2(ϕ(xn)), n = 1, 2, . . . .In the second case, p = 2, q = 1, we obtain in our analogous fashion, the followingexpressions for (xn)n≥0 and (ϕ(xn))n≥0(23) xn+1 = xn − f(xn)[xn,xn;f ] −[xn,xn,ϕ(xn);f ][xn,xn;f ][xn,ϕ(xn);f ]2 f2(xn)n = 1, 2, . . . , or, equivalently,(24) xn+1 = ϕ(xn)− f(ϕ(xn))[xn,ϕ(xn);f ]−[xn,xn,ϕ(xn);f ][xn,xn;f ][xn,ϕ(xn);f ]2 f(xn)f (ϕ(xn)) , n = 1, 2, . . . .In order to study the convergence of methods (21) or (23) we need to determinethe third order derivative of f−1.By (3) for k = 3, we get:[f−1(y)]′′′ = 3[f′′(x)]2−f ′′′(x)f ′(x)[f ′(x)]5 , y = f(x).Regarding methods (21) or (22) the following result holds.Theorem 4. If the initial approximation x1 and functions f and ϕ obeyi4. equation (1) has a root x̄ ∈]c, d[;ii4. function f is derivable up to the order 3 on [c, d];iii4. f ′(x) 6= 0, ∀x ∈ [c, d];iv4. there exists r > 0 such that ∆ = [x̄− r, x̄+ r] ⊆ [c, d];v4. a = maxx∈∆ |f ′(x)| ;vi4. function ϕ verifies (18), where 0 < b < 1;vii4. ρ1 = ab[ma6 ]1/2 | x̄− x1 |< 1, wherem = maxx∈∆∣∣ 3[f ′′(x)]2−f ′(x)f ′′′(x)[f ′(x)]5∣∣,then the elements of the sequence (xn)n≥0 generated by (21) remain in ∆ and, more-over, the following relations hold:j4) lim xn = x̄;jj4) |x̄− xn+1| ≤ 1ab√6maρ3n1 , n = 1, 2, . . . , i.e., the sequence (xn)n≥1 has the r-convergence order 3.Proof. We assume first that hypothesis iii4 implies that function f : [c, d] −→ Fadmits an inverse f−1 : F −→ [c, d] and also that the root x̄ of equation (1) is uniquein the interval [c, d]. Conditions ii4 and iii4 ensure that function f−1 is derivable upto the order 3 on F.By (18) and hypothesis vi4 we have that for any x1 ∈ ∆, ϕ(x1) ∈ ∆.The element x2 is given byx2 = H(f(x1), 1; f(ϕ(x1)), 2; f−1 | 0)and by (5) we have:x̄ = x2 + (−1)3[0, f(x1), f(ϕ(x1)), f(ϕ(x1)); f−1]f(x1)f2(ϕ(x1)).7 On a Steffensen-Hermite type method 93From this relation, taking into account the mean value formula for divided differ-ences, and also hypotheses v4–vii4, we get that(25) |x̄− x2| ≤ ma3b26 |x̄− x1|3.Relation ρ1 < 1 and condition (25) imply that |x̄− x2| < |x̄− x1| , i.e., x2 ∈ ∆.In general, assuming that an element xk of the sequence (xn)n≥1 belongs to theinterval ∆, obviously ϕ(xn) ∈ ∆, and, similarly as above, we can prove that|xn+1 − x̄| ≤ Ma3b26 |x̄− xn|3, k = 1, 2, . . . ,The above relations imply jj4. and then obviously j4. Regarding the sequence generated by (23), resp. (24), the following result holds:Theorem 5. If the initial approximation x1, and the function f, ϕ obeyi5. equation (1) has a root x̄ ∈]c, d[;ii5 function f admits derivatives of orders up to 3 on [c, d];iii5 f ′ (x) 6= 0 for all x ∈ [c, d];iv5 there exists r > 0 such that ∆ = [x̄− r, x̄+ r] ⊆ [c, d];v5. a = max{|f ′(x)| : x ∈ ∆};vi5 function ϕ verifies (18) with 0 < b < 1;vii5 ρ1 = a√mab6 |x̄− x1| < 1, wherem = maxx∈∆∣∣ 3[f ′′(x)]2−f ′(x)f ′′′(x)[f ′(x)]5∣∣,then the sequence (xn)n≥1 generated by (23) is convergent, and the following relationshold:j5. lim xn = x̄;jj5. |x̄− xn+1| ≤ 1a√6mab q3n1 , n = 1, 2, . . . .Proof. The proof of this theorem can be done in an analogous fashion, as forTheorem 4. REFERENCES[1] Costabile, F., Gualtieri, I.M. and Luceri, R., A new iterative method for the computationof the solution of nonlinear equations, Numer. Algorithms, 28, pp. 87–100, 2001.[2] Frontini, M., Hermite interpolation and a new iterative method for the computation of theroots of non-linear equations, Calcolo, 40, pp. 109–119, 2003.[3] Grau, M. An improvement to the computing of nonlinear equation solutions, Numer. Algo-rithms, 34, pp. 1–12, 2003.[4] Ostrowski, A., Solution of Equations in Euclidean and Banach Spaces, Academic Press, NewYork and London, 1973.[5] Păvăloiu, I., Optimal efficiency index for iterative methods of interpolatory type, ComputerScience Journal of Moldova, 1, 5, pp. 20–43, 1997.[6] Păvăloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonicwequences, Calcolo, 32, 1–2, pp. 69–82, 1995.94 Ion Păvăloiu 8[7] Păvăloiu, I., Optimal problems concerning interpolation methods of solution of equations,Publications de L’Institut Mathematique, 52 (66), pp. 113–126, 1992.[8] Păvăloiu, I., Optimal effiency index of a class of Hermite iterative methods, with two steps,Rev. Anal. Numér. Théor. Approx., 29, 1, pp. 83–89, 2000.[9] Păvăloiu, I., Local convergence of general Steffensen type methods, Rev. Anal. Numér. Théor.Approx., 33, 1, pp. 79–86, 2004.[10] Traub, J.F., Iterative Methods for Solutions of Equations, Prentice-Hall Inc., Englewood Cliffs,New Jersey, 1964.[11] Turowicz, B.A., Sur les derivées d’ordre superieur d’une function inverse, Ann. Polon. Math.,8, pp. 265–269, 1960.Received by the editors: March 11, 2006.

On a Steffensen-Hermite type method for approximating the solutions of nonlinear equations

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On a Steffensen-Hermite type method for approximating the solutions of nonlinear equations

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