We study the monotone convergence of two general methods of Aitken-Steffenssen type. These methods are obtained from the Lagrange inverse interpolation polynomial of degree two, having controlled nodes. The obtained results provide information on controlling the errors at each iteration step

Ion Păvăloiu

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 35 (2006) no. 2, pp. 173–182ictp.acad.ro/jnaatBILATERAL APPROXIMATIONS OF THE ROOTS OF SCALAREQUATIONS BY LAGRANGE-AITKEN-STEFFENSEN METHODOF ORDER THREE∗ION PĂVĂLOIUAbstract. We study the monotone convergence of two general methods ofAitken-Steffenssen type. These methods are obtained from the Lagrange inverseinterpolation polynomial of degree two, having controlled nodes. The obtainedresults provide information on controlling the errors at each iteration step.MSC 2000. 65H05.Keywords. Aitken-Steffenssen methods, Lagrange inverse interpolation.1. INTRODUCTIONIt is well known that the Steffensen, Aitken, and Aitken-Steffensen methodsare obtained from the inverse Lagrange interpolation polynomial of degree one,with controlled nodes [8]–[12], [6]. Consider the equation(1.1) f(x) = 0wheref : [a, b]→ R, a, b ∈ R, a < b.We also consider the following three equations, each of them equivalent withequation (1.1):(1.2) x = g(x), g : [a, b]→ [a, b]andx = g1(x), g1 : [a, b]→ [a, b],(1.3)x = g2(x), g2 : [a, b]→ [a, b].The Steffensen method in given by relations(1.4) xn+1 = xn −f(xn)[xn, g(xn); f ], x0 ∈ [a, b], n = 0, 1, 2, . . . ,∗This work has been supported by MEdC-ANCS under grant 2CEEX-06-11-96.†“Tiberiu Popoviciu” Institute of Numerical Analysis, P.O. Box. 68-1, Cluj-Napoca,Romania, e-mail: pavaloiu@ictp.acad.ro.174 Ion Păvăloiu 2and, analogously, the Aitken method is of the following form:(1.5) xn+1 = g1(xn)−f(g1(xn))[g1(xn), g2(xn); f ], x0 ∈ [a, b], n = 0, 1, 2, . . . .Finally, the Aitken-Steffensen method is given by the relations:(1.6) xn+1 = g1(xn)−f(g1(xn))[g1(xn), g2(g1(xn)); f ], x0 ∈ [a, b], n = 0, 1, 2, . . . .The order of convergence of all three methods (1.4)–(1.6) is at least two, andthis order depends on the functions g and g1, g2 respectively. Essentially,the methods (1.4)–(1.6) are obtained from the method of chord where theinterpolation nodes depend on the functions g and respectively g1, g2. Inpapers [1], [8], [9], [12] some conditions had to be considered in order thatall the three methods (1.4)–(1.6) generate two sequences (un)n≥0 and (vn)n≥0with the properties:α) The sequence (un)n≥0 is increasing and the sequence (vn)n≥1 is decreas-ing:β) lim un = lim vn = x̄, where x̄ is the root of equation (1.1), x̄ ∈ [a, b].Practically, such sequences are very interesting, because by inequalitiesmax{x̄− un, vn − x̄} ≤ vn − un, n = 0, 1, 2, . . . ,the errors of approximation at every step of iteration may be controlled.Let a1, a2, a3 ∈ [a, b] be three nodes of interpolation, and let b1, b2, b3 thevalues of the function f , i.e.b1 = f(a1), b2 = f(a2), b3 = f(a3).Suppose that the function f : [a, b]→ F, is bijective where F = f([a, b]).Then there exists f−1 : F → [a, b] and the following equality holds [12],[14]:f−1(y) = a1 + [b1, b2; f−1](y − b1)(1.7)+[b1, b2, b3; f−1](y − b1)(y − b2)+[y, b1, b2, b3; f−1](y − b1)(y − b2)(y − b3),for every y ∈ F.If x̄ ∈ [a, b] is the root of equation (1.1), then x̄ = f−1(0), and by (1.7) oneobtains the following representation of x̄ :x̄ = a1 − [b1, b2; f−1]b1 + [b1, b2, b3; f−1]b1b2(1.8)−[0, b1, b2, b3; f−1]b1b2b3.By relations (see [12])[b1, b2; f−1] =1[a1, a2; f ]3 Lagrange-Aitken-Steffensen method 175and[b1, b2, b3; f−1] = −[a1, a2, a3; f ][a1, a2; f ][a1, a3; f ][a2, a3; f ],using (1.8), one obtains the following approximation for x̄ :(1.9) a4 = a1 −f(a1)[a1, a2; f ]− [a1, a2, a3; f ]f(a1)f(a2)[a1, a2; f ][a1, a3; f ][a2, a3; f ]and the error:(1.10) x̄− a4 = −[0, b1, b2, b3; f−1]f (a1) f(a2)f (a3) .If f ∈ C3([a, b]) and f ′(x) 6= 0, ∀x ∈ [a, b], then f−1 ∈ C3(F ) and thefollowing equality holds (see [12], [16]):(1.11) [f−1(y)]′′′ = 3[f′′(x)]2−f ′(x)f ′′′(x)[f ′(x)]5 ,where y = f (x) .Using the mean value formula for divided differences (see [12]) it followsthat there exists η ∈ F such that(1.12) [0, b1, b2, b3; f−1] = (f−1)′′′(η)6 .Because f is bijective, it follows that there exists ξ ∈ [a, b] such that η =f(ξ), and by (1.11) and (1.12) one obtains:(1.13) [0, b1, b2, b3; f−1] = 3[f′′(ξ)]2−f ′(ξ)f ′′′(ξ)6[f ′(ξ)]5 .By (1.9), if one considers particular nodes a1, a2, a3 it is possible to obtaindifferent methods of Steffensen type, of Aitken type or of Aitken-Steffensentype.Let xn ∈ [a, b] be an approximation of the root x̄ of equation (1.1).If one considers a1 = xn, a2 = g(xn), a3 = g(g(xn)), then it follows thefollowing method of Steffensen type [8], [9], [12]:xn+1 = xn −f(xn)[xn, g(xn); f ](1.14)− [xn, g(xn), g(g(xn)); f ]f(xn)f(g(xn))[xn, g(xn); f ][xn, g(g(xn)); f ][g(xn), g(g(xn)); f ],x0 ∈ [a, b], n = 0, 1, 2, . . . .If a1 = xn, a2 = g1(xn), a3 = g2(g1(xn)), then one obtains the followingmethod of Aitken-Steffensen type:xn+1 = xn −f(xn)(xn, g1(xn); f ](1.15)− [xn, g1(xn), g2(g1(xn)); f ]f(xn)f(g1(xn))[xn, g1(xn); f ][xn, g2(g1(xn)); f ][g1(xn), g2(g1(xn)); f ],176 Ion Păvăloiu 4n = 0, 1, . . . , x0 ∈ [a, b], and finally, for a1 = xn, a2 = g1(xn), a3 = g2(xn) oneobtains the following method of Aitken type:xn+1 = xn −f(xn)[xn, g1(xn); f ](1.16)− [xn, g1(xn), g2(xn); f ]f(xn)f(g1(xn))[xn, g1(xn); f ][xn, g2(xn); f ][g1(xn), g2(xn); f ],n = 0, 1, . . . , x0 ∈ [a, b].Using the symmetry of Lagrange polynomial with respect to nodes, by per-mutations of a1, a2, a3 in methods (1.14)–(1.16), one obtains the same resultsfor xn+1.In [14] conditions are given in order that method (1.14) generates sequencesapproximating the root of equation (1.1) bilaterally. In this paper we studymethods (1.15) and (1.16) and we obtain same conditions in order that thesequences generated by there methods are bilateral approximations of the rootx̄ of equations (1.1).2. THE CONVERGENCE OF AITKEN-STEFFENSEN METHODIn the following we study the method (1.15) and we search the conditionson the sequences (xn)n≥0, (g1(xn))n≥0 and (g2(g1(xn))n≥0 generated from thismethod in order that they are monotonic sequences, bilaterally approximatingthe root x̄ of equation (1.1).We need the following hypothesis:a) g1 is increasing on [a, b];b) g2 is continuous and decreasing on [a, b];c) equation (1.1) has a solution x̄ ∈ [a, b] and g1(x̄) = g2(x̄) = x̄,d) function f is in C3([a, b]), and for every x ∈ [a, b] the following relationis fulfilled:(2.1) 3[f ′′(x)]2 − f ′(x)f ′′′(x) < 0;e) function g1 satisfies inequality|g1(x)− g1(x̄)| ≤ L |x− x̄| , ∀x ∈ [a, b],where 0 < L < 1.Concerning the convergence of sequence (xn)n≥0 generated by (1.15), the fol-lowing Theorem holds:Theorem 2.1. Let x0 ∈ [a, b] and f, g1, g2 verify the following conditions:i1) f is increasing on [a, b];ii1) f is convex on [a, b];iii1) functions g1, g2 and f verify hypotheses a)− e);iv1) x0 > x̄ and g2(g1(x0)) ≥ a.5 Lagrange-Aitken-Steffensen method 177Then the sequences (xn)n≥0, (g1(xn))n≥0 and (g2(g1(xn))n≥0 generated by(1.15) verify the properties:j1) sequences (xn)n≥0 and (g1(xn))n≥0 are decreasing and bounded frombelow by x̄ ;jj1) sequence (g2(g1(xn))n≥0 is increasing and bounded from above by x̄;jjj1) at every iteration step the following inequalities hold:(2.2) xn − x̄ ≤ xn − g2(g1(xn)), n = 0, 1, . . . ;jv1) lim xn = lim g1(xn) = lim g2(g1(xn) = x̄.Proof. By hypotheses a) and x0 > x̄ it follows g1 (x0) > g1(x̄), i.e g1(x0) >x̄. Hypothesis e) implies g1(x0) < x0. From iv1) it follows g2(g1(x0) ≥ a andfrom b) and g1(x0) > x̄ one obtains g2(g1(x1)) < x̄. Now using i1) it follows:f(x0) > 0, f(g1(x0)) > 0andf(g2(g1(x0)) < 0,and by considering i1) and ii1) for n = 0 in (1.15) it follows that x1 < x0.Using (1.13), by hypotheses d) and (1.10) for a4 = x1 and b1 = f(x0), b2 =f(g1(x0)), b3 = f(g2(g1(x0)) it follows x̄−x1 < 0, i.e x1 > x̄. By hypotheses a)and x1 < x0 it follows g1(x1) < g1(x0) and then, by b) one obtains g2(g1(x1)) ≥g2(g1(x0)), and g2(g1(x1)) < x̄. Let xm, m ∈ N be an element of sequence(xn)n≥0 generated by (1.15) and suppose that xm > x̄. Then one obtains therelations:(2.3) g2(g1(xm)) < g2(g1(xm+1)) < x̄ < g1(xm+1) < xm+1 < g1(xm) < xm.Inequality xm+1 < g1(xm) follows from equality:xm+1 = xm −f(xm)[xm, g1(xm); f ]− [xm, g1(xm), g2(g1(xm)); f ]f(xm)f(g1(xm)[xm, g1(xm); f ][xm, g2(g1(xm)); f ][g1(xm), g2(g1(xm)); f ]= g1(xm)−f(g1(xm)[xm, g1(xm); f ]− [xm, g1(xm), g2(g1(xm)); f ]f(g1(xm))f(xm)[xm, g1(xm); f ][xm, g2(g1(xm)); f ][g1(xm), g2(g1(xm)); f ]and from the hypothesis of the theorem.From relations (2.3) it follows (2.2) and conclusions j1) and jj1). Conclusionjv1) is obvious. The theorem is proved. Remark 2.2. If function f is concave and decreasing on [a, b], then functionh : [a, b]→ R defined by h(x) = −f(x) in convex and increasing. The relation(2.1) is also verified for h. Consequently the sequence generated by (1.15) forfunction h, verifies all the conditions of Theorem 2.1, and then the conclusionsof this theorem hold.178 Ion Păvăloiu 6An analogous proof with that from Theorem 2.1 is valid for the following:Theorem 2.3. Let x0 ∈ [a, b] and f, g1, g2 verify the following conditions:i2) function f is increasing on [a, b];ii2) function f is concave on [a, b];iii2) functions g1, g2 and f verify the hypotheses a)− e);iv2) x0 < x̄ and g2(g1(x0) ≤ b.Then sequences (xn)n≥0, (g1(xn))n≥0 and (g2(g1(xn≥0))) generated by (1.15)have the following properties:j2) sequences (xn)n≥0 and (g1(xn))n≥0 are increasing and bounded fromabove by x̄;jj2) sequence g2(g1(xn)) is decreasing and bounded from below by x̄;jjj2) the following relations hold:(2.4) x̄− xn ≤ g2(g1(xn))− xn, n = 0, 1, 2, . . . ,jv2) lim xn = lim g1(xn) = lim g2(g1(xn)) = x̄.Remark 2.4. If function f is decreasing and convex, then h1 : [a, b] → Rdefined by h1(x) = −f(x) is increasing and concave, h1verifies all hypothesesof Theorem 2.3 and consequently all the conclusions j2)− jv2) hold.3. CONVERGENCE OF AITKEN TYPE METHODIn order to study the convergence of the sequences generated by (1.16) wemust suppose that functions f, g1 and g2 verify hypotheses a)-e) of section 2.The following theorem holds:Theorem 3.1. Let x0 ∈ [a, b] and the functions f, g1 and g2 verify theconditions:i3) f is increasing an [a, b];ii3) f is convex an [a, b];iii3) f, g1 and g2 verifies the hypotheses a)− e);iv3) x0 > x̄ and g2(x0) > a.Then sequences (xn)n≥0, (g1(xn))n≥0 and (g2(xn))n≥0 generated by (1.16)have the properties:j3) sequences (xn)n≥0 and (g1(xn))n≥0 are decreasing and bounded frombelow by x̄ ;jj3) sequence (g2(xn))n≥0 is increasing and bounded from above by x̄;jjj3) the following relations hold,xn − x̄ ≤ xn − g2(xn), n = 0, 1, . . . ;jv3) lim xn = lim g1(xn) = lim g2(xn) = x̄.7 Lagrange-Aitken-Steffensen method 179Proof. Let be xm ∈ [a, b], xm > x̄ and g2(xm) > a, where m ∈ N. By a) itfollows that g1(xm) > x̄ and by b) g2(xm) < x̄. Using e) one obtains g1(xm) <xm. By hypothesis i3) and ii3), and by the above relations and (1.16), forn = m it follows xm+1 < xm.For a1 = xm, a2 = g1(xm) and a3 = g2(xm) in (1.10), by (1.13) and hy-pothesis d) it follows that xm+1 > x̄. Observe that xm+1 in (1.16) may byexpressed in the following way:xm+1 = g1(xm)−f (g1(xm))[g1(xm), xm; f ]−− [xm, g1(xm), g2(xm); f ]f(g1(xm))f(xm)[xm, g1(xm); f ][xm, g2(xm); f ][g1(xm), g2(xm); f ]and then xm+1 < g1(xm). By relation xm+1 < xm it follows that g2(xm+1) >g2(xm). Consequently it follows that for every m ∈ N, the following relationshold:g2(xm) < g2(xm+1) < x̄ < g1(xm+1) < xm+1 < g1(xm) < xm.By these relations conclusions jjj3) and jv3) also follow. Remark 3.2. If function f is decreasing and concave, then function h(x) =−f(x), x ∈ [a, b] verifies all the hypothesis of Theorem 3.1 and consequentlythe sequence (xn)n≥1 generated by (1.16) verifies all the conclusions in Theo-rem 3.1.Analogously, the following result can be provedTheorem 3.3. Let x0 ∈ [a, b] and functions f, g1, g2 have the followingproperties:i4) f is increasing on [a, b];ii4) f is concave on [a, b];iii4) f, g1, and g2 verify hypothesis a)-e);iv4) x0 < x̄ and g2(x0) ≤ b.Then sequences (xn)n≥0, (g1(xn))n≥0 and (g2(xn))n≥0 generated by (1.16)verify the properties:j4) (xn)n≥0 and (g1(xn))n≥0 are increasing and bounded from above by x̄;jj4) sequence (g2(xn))n≥0 is decreasing and bounded from below by x̄;jjj4) the following relations hold:x̄− xn ≤ g2(xn)− xn, n = 0, 1, 2, . . . ,jv4) lim xn = lim g1(xn) = lim g2(xn) = x̄.Remark 3.4. If function f is decreasing and convex, then function h(x)= −f(x), x ∈ [a, b] is increasing and concave. It follows that sequence (xn)n≥0generated by (1.16) verifies the hypotheses and all the conclusions of Theorem3.3.180 Ion Păvăloiu 84. THE DETERMINATION OF THE AUXILIARY FUNCTIONSIn the following, for every situation concerning the monotonicity and con-vexity of function f, functions g1 and g2 can be determined such that conditionsa), b), c) and e) should be verified. In the following we thoroughly present thecase in which function f is increasing and convex.Supposing that f ′(x) > 0 for every x ∈ [a, b], one considers the functions:(4.1) g1(x) = x− f(x)f ′(b) ,and(4.2) g2(x) = x− f(x)f ′(a) .Theng′1(x) = 1−f ′(x)f ′(b) ≥ 0for every x ∈ [a, b] and, consequently g1 is increasing. Analogouslyg′2(x) = 1−f ′(x)f ′(a) ≤ 0for every x ∈ [a, b] and then g2 is a decreasing function. It follows that g1and g2 verify hypotheses a) and b). The hypothesis e) is also verified becauseg′1(x) < 1, for every x ∈ [a, b]. In iv1) the inequality g2(g1(x0) ≥ a holds.Because g1(x0) < x0 and x0 > x̄, the above inequality is verified if g2(x0) ≥ a.This follows, because g2(g1(x0)) > g2(x0) (g2 is a decreasing functions). Thismeans that the following relation must hold:x0 − f(x0)f ′(a) ≥ ai.e.x0 ≥ a+ f(x0))f ′(a) .Because f(x0) > 0 and f ′(a) > 0, for the veridicity of last inequality it issufficient thata+ f(x0)f ′(a) ≤ x̄,where x̄ is the root of equation (1.1). This last inequality may be realized ifx0 is sufficiently close to x̄.Consequently, the hypotheses of theorems 2.1 and 3.1 are realized for g1and g2 considered above. The other cases may be similarly analyzed.5. ORDER OF CONVERGENCEIn the following we prove that every method (1.15) and (1.16) have the orderof convergence three. The order of convergence for method (1.14) was treatedin [14], and this order is at least three. Assume the following hypotheses:9 Lagrange-Aitken-Steffensen method 181α) function g2 verifies relation|g2(x)− g2(x̄)| ≤ p |x− x̄| ,for every x ∈ [a, b], where p > 0, p ∈ R;β) m ≤ |f ′(x)| ≤ M, for every x ∈ [a, b], where m > 0 and M > 0 are realnumbers.γ)∣∣3[f ′(x)]2 − f ′(x)f ′′′(x)∣∣ ≤ q, for every x ∈ [a, b], where q > 0, q ∈ R .In hypotheses α), β), γ) and e), by (1.10) and (1.13), for sequence (xn)n≥0generated by (1.15), one obtains:|x̄− xn+1| ≤ qM3L2p6m5 |x̄− xn|3 , n = 0, 1, 2, . . .and this means that the order of convergence for method (1.15) is at leastthree.With the same hypotheses, for tree sequence (xn)n≥0 generated by (1.16) itfollows:|x̄− xn+1| ≤ qM3Lp6m5 |x̄− xn|3 , n = 0, 1, 2, . . .i.e. the order of convergence of (1.16) is also at least three.REFERENCES[1] Balázs, M., A bilateral approximating method for finding the real roots of real equations,Rev. Anal. Numér. Théor. 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Bilateral approximations of the roots of scalar equations by Lagrange-Aitken-Steffensen method of order three

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Bilateral approximations of the roots of scalar equations by Lagrange-Aitken-Steffensen method of order three

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