The Himalayan Vol. 6 & 7, April 2017 (112-114) ISSN 2542-2545 Interval and Speed of Convergence on Iterative Methods

Iswarmani Adhikari Department of Mathematics, Prithvi Narayan Campus, Pokhara Email: [email protected] Abstract : The iterative method is a tool of solving the non-linear equations to get their approximate roots with some errors of tolerance. Repetition of the similar process is applied successively on such iterations to compute a of increasingly accurate estimates of the roots. In this paper, the construction of an iterative method for solving an equation, its convergence and the determination of interval of convergence for the approximate choice of initial guess and the speed of convergence are highlighted. Keywords: Iteration, convergence, convergence interval, speed of convergence

1. Introduction Let x0 be the initial guess. Let the tangent at p(x0, f(x0)) meets the x-axis as x , then it becomes the better approximation of The rate at which a convergent sequence approaches its 1 the root than x . is the speed of convergence. The set of all values of 0 Slope of the tangent line = x for which the sequence converges gives the interval of \ f'(x ) = ------(3) convergence. A power series often converge or diverge, 0 Then, (2) becomes depending upon the values of x. If converges, the nature of + = convergence is also different. or, x - x = ax 2 - x Finding one or more roots of an equation f(x) = 0 ------0 1 0 0 or, x = 2x - ax 2. --- (1) is one of the most commonly occurring problems of 1 0 0 \ x = x (2-ax ) ------(4) , specially on the numerical methods. 1 0 0 Equivalent to say, x =x (2-ax ) ------(5) for For such equations, the explicit solutions are not available n+1 n n n≥0, for the iterative methods. and their better approximations are obtained by different Let r = 1 - ax ------(6) then x = x (1+r ) iterative methods. An iterative method is a powerful device n n n+1 n n where the error be of solving and finding the roots of the non-linear equations. e =- x = = Iteration means the act of repeating the similar process to n n \ e = ------(7) achieve the desired goal where the result of one iteration is n From (6), we can write used as the starting point for the next operation. Successive r = 1 - ax approximations are used in each step to get more accurate n+1 n+1 = 1 - ax (1+r ), from (5) solutions to a linear system and it involves a large number of n n = 1-(1-r ) (1+r ), from (6) iterations of arithmetic operations. n n 2 = 1 - 1 + rn 2 \ rn+1= rn ------(8) 2. Convergence of xn Here, Consider the equation be f(x) = a-= 0 ------(2) for given a >0. 2 n = 0 Þ r1 = r0

2 2 2 2 2 n = 1 Þ r2 = r1 = (r0 ) = r0

3 2 4 2 2 n = 2 Þ r3 = r2 = (r0 ) = r0 , and so on.

2n \ rn = r0 ------(9) for n ³ 0. Thus from (9), we can say,

rn ® 0 iff |r0| < 1

iff -1 < r0 < 1 112 Interval and Speed of Convergence on Iterative Methods

n+1 iff -1 < 1 - ax0 < 1, from (6) −2 | xn+1 | 10 lim a n iff -2 < -ax0 < 0 n→∞ lim −2 .a | xn | = n→∞ 10 iff 0 < ax0 < 2 \ we get the convergence for α = 2. iff 0 < x0 < ------(10) for a > 0. Thus for the terms of the sequence be like 10-2, Hence, for the convergence of xn to , it is necessary 10-4, 10-8, 10-16, 10-32, 10-64, … which shows the very fast and sufficient that x0 be chosen in between 0 and ; which tells us the interval of convergence on different iterative convergence of order 2, quadratically convergent. methods. Relative error Speed of Convergence The magnitude of the difference between the exact value The speed of convergence of a sequence can be determined and the approximation is absolute error where the relative as follows: error is simply the ratio of absolute error to the exact value which can also be expressed in terms of percentage. Let the sequence of real numbers be x0, x1, x2, … converges to α. The speed of convergence is the rate at To examine the rate of decrease on the relative error, let us which the numbers are converging to α. examine the speed of convergence from (6). For this we write the above mentioned result be,, If, lim = λ < ∞, then the sequence converges n→∞ e = = = = ae 2 linearly to α where the constant λ is called the asymptotic n+1 n or, = e 2a2 = error. n 2 Þ Rel(xn+1) = [Rei(xn)] , for n ³ 0, where Rel(xn) denotes the | xn+1 −a | If, lim = λ < ∞, relative error in x . n→∞ | x −a |2 n n So, to know how fast the relative error will decrease, let Rel then the sequence converges quadratically to α. (x0) = 0.1,where, 2 -2 Rel(x1) = [Rel(x0)] = 0.01 = 10 Rel(x ) = [Rel(x )]2 = 0.0001 = 10-4 | xn+1 −a | 2 1 Thus, if lim = λ < ∞, 2 -8 n→∞ p Rel(x ) = [Rel(x )] = 10 | xn −a | 3 2 Rel(x ) = [Rel(x )]2 = 10-16 … and so on. α 4 3 then the sequence converges to of order p. Here, the relative errors in each iterations double the Examples: number of significant digits, which provides the nature of the rate of its decrease in the relative errors. i) Let xn = for some k > 0. Clearly, it converges to 0, where the rate of 3. Conclusion convergence be given by 1 An equation can be solved with the better approximations k by the iterative approach with some reasonable errors. For (n +1) a k | xn+1 | lim lim  n  the solution procedure, iterative method is a better tool lim = n→∞ = n→∞   n→∞ a 1   | xn | a  n +1 for different non-linear equations to get their approximate n k We get the convergence for α = 1, but not for α = 2. solution where the repetition of the similar process is carried Hence, the above sequence converges linearly to 0. out successively to compute a sequence of increasingly accurate estimation of their solutions. In the convergence In fact, xn = , xn = , xn = , … all converge linearly to zero. analysis of the solution of non-linear equations, the initial guess or the initial approximation within the interval of −2n ii) Let xn = 10 , which also converges to 0. convergence and the speed of convergence play the vital For the rate of convergence, role .

113 The Himalayan Physics Vol. 6 & 7, April 2017 Iswarmani Adhikari References 1. Atkinson, Kendall E. (2008). An Introduction to . John Wiley and Sons. Inc. U.K. 2. Brain Bradie (2007). A Friendly Introduction to Numerical Analysis. Prentice-Hall, Eaglewood Cliffs, N.J. 3. Faires, J. Dougles and Burden, Richard L. (2002). Numerical Methods. Brooks Cole, Canada. 4. Goldstine, H. (1977). A History of Numerical Analysis. Springer-verlang, New York. 5. Scheid, Francis (1968). Theory and Problems of Numerical Analysis. McGraw-Hill, New York. 6. Süli, Endre, and Mayers, David F. (2003). An Introduction to Numerical Analysis. Cambridge University Press, New York.

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