A New Approach to Find Roots of Nonlinear Equations by Hybrid Algorithm to Bisection and Newton-Raphson Algorithms

Transcription

1 A New Approach to Find Roots of Nonlinear Equations by Hybrid Algorithm to Bisection and Newton-Raphson Algorithms Abed Ali H. Altaee 1, Haider K. Hoomod 2, Khalid Ali Hussein 3 1(Department of Mathematics/ Al-Mustansiriyah University/ Baghdad Iraq) 2,3(Department of computer/al-mustansiriyah University/ Baghdad Iraq ) kah682001@yahoo.com, drhaider @yahoo.com Abstract The root-finding problem is one of the most important computational problems and applications. In this paper we introduced a new algorithm to finding real roots of single nonlinear equations by hybrid between the Bisection algorithm and Newton-Raphson algorithm with described and comparison between them. Keyword: Numerical Algorithms, bisection Method, Newton-Raphson Method, Hybrid Algorithm, finding roots. جذور إلياد نهج جديد المعد الت غير الخطية من قبل خوارزمية التنصيف ونيوتن رافسون هاينة من خوارزميدت د.عبدعلي حمودي الطائي د.حيدركاظم حمود خالد علي حسين الجامعة المستنصرية / كلية التربية / قسم علوم الحاسبات الملخص : أيجاد جذور المعادلة الالخطية تعتبر احد اهم المشاكل الحسابية وتطبيقاتها. في هذا البحث قدمنا خوارزمية جديدة اليجاد الجذور الحقيقية للمعادالت الالخطية عن طريق مزيج بين خوارزمية التنصيف وخوارزمية نيوتن-رافسون مع الوصف والمقارنة بينهم. 75

2 1. Introduction Numerical analysis is a very important branch of computer science that deals with the study of algorithms that use numerical approximation in mathematical analysis. It involves the study of methods of computing numerical data. It has its applications in all fields of engineering and the sciences [1]. The algorithms in that study uses in several applications like as, uses bisection method in electrical engineering [2], and implemented on an electrical circuit element [3], also uses in image processing to adaptive filters, like as [4] using Newton-raphson method to improve two novel algorithms CB (Center-to-Boundary) and BB (Boundary-to-Boundary) filters. In this paper, we introduce a new algorithm to find the real roots of single nonlinear equation, its Hybrid algorithm. It is organized as follows: section 2 establishments to rootfinding, Bisection and Newton-Raphson methods, section 3 putting hybrid algorithm, section 4 conclusion and results, section 5 future work. 2. Root-finding Algorithm The root-finding problem is one of the most important computational problems. It rises in a wide variety of practical applications in mathematics, physics, chemistry, biosciences, engineering, etc. As a matter of fact, determination of any unknown appearing implicitly in scientific or engineering formulas gives rise to a root-finding problem. A root-finding algorithm is a numerical method to finding a value x such that f(x)=0, for given function f. Numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards a limit (the so-called "fixed point") which is a root. The first values of this series are initial guesses. The method computes subsequent values based on the old ones and the function f Bisection Method: The Bisection (Binary Search) method which is based on the Intermediate value Theorem (IVT).The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method [5][7] [8]. If f(x) is continuous on [a,b] with f(a) and f(b) of opposite sign.the main idea of bisection method that[9] : Let c= (a+b ) / 2 (midpoint of [a,b] ), compute f(c) 76

3 If f ( c ) =0, then c is a root If f( a ). f( c ) < 0, a root exists in [a,c] Else the root exists in [c,b] Newton Raphson Method: In numerical analysis,newton-raphson method is a very popular numerical method used for finding successively better approximations to the zeroes of a real-valued function f ( x ) = 0.[6] X n+1 = x n - f ( x n ) / df (x n )..(1) This method is distinguished from the methods by the fact that it requires the evaluation of both the function f( x) and the derivative df (x) at arbitrary points x, where df is derivative of function f Newton-Raphson Algorithm:[6] Given f, df, x 0 (initial approximation) and δ (tolerance). Step 1 : i=1 Step 2 : x = x 0 - f ( x o ) / df(x o ) Step 3 : If x x 0 < δ, then go to step 6 Step 4 : x 0 = x Step 5 : i = i + 1, go back to step 2 Step 6 : stop iteration. 3. Hybrid algorithm This algorithm is a new approach to compute the roots of nonlinear equations f(x)=0, by propose hybrid algorithm between the Bisection algorithm and Newton-Raphson algorithm. It s take a first approximation by apply two times the Bisection method and complete a correct approximation by use the Newton-Raphson method. 77

4 3.1 Proposed Hybrid Algorithm Given f, df, a, b, and δ=10-6 (tolerance) Step 1 : for i=1 to 2 Step 2 : x i = (a+b) / 2 Step 3 : If f (x i ) = 0 or f (x i ) < δ, then step 10 Step 4 : If f (a) * f (x i ) < 0, then b= x i Step 5 : Else a = x i Step 6 : end for Step 7 : x = x i - [ f ( x i ) / df (x i ) ] Step 8 : If f (x) < δ, then go to step 10 Step 9 : x i = x, and go to step 7 Step 10 : stop iteration. 78

5 Start Given f,df,a,b and δ for i=1 To 2 x i = (a+b)/2 f(x i )=0 or f(x i ) < δ No Yes f(a) * f(x i ) < 0 No Yes a=x i end for b=x i for i=1 To N x=x i - [f(x i ) / df(x i )] f(x i )< δ? Yes No x i =x end for Stop Hybrid algorithm fig.(1) flowchart of Hybrid Algorithm 79

6 4. Results and discussion We implement the numerical algorithms on Dell i7 core Intel (4 cores) computer using Matlab ver (R2011a) 32-bit. The case study is real function (have real roots) as a form quartic function f(x)= a x 4 + b x 3 + c x 2 + d x + e, we take as for example f(x)= x 4 +3x 3 15x 2-2x + 9, with tolerance δ=10-6, where i= number of iterative, a & b are initials values and c= (a + b) / 2. Tables (1), (2), and (3) show the results of implement the above three algorithms. From these results, the Hybrid method is better than Bisection method with respect to number of iterative and elapsed time, but the result between Hybrid method and Newton-Raphson method is convergent. The serial Bisection method needs log 2 [(b-a)/ɛ] function evaluations, additions and multiplications to enclose the zero in an interval of length Ɛ. The serial Newton-Raphson method needs log 2 Ω, where Ω is the number of iterative operations needs to reach f(x) < δ (tolerance). So the number of operations of Hybrid is needs log 2 [(b-a) / Ɛ] + log 2 Ω, thus the number of operations. Table (1) result of Bisection method for 4 roots sequentially Interval =[-6,-4] Interval=[-2,0] Interval =[0,2] Interval =[2,4] i c F( c ) c F( c ) c F( c ) c F( c )

7 Elapsed time is Elapsed time is Elapsed is Elapsed time is Table (2) result of newton-raphson method for 4 roots sequentially Interval =[-6,-4] Interval=[-2,0] Interval =[0,2] Interval =[2,4] i x F( x ) x F( x ) x F( x ) x F( x ) Elapsed time is Elapsed time is Elapsed time is Elapsed time is Table (3) result of Hybrid algorithm for 4 roots sequentially Interval =[-6,-4] First approximate Second approximate i c F( c ) a b c x F(c ) df(c ) Elapsed time is Interval =[-2,0] First approximate Second approximate i c a b F( c ) c x F(c) df(c) Elapsed time is Interval =[0,2] First approximate Second approximate i c a b F( c ) c x F(c) df(c) Elapsed time is Interval=[2,4] First approximate Second approximate i c a b F( c ) c x F(c) df(c) 81

8 Elapsed time is Future Work Convert the Hybrid Algorithm to parallel numerical algorithm. Convert the bisection and Newton-Raphson Algorithms to parallel numerical algorithms with comparison with parallel Hybrid algorithm. Implement Hybrid Algorithm in different application field, for example ciphering, image processing, electrical engineering, etc. References 1. Soram R., and others, On The Rate of Convergences of Newton-Raphson Method, the international journal of engineering and sciences (IJES), vol.2, Issue 11, pp. 5-12, October Autar Kaw, Bisection Method of Solving a Nonlinear Equation More Examples in Electrical Engineering, A. Ersoz and M. Kurban, Bisection Method and Algorithm for Solving the Electrical Circuits, International Eurasian Conference Mathematical and Sciences in 2013,Sarajevo,Bosnia-Herzegovina., 4. K. K. Lavania, Shivali, R. Kumar, Image Enhancement Using Filtering Techniques, International Journal on Computer Science and Engineering (IJCSE), Vol.4, NO. 01, pp , R.L. Burden, J.D. Faires, Numerical Analysis, 9 th Edition, Brooks/Cole, R.L. Burden; J.D. Faires, Numerical Analysis (3rd ed.), PWS Publishers, Boston, MA, Otto S.R., Denier J.P, An Introduction to programming and Numerical Methods in Matlab, springer, Yang W.Y., Cao W., Chung T.S., Marris J., Applied Numerical Methods Using Matlab, Ajohn wily & Sons. Inc., A.W.Al-Khafaji and J.R.Tooley, Numerical Methods in Engineering Practice, CBS. Publishing Japan Ltd.,