NAME WAN NOR AMIRA BINTI MIOR MOHAMAD SAZALI

NAME: WAN NOR AMIRA BINTI MIOR MOHAMAD SAZALI STUDENT ID: 2017696336 SUPERVISOR: DR. RIVAIE BIN MOHD ALI

INTRODUCTION • Numerical analysis: • Study of quantitative approximations to the solutions of mathematical problems including consideration of and bounds to the error involved (Merriam-Webster, 1853). • Area of mathematics and computer science that creates, analyzes and implements algorithms for obtaining numerical solution to problems involving continuous variables (Atkinson, 2007).

PROBLEM STATEMENT •

OBJECTIVES • To analyze the performance of Osada’s method, Schroder’s method, Jarratttype method and Sharma method. • To compare the accuracy and error of all method. • To determine the most suitable method to find multiple root.

LITERATURE REVIEW •

METHODOLOGY

RESULTS AND DISCUSSION Method F 1 F 2 F 3 F 4 F 5 F 6 -2. 3 -1. 3 -0. 3 0. 3 1. 3 2. 3 Osada method Jarratt-type method Schroder's method Sharma and Sharma method 5 7 3 6 3 2 3 3 7 5 6 9 4 3 2 2 3 4 3 3 16 16 3 3 5 5 3 3 3 1 3 2 17 32 17 37 16 13 11 10 15 13 16 15 12 13 11 10 11 11 4 7 11 11 7 4 9 8 12 12 8 9 16 15 14 15 8 6 18 27 7 5 6 5 10 24 9 7 5 8 7 6 F 9 4 8 4 4 4 10 10 4 4 7 4 6 4 3 5 11 5 18 7 10 3 4 3 10 7 4 11 6 4 3 3 5 6 3 3 16 16 3 3 4 3 6 6 3 4 4 3 3 3 2 3 ü The best method Osada method is ü Least number of iteration

Method F 1 F 2 F 3 F 4 F 5 Osada method Jarratt-type method Schroder's method Sharma and Sharma method -2. 3 5 23 8 11 -1. 3 7 38 6 6 -0. 3 3 23 18 22 0. 3 6 43 27 11 1. 3 3 22 7 11 2. 3 2 19 5 4 -2. 3 3 15 6 4 -1. 3 3 15 5 3 -0. 3 7 20 11 F 0. 3 5 18 24 F 1. 3 6 21 10 4 2. 3 9 20 8 12 -2. 3 4 18 6 6 -1. 3 3 18 9 5 -0. 3 2 17 7 4 0. 3 2 16 6 4 1. 3 3 17 F 5 2. 3 4 17 10 6 -2. 3 3 4 4 3 -1. 3 4 7 8 4 -0. 3 25 16 4 25 1. 3 4 7 8 4 2. 3 3 4 4 3 -2. 3 3 12 5 5 -1. 3 3 12 5 3 -0. 3 5 16 10 9 1. 3 3 12 5 3 2. 3 3 13 5 5 -2. 3 3 23 8 F -1. 3 1 22 4 5 ü The best method Osada method is ü Least number of iteration

Method F 1 F 2 F 3 F 4 F 5 F 6 -2. 3 -1. 3 -0. 3 0. 3 1. 3 2. 3 Osada method Jarratt-type method Schroder's method Sharma and Sharma method 5 7 3 6 3 2 24 39 23 44 23 20 16 16 20 18 21 20 24 24 23 22 4 7 21 21 7 4 8 6 18 27 7 5 6 5 11 24 10 8 6 9 8 7 11 6 22 11 11 4 4 3 F F 4 12 6 5 4 4 5 6 3 4 34 34 4 3 5 3 9 9 3 5 F F 6 3 2 10 3 3 7 5 6 9 4 3 2 2 3 4 33 33 4 3 3 3 5 5 3 3 3 1 3 2 14 12 16 16 12 14 26 25 26 24 22 23 F 10 4 8 4 5 5 10 10 5 5 8 4 6 4 3 5 ü The best method Osada method is ü Least number of iteration

Method F 1 F 2 F 3 F 4 F 5 F 6 Osada method Jarratt-type method Schroder's method Sharma and Sharma method -2. 3 1. 7188 1. 6563 1. 4531 1. 8593 -1. 3 1. 7656 1. 6719 1. 4375 1. 8906 -0. 3 1. 7500 1. 7813 1. 5156 1. 9844 0. 3 1. 8281 2. 5213 2. 3750 2. 0000 1. 3 1. 9219 1. 6719 1. 4844 1. 9688 2. 3 1. 7031 1. 7500 1. 5000 1. 9062 -2. 3 1. 7344 1. 5469 1. 4688 2. 4219 -1. 3 1. 9531 1. 5938 1. 3750 2. 0938 -0. 3 1. 8594 1. 5938 1. 4063 2. 3594 0. 3 1. 7696 1. 5469 1. 5000 2. 2656 1. 3 1. 8125 1. 5938 1. 5313 2. 2969 2. 3 1. 6563 1. 7188 1. 3281 2. 2656 -2. 3 1. 8125 1. 6563 1. 4688 1. 9531 -1. 3 1. 7969 1. 5469 1. 4219 2. 0313 -0. 3 1. 8594 1. 5000 1. 5156 2. 0156 0. 3 1. 7500 1. 6406 1. 5000 1. 9375 1. 3 1. 8281 1. 5938 2. 1094 2. 3 1. 7500 1. 5469 F 1. 4375 -2. 3 1. 7696 1. 5313 1. 5625 1. 8281 -1. 3 1. 9063 1. 5938 1. 5313 1. 9688 -0. 3 1. 7500 1. 6406 1. 4844 1. 9219 0. 3 2. 0781 1. 8125 1. 5469 1. 9531 1. 3 1. 8281 1. 7656 1. 5313 1. 8438 2. 3 1. 7696 1. 6406 1. 5469 1. 9375 -2. 3 1. 8438 1. 8125 2. 2813 1. 6094 1. 3906 -1. 3438 1. 9219 -0. 3 1. 6563 1. 6250 1. 4219 1. 9844 0. 3 1. 8438 1. 7500 1. 4688 1. 8125 1. 3 1. 7031 1. 6250 1. 3750 1. 8438 2. 3 1. 6719 1. 5313 1. 4375 1. 8906 -2. 3 2. 8281 1. 7344 1. 8594 2. 0938 -1. 3 2. 1094 1. 7188 1. 5938 2. 2813 -0. 3 2. 0938 1. 6719 1. 7344 2. 2969 0. 3 1. 9688 1. 7344 1. 5781 2. 2188 1. 3 1. 9531 1. 5781 1. 7813 2. 3594 2. 3 2. 0156 1. 6719 1. 7031 2. 3125 2. 1406 ü The best method is Schroder’s method ü Least CPU time

Method F 1 F 2 F 3 F 4 F 5 F 6 Osada method Jarratt-type method Schroder's method Sharma and Sharma method -2. 3 2. 0313 1. 5938 1. 4688 1. 9063 -1. 3 2. 0000 1. 4844 1. 3906 1. 8750 -0. 3 1. 7656 1. 5938 1. 4219 1. 9531 0. 3 1. 7500 1. 5625 1. 3906 1. 8125 1. 3 1. 7813 1. 5625 1. 3476 1. 9375 2. 3 1. 7500 1. 4844 1. 2656 1. 9531 -2. 3 1. 8438 1. 6719 1. 4375 2. 3594 -1. 3 1. 8750 1. 7031 1. 3750 2. 5156 -0. 3 1. 8750 1. 7696 1. 3750 F 0. 3 1. 6875 1. 6406 1. 4688 F 1. 3 1. 7031 1. 6094 1. 5938 2. 2813 2. 3 1. 8125 1. 5938 1. 3906 2. 2969 -2. 3 1. 9063 1. 5938 1. 4688 1. 9063 -1. 3 1. 9063 1. 6094 1. 4531 2. 5000 -0. 3 1. 8750 1. 6250 1. 4844 2. 0313 0. 3 1. 8906 1. 6406 1. 4063 2. 1563 1. 8750 1. 6719 2. 1094 2. 3 1. 9375 1. 6875 F 1. 5000 -2. 3 1. 7188 1. 5938 1. 7969 1. 8750 -1. 3 1. 8906 1. 6094 1. 5938 1. 7344 -0. 3 1. 8281 1. 6875 1. 4063 1. 9688 0. 3 2. 0781 1. 9063 1. 7188 1. 3906 1. 8906 1. 3 1. 7969 1. 5000 1. 4844 1. 9219 2. 3 1. 7656 1. 6719 1. 3594 1. 8125 -2. 3 1. 7344 1. 3906 1. 8750 -1. 3 1. 7500 1. 6875 1. 5469 1. 4219 1. 9375 -0. 3 1. 8125 1. 5781 1. 3906 1. 8281 0. 3 1. 7656 1. 5938 1. 4219 1. 9063 1. 7813 1. 6250 1. 4688 1. 9219 2. 3 1. 7500 1. 5781 1. 4219 1. 9531 -2. 3 1. 9844 1. 5781 1. 6250 F -1. 3 1. 9375 1. 6719 1. 6406 2. 4375 -0. 3 1. 9063 1. 6719 2. 2656 0. 3 2. 0156 1. 6719 1. 6250 2. 0938 1. 3 1. 9063 1. 6250 1. 6719 2. 2344 2. 3 1. 8906 1. 7031 1. 6719 2. 3438 ü The best method is Schroder’s method ü Least CPU time

Method F 1 F 2 F 3 F 4 F 5 F 6 -2. 3 -1. 3 -0. 3 0. 3 1. 3 2. 3 Osada method Jarratt-type method Schroder's method Sharma and Sharma method 1. 7656 1. 7344 1. 8594 1. 9063 1. 7500 1. 8750 1. 5938 1. 6094 1. 6563 1. 5781 1. 5938 1. 6250 1. 7500 1. 5469 1. 6406 1. 5781 1. 6406 1. 6875 1. 7344 1. 6250 1. 6406 1. 6875 1. 4219 1. 5625 1. 6094 1. 7031 1. 6250 1. 4531 1. 6563 1. 3438 1. 4219 1. 4375 1. 4844 1. 3594 1. 3750 1. 6875 1. 6719 1. 5625 1. 7344 1. 5626 1. 5625 1. 3906 1. 5625 1. 6406 1. 4844 1. 9688 1. 9844 1. 9063 1. 8906 1. 9688 1. 8438 2. 5156 2. 4531 F F 2. 2969 2. 3125 1. 9375 1. 8281 2. 0938 2. 1250 1. 9844 2. 0156 1. 9531 1. 9375 1. 9063 1. 8438 1. 9844 2. 0156 1. 8438 2. 1719 1. 8750 1. 9844 2. 0000 F F 2. 1719 2. 3438 2. 4063 2. 3281 1. 7500 1. 8906 1. 7500 1. 8594 1. 9063 1. 7969 1. 8438 1. 9375 1. 8906 1. 9375 2. 0156 1. 8125 1. 8594 1. 8750 1. 8438 1. 9219 1. 8125 1. 8438 1. 7500 1. 8281 1. 7813 1. 7344 1. 7188 1. 9531 2. 0156 2. 0469 2. 1563 1. 9375 2. 0000 1. 5625 1. 5156 1. 5469 1. 6875 1. 5781 1. 6719 1. 6406 1. 6563 1. 6094 1. 7031 1. 6563 1. 6719 F 1. 6563 1. 4063 1. 3125 1. 5625 1. 4219 1. 4063 1. 4531 1. 4219 1. 4844 1. 4219 1. 3594 1. 3750 1. 7344 1. 7500 1. 6406 1. 6719 1. 6563 1. 7188 ü The best method is Schroder’s method ü Least CPU time

CONCLUSION AND RECOMMENDATION Conclusion: q The best method to find multiple roots in term of number iteration is Osada method. q The best method to find multiple roots in term of CPU time is Schroder’s method. Recommendation: q Use larger interval q Use more different function

GANT CHART

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AUTHOR BIOGRAPHY Name : Wan Nor Amira binti Mior Mohamad Sazali Matric number : 2017696336 Programme : Bachelor Of Science (Hons. ) Computational Mathematics : Faculty of Computer and Mathematical Science Universiti Teknologi Mara : Performance Comparison of Two-steps Method for Finding Multiple Roots of Nonlinear Equation Supervisor : Dr. Mohd Rivaie bin Mohd Ali Email : amirasazali 98@gmail. com Address : No. 244 Laluan 1/6 Taman Kledang 31100 Sungai Siput (U) Perak. Phone number : 0195156573 Faculty Final year project title