F06 - Numerical Methods - Lecture Notes (Teacher_s Version)

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Numerical Methods (Teacher’s Version) Page 1 of 33 National Junior College 2016 – 2017 H2 Further Mathematics Topic F6: Numerical Methods (Lecture Notes) Key Questions to Answer:  What is a continuous function?  How to locate the roots of an equation by means of simple graphical or numerical methods?  How to use the method of linear interpolation and Newton Raphson method to find an approximation to a root of an equation? - What are the cases where the method fails to converge?  How to use iterations involving recurrence relations of the form  1 Fn nx x  to determine a root to a prescribed degree of accuracy? - What are the cases where the method fails to converge?  How to approximate the integral of a function using the trapezium rule and Simpson’s rule?  How to approximate solutions of first order differential equations using Euler method and improved Euler method respectively? Background In this topic, we will learn to use various numerical methods to approximate the roots of an equation which cannot be solved exactly ., for example solving cosx x . For many engineering and design problems, we cannot have an analytic solution but there is no real need for one. In such cases, all that is needed an accurate approximation to the actual solution. We term this a numerical solution to the problem, and its associated method a numerical method. §1 Continous Function A function f defined on  ,a b is continuous if for all points  0 ,x a b , for each 0  . there exists 0  such that whenever x a   , we have    f f x a   . Informally to say, a function f defined on  ,a b is continuous if it can be sketched from one end to the other with one stroke of the pen. Can you list some functions which are continuous? A function can fail to be continuous at a point x a for any one of the following three reasons.   f a does not exist   limf x a x  does not exist     limf f x a x a   www.KiasuExamPaper.com 390

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Numerical Methods (Teacher’s Version) Page 2 of 33 §2 Location of Roots Let  fy x be a function of ݔThe number  is said to be a solution (or root) of the equation  f 0x  when  satisfies  f 0.  Consider a continuous curve  f .y x If  1f x and  2f x have opposite signs, the curve must cross the axisx between 1 2 and .x x In general, if  f x is continuous and (i)    1 2f f 0x x  (i.e.  1f x and  2f x have opposite signs), then the equation  f 0x  has an ODD number of real roots between 1x and 2x (Some of which may be repeated). (ii)      1 2 1f f 0 (i.e. f andx x x  2f x have the same sign), then the equation  f 0x  has either NO real roots or an EVEN numbe