Chapter 2 - Solution of Nonlinear Equation PDF [PDF]

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Chapter 2: Solution of nonlinear equation 2.1



Bisection method In general, if a function f (x ) is real and continuous in the interval from x a to xb and



f ( xa ) and f ( xb ) have opposite signs, that is f ( xa )  f ( xb )  0 then, there is at least one real root between x a and xb Example: x2  4x  4  0



( x  2) 2  0



x2 The bisection method is one type of incremental search method in which the interval is always divided in half. Steps of Bisection method. 1. Choose lower x a and upper xb guesses for the root. 2. An estimate of the root xm is determined by xm 



x a  xb . 2



3. Calculate the functions values of x a and xm . 4. Make the following evaluations to determine which subinterval the root lies: a) If f ( xa )  f ( xm )  0 , therefore, xm  xb b) If f ( xa )  f ( xm )  0 , therefore, xm  xa c) If f ( xa )  f ( xm )  0 ,then the solution/root equals to xm 5. If the difference | xb  xm | ≤  , then accept xm as the solution/root.



Faculty of Engineering and Built Environment, SEGi University Kota Damansara Prepared by: Fatin Nur Diana binti Abu Samah



2.2



Newton-Raphson Method The Newton-Raphson method uses a straight line approximation to the function whose zero we wish to find, in this case the line is the tangent to the curve. The approximation to the zero is the value of x where the tangent crosses the x-axis requires additional about the function. (its derivatives) Given an initial estimate of the zero, x0 the value of the function x0 , y0  f ( x0 ) and the value of the derivative at x0 , y '0  f ' ( x0 ) the x-intercept of the tangent line, which is the new approximation to the zero is,



f ( x0 ) f ' ( x0 )



x1  x0 



The process continues until the | xi  xi 1 | ≤  , then accept xi1 .At the ith stage, we have



xi 1  xi 



f ( xi ) f ' ( xi )



Steps of the Newton-Raphson method: 1. Find the first derivative of the nonlinear function, f ' ( x) 2. Use an initial guess of the root, xi to estimate the new value of root, xi 1 as



xi 1  xi 



f ( xi ) f ' ( xi )



3. Calculate the difference, | xi 1  xi |. 4. STOP the process until the difference | xi 1  xi | ≤  , then accept xi as the root.



Faculty of Engineering and Built Environment, SEGi University Kota Damansara Prepared by: Fatin Nur Diana binti Abu Samah



2.3



Simple Fixed-Point Method In general, a new formula was developed for simple fixed-point iteration by rearranging the function f ( x)  0 , so that x is on the left-hand side of equation. x  g (x )



Steps of the Simple Fixed-Point method: 1. Develop new possible formulas from a given function, x  g (x ) 2. Label the left-hand side as xi1 and the right-hand side as xi 3. Use an initial guess of the root, xi to estimate the new value of root, xi 1 4. Calculate the difference, | xi 1  xi |. 5. STOP the process until the difference | xi 1  xi | ≤  , then accept xi as the root.



Faculty of Engineering and Built Environment, SEGi University Kota Damansara Prepared by: Fatin Nur Diana binti Abu Samah