BSM510 Numerical Analysis

Transcription

1 BSM510 Numerica Anaysis Roots: Bracketing methods : Open methods Prof. Manar Mohaisen Department of EEC Engineering

2 Lecture Content v Introduction v Bracketing methods v Open methods v MATLAB hints 2

3 Introduction f() roots f() v Roots of a function The points at which Numerica methods q Bracketing methods: initia guess that brackets the root q Open methods: one/more initia guesses but no need for them to bracket the root v Loca/goba maimums f '( ) = 0 and f ''( ) < 0 The points at which f( ) = 0 v Loca/goba minimums The points at which f '( ) = 0 and f ''( ) > 0 3

4 Bracketing methods v Method I: Incrementa search Check the occurrence of sign change in the given range with a predefined increment ength q Large increment ength might ead the search to miss roots q Sma increment ength sows down the search Check Fig. 5.4 in the tet book (run the code) 2 1 ns = 10, brackets = 5 ns = 50, brackets = 5 ns = 100, brackets = 9 f()

5 Bracketing methods contd. v Method II: Bisection Good method to ocate the interva of a root Sow q It is foowed by a fast converging method to find the vaue of the root Stage I q Make a guess on the interva of the root Stage II q Cacuate the midpoint r q If r and have the same sign = r, don t modify u q If r and have opposite signs Continue u = r, don t modify q same as in stage II ti a tight interva is found I = [, ] r = e = a u + 2 u new r r - od r new 100% 5

6 Bracketing methods contd. v Method III: Fase position Same as the bisection method with the difference that r = - u f( u)( - u) f( ) - f( ) u q The computed r repaces whether or u q The resuting function vaue has same sign of f( r ) q or u aways bracket the true root 6

7 Bracketing methods contd. v Numerica eampe Function æ ö f( ) = sin ç 2p + cos( p ) è ø f() v Bisection Fase position method is very simiar f( ) f( ) < 0 = r u r f ( ) f ( ) > 0 = r f( ) f( ) < 0 = r r u r f ( ) ( ) 0 f r > = f( ) f( ) < 0 = r u r f ( ) f ( ) > 0 = f( ) f( ) < 0 = r r r r u r Iteration u r e a % % % 7

8 Bracketing methods contd. v Comparison between methods A methods q Generay, converge to the root s interva, more iterations à tighter interva Incrementa search q Locates the interva(s) of sign change q The choice of increment ength is critica (sma à sow, argeà miss a toot) Bisection (modification of IS) q The interva is aways divided in haf to ocate the root ocation q Sow but certainy converges q Convergence is generay independent of the function s curvature Fase Position (modification of bisection) q Generay, faster convergence q However, for functions with significant curvature Fase position method is sower than bisection method 8

9 Open methods v Bracketing methods Require two initia points bracketing the root Converge Sow v Open methods Require one or two points Empoy a formua to predict the root Therefore, q Since root is not bounded, Methods might diverge If they converge q They converge fast 9

10 Open methods contd. v Method I: Simpe fied-point iteration Step I: Rearrange the function f()=0 so that is on the eft side q f() = 0 means the ocation of a root q Eampe = g( ) f( ) = 2 - f( ) = 0 = g( ) = 2 Step II: a new estimate of the root is then given by i g( ) 1 i + = f() = e - - f() root: e - =

11 Open methods contd. v Method I: Simpe fied-point iteration Aternative graphica approach q Step I: Spit f()=0 into two functions y 1 () = y 2 () q Step II: The root is the intersection of these two functions y 1 (), y 2 ()

12 Open methods contd. v Method I: Simpe fied-point iteration Convergence q Error at (i+1)th iteration q If E ' i 1 g + = E g'( ) ³ 1 ( ) i The method diverges q If '( ) 1 g < The method converges 12

13 Open methods contd. v Method I: Simpe fied-point iteration Root for the function f( ) = e- - f( ) = 0 = e - i i e a % e t % e t i / e t i

14 Open methods contd. v Method II: Newton-Raphson method The update formua f( ) - 0 f( ) ( ) f '( ) = i i i - = - i+ 1 i ' i i+ 1 f i q The 0 is the epected vaue of f( i+1 ). Eampe f( ) = e- - e- - = i+ 1 i e- 1 i i e t % <

15 Open methods contd. v Method II: Newton-Raphson method NR method performs poory for the functions in the figure q Remedy: Have a guess cose to the root Good aspect q The error of the i+1 th iteration is roughy proportiona to the square of the error of the ith iteration - this is caed quadratic convergence 15

16 Open methods contd. v Method III: Secant methods Drawback of NP method q The derivative might be hard to compute or to evauate The derivative is therefore approimated as (using 2 points) f( ) - f( ) f( )( - ) f ( = = - ( ) ( ) ' i-1 i i i-1 i - i+ 1 i i-1 i f - f i-1 i Same can be done with a point and a fraction of it as foows d f( ) ( ) ( ) i i i+ 1 = i - f + d - f i i i q Other properties are same as those of NR method 16

17 MATLAB Code v fzero fzero(@() func, 0) q Tries to find a zero of the function func near the point 0 Eampe q fzero(@() ep(-) -, 0.8) v roots roots(c) q Find the roots of the poynomia whose coefficients are the vector c Eampe q roots([1-2 -1]) q ans: ,

18 Lecture Summary v Introduction v Bracketing methods v Open methods v MATLAB hints 18