Solution of Equations

Transcription

1 Solution of Equations

2 Outline Bisection Method Secant Method Regula Falsi Method Newton s Method

3 Nonlinear Equations This module focuses on finding roots on nonlinear equations of the form f()=0. Due to the nonlinearity of f()=0 the solution methods used are usually iterative.

4 Convergence Rate In any iterative process we are interested in the convergence rate. We will define this in terms of: k 1 lim c k k If there eists a value of p such that there is a non-zero constant c which satisfies this equation, then p is the order of convergence. p=1 (linear), p=2 (quadratic), p=3 (cubic) p

5 Bisection Method The simplest method to solve f()=0 is the bisection method. This method begins from an initial bracket where f() is negative on one side of the bracket and positive on the other. f ( ) f ( 1 ) 0 o

6 Bisection Method +ve New interval +ve -ve 1/2h 1/2h

7 Bisection Method At each step in the process a point in the middle of the interval is considered: Three cases eist: f ( 2 f ( 2 f ( 2 ) 0, ) ) is is 2 then the o root has the same sign has the same sign as in the interval ( in the interval ( 1 2 as 0 2,, 1 2 ) ) o has been found f ( o f ( 1 ) then the root ) then the root

8 f() Bisection Method: Eample Bisection method

9 Bisection Method: Eample o 1 f(o) f(1) 2 f(2) f ( ) ( 1)( 2)( 4) 1

10 Rate of Convergence After k iterations the size of the interval bracketing the root has decreased to: 1 k Therefore the number of iterations required to achieve a certain tolerance 0 is: Method has linear convergence. 2 k log

11 f() Secant Method The secant method is not a bracketing method. It starts from two initial points and draws a straight line between these points. 10 Secant method oldest previous

12 f() Secant Method A new value is chosen by finding where this straight line crosses the - k k k1 ais: f ( )( ) 10 k1 k f ( k ) f ( Secant method k1 ) oldest previous

13 f() Secant Method This new value replaces the oldest value being used in the calculation. 10 Secant method oldest previous

14 Secant Method: Convergence Because the secant method is not a bracketing method it may not converge. But when the method converges it can be shown to have an order of convergence which is: Sometimes it is good to start finding a root using the bisection method then once you know you are close to the root you can switch to the secant method to achieve faster convergence.

15 f() Secant Method: Failure If the function is very flat the secant method can fail, for eample: 1 Secant method 1 f ( ) ( 1) oldest first iteration second iteration new previous

16 Secant Method: Failure The numerical values associated with the failure eample are: o 1 f(o) f(1) 2 f(2)

17 Regula Falsi To avoid the failure of the secant method the regular falsi method can be used. This is a variant of the secant method which maintains a bracket around the solution. This method is also convergent, but in return it will be slower than the secant method.

18 f() Regula Falsi ve ve -ve -1

19 Regula Falsi The regula falsi method proceeds in the same manner as the secant method ecept that the initial points must bracket the root and the values are updated to maintain that property.

20 f() Regula Falsi: Eample Regula falsi method

21 Regula Falsi: Eample o 1 f(o) f(1) 2 f(2)

22 f() Regula Falsi: Eample 2 When regula falsi applied to the case which caused the secant method to fail we have: Regula falsi method

23 Regula Falsi: Eample 2 Convergence in this case is slow but guaranteed. o 1 f(o) f(1) 2 f(2)

24 Newton s Method The final method we will mention for finding the roots of a single equation is Newton s method. Unlike the others it requires information about the derivative f (). From Taylor series we have: f ( ) 0 f ( ) f '( ) f ''( ) ( )

25 Newton s Method If we truncate the series after the first derivative and set f(+) to zero this gives: k1 k f ( f '( Method has quadratic convergence. k k ) )

26 f() Newton s Method: Eample Newton method tangent

27 Newton s Method: Eample The numerical values associated with the previous figure are: o f(o) f'(o) E

28 f() Newton s Method: Eample 2 Newton s method will also work on the case that gave the secant method problems. Newton method

29 Newton s Method: Eample 2 Numerical values... o f(o) f'(o) E

30 THANK YOU Submitted by: Rakesh Kumar, (Deptt. Of Mathematics), P.G.G.C.G, Sec-11, Chandigarh.