Fixed point iteration Numerical Analysis Math 465/565

Transcription

1 Fixed point iteration Numerical Analysis Math 465/565 1

2 Fixed Point Iteration Suppose we wanted to solve : f(x) = cos(x) x =0 or cos(x) =x We might consider a iteration of this type : x k+1 = cos(x k ) Try this : Enter any number into your calculator and repeatedly hit the cos function. You will see the value converge to the solution of cos(x) =x. This value x is the fixed point of the function cos(x). 2

3 Fixed point iteration 3

4 Fixed point iteration 4

5 Fixed point iteration 5

6 Convergence of fixed point iteration 6

7 Convergence of fixed point iteration Let g(x) satisfy the Lipschitz condition 6

8 Convergence of fixed point iteration Let g(x) satisfy the Lipschitz condition g( ) g( ) apple, 0 apple < 1 6

9 Convergence of fixed point iteration Let g(x) satisfy the Lipschitz condition g( ) g( ) apple, 0 apple < 1 for all values and initial guess x 0 satisfies in the interval [a, b]. Suppose the 6

10 Convergence of fixed point iteration Let g(x) satisfy the Lipschitz condition g( ) g( ) apple, 0 apple < 1 for all values and initial guess x 0 satisfies in the interval [a, b]. Suppose the x 0 g(x 0 ) apple 1 b a 2 1 6

11 Convergence of fixed point iteration Let g(x) satisfy the Lipschitz condition g( ) g( ) apple, 0 apple < 1 for all values and initial guess x 0 satisfies in the interval [a, b]. Suppose the x 0 g(x 0 ) apple 1 b a 2 1 Initial guess is close enough to a root. 6

12 Convergence of fixed point iteration Let g(x) satisfy the Lipschitz condition g( ) g( ) apple, 0 apple < 1 for all values and initial guess x 0 satisfies in the interval [a, b]. Suppose the Then x 0 g(x 0 ) apple 1 b a All of the iterates will be in [a, b], Initial guess is close enough to a root. 2. The iterates will converge to a fixed point x, and 3. The point x is the only root in [a, b]. 6

13 Convergence of the fixed point method 1. Proof by induction : Since x 1 = g(x 0 ), we have that x 1 x 0 apple (1 ) apple Assume that for each x k,wehave x k x 0 apple. We have that x k+1 x k = g(x k ) g(x k 1 ) Use the Lipschitz condition recursively, and our requirement on x 0 to get x k+1 x k apple x k x k 1 apple 2 x k 1 x k 2 apple......apple k x 1 x 0 apple k (1 ) Then show that x k+1 x 0 apple ( k + k )(1 ) =(1 apple k+1 ) 7

14 Convergence of fixed point method 2. (existence) First, we have to prove that the sequence x k is a Cauchy sequence. This allows us to show that the sequence has a limit in our interval [a, b], i.e. In fact, we have that lim x k = x, x 2 [a, b] k!1 x k x apple k 8

15 Convergence of the fixed point method 3. (uniqueness) Suppose we have a second root in [a, b]. Then assuming 6= 0, we have = g( ) g( ) apple < This contradiction implies that =. Corallary : If g 0 (x) apple < 1, and g(x 0 ) x 0 apple (1 ) then the conclusion of the theorem still holds. Proof : Use the Mean Value Theorem. 9

16 Fixed point iteration To reformulate a root finding problem of the type f(x) = 0 into a fixed point problem, we can define g(x) = f(x)+x. 10

17 Fixed point iteration In general, a fixed point iteration can be used to solve equations of the form : To reformulate a root finding problem of the type f(x) = 0 into a fixed point problem, we can define g(x) = f(x)+x. 10

18 Fixed point iteration In general, a fixed point iteration can be used to solve equations of the form : g(x) =x To reformulate a root finding problem of the type f(x) = 0 into a fixed point problem, we can define g(x) = f(x)+x. 10

19 Fixed point iteration In general, a fixed point iteration can be used to solve equations of the form : g(x) =x To reformulate a root finding problem of the type f(x) = 0 into a fixed point problem, we can define g(x) = f(x)+x. Then g(x) =x, f(x)+x = x, f(x) =0 10

20 Fixed point iteration In general, a fixed point iteration can be used to solve equations of the form : g(x) =x To reformulate a root finding problem of the type f(x) = 0 into a fixed point problem, we can define g(x) = f(x)+x. Then g(x) =x, f(x)+x = x, f(x) =0 There are often several choices one can make to reformulate the problem. There is no guarantee that any of them will give a convergent method, however. 10

21 Example Use fixed point iteration to find the root of the polynomial f(x) =x 3 + x 2 3x 3 11