數值方法 2008 Applied Mathematics, NDHU1  Bisection method for root finding  Binary search  fzero Lecture 4

數值方法 2008 Applied Mathematics, NDHU2 Drawbacks of Newton method x=linspace(-5,5);plot(x,x.^2-2*x-2); hold on;plot(x,-3,'r') Tangent line with zero slope

數值方法 2008 Applied Mathematics, NDHU3 Perturb current guess if zero derivative is detected

數值方法 2008 Applied Mathematics, NDHU4 Failure f=inline('x.^3-2*x+2'); x=linspace(-2,2);plot(x,f(x));hold on plot(x,0,'g') plot([0 1],[0 f(1)],'r'); plot([1 0],[0 f(0)],'r'); The tangent lines of x^3 - 2x + 2 at 0 and 1 intersect the x-axis at 1 and 0, respectively, illustrating why Newton's method oscillates between these values for some starting points.

Bisection method A root is within an interval [L,R] The bisection method Cut [L,R] into equal-size subintervals. such as [L,R] to [L,M] U [M,R] Determine which interval contains a root Then select it as the searching interval Repeat the same process until halting condition holds. 數值方法 2008 Applied Mathematics, NDHU5

Different signs Let f be continuous in the interval [L,R] These exists at least one root in [L,R] if f(L) and f(R) have different signs, equivalently f(L) f(R) < 0 數值方法 2008 Applied Mathematics, NDHU6

7 Bisection Method LRM=(L+R)/2 f(L) and f(M) have the same sign: L  M f(R) and f(M) have the same sign: R  M LRM L R M L  M R  M

數值方法 2008 Applied Mathematics, NDHU8 Bisection method 1. Create an inline function, f 2. Input two guesses, L < R 3. If f(L)f(R) > 0, return 4. Set M to the middle of L and R 5. If f(L)f(M) < 0, R = M 6. If f(R)f(M) < 0, L = M 7. If the halting condition holds, exit, otherwise goto step 4.

數值方法 2008 Applied Mathematics, NDHU9 Flow chart Input L,R,f with f(L)f(R) < 0 R=M abs(f(M)) < epslon T F f(L)f(M)<0 L=M M=0.5*(L+R) c=bisection(f,L,R) T

數值方法 2008 Applied Mathematics, NDHU10 Flow chart if f(L)f(R) > 0 return M=0.5*(L+R) R=M f(L)f(M)<0 L=M M=0.5*(L+R) abs(f(M)) < epslon M=bisection(f,L,R) T T return

數值方法 2008 Applied Mathematics, NDHU11 Flow chart if f(L)f(R) > 0 return M=0.5*(L+R) R=M f(L)f(M)<0 L=M M=0.5*(L+R) abs(f(M)) > epslon M=bisection(f,L,R) T T return

數值方法 2008 Applied Mathematics, NDHU12 Halting condition abs(f(M)) < epslon Absolute f(M) is small enough to approach zero.

數值方法 2008 Applied Mathematics, NDHU13 Non-decreasing sequence x = y L R

數值方法 2008 Applied Mathematics, NDHU14 Binary search Binary search is a general searching approach Let x represent a non-decreasing sequence x(i) is less or equal than x(j) if i < j Let y be an instance in sequence x Determine i such that x(i) = y

數值方法 2008 Applied Mathematics, NDHU15 Random sequence >> x=round(rand(1,15)*100) x = Columns 1 through Columns 10 through

數值方法 2008 Applied Mathematics, NDHU16 Searching for non-decreasing sequence x = y L=1 R=15 M=8

數值方法 2008 Applied Mathematics, NDHU17 Searching for non-decreasing sequence x = y R=15 L=8 M=12

數值方法 2008 Applied Mathematics, NDHU18 Searching for non-decreasing sequence x = y L=8 R=12 M=10 Halt since x(c) equals y

數值方法 2008 Applied Mathematics, NDHU19 Searching Ex. y=76 The answer i=10 indicates an index where x(i) equals y >> x(10) ans = 76

數值方法 2008 Applied Mathematics, NDHU20 Flow chart Input x,y L=1;R=length(x) L=M Halting condition T F x(M)<y R=M M=ceil(0.5*(L+R))

數值方法 2008 Applied Mathematics, NDHU21 Flow chart L=1;R=length(x) M=ceil(0.5*(L+R)) L=M x(M)<y R=M M=ceil(0.5*(L+R)) Halting condition c=bi_search(x,y) T T return

數值方法 2008 Applied Mathematics, NDHU22 Discussions What is a reasonable halting condition? Index L must be less than index R Halt if L >= R or x(M) == y

數值方法 2008 Applied Mathematics, NDHU23 MATLAB: fzero x=linspace(-5,5);plot(x,sin(x.*x)); hold on; plot(x,0,'r') x = plot(x,0,'or')

數值方法 2008 Applied Mathematics, NDHU24 fzero x=linspace(-5,5);plot(x,sin(x.*x)); hold on; plot(x,0,'r') x = plot(x,0,'or')

Exercise 4 due to 10/22 1. Draw a flow chart to illustrate the bisection method for root finding 2. Implement the flow chart 3. Give two examples to verify the matlab function 數值方法 2008 Applied Mathematics, NDHU25