수치해석 MATLAB Bisection Method, False Position Method

 1. Locate the nontrivial root of the non-algebraic equation, sin(x)=x2, where x is in radians, using:

a) The bisection method, with an initial interval (0.5, 1). Perform the computation to achieve an approximate percent relative error εa to be within εa<2×10-4%. Estimate the number of bisection iterations required to attain a desired absolute error, Ea,d of 1.75×10-6.

b) The method of false-position, with an initial interval (0.5, 1). Perform the computation to meet the same requirement (εa<2×10-4%). For both methods, plot a true error, εt and an approximated error, εa (log scale) versus iteration numbers and discuss the rate of convergence.

주어진 방정식 ;

a) Bisection Method

출력데이터 (iteration number, xl, xu, ea, et)

0	0.50000000	1.00000000
1	0.75000000	1.00000000	100.00000000	100.00000000
2	0.87500000	1.00000000	14.28571429	14.45448044
3	0.87500000	0.93750000	6.66666667	0.19689385
4	0.87500000	0.90625000	3.44827586	6.93189945
5	0.87500000	0.89062500	1.75438596	3.36750280
6	0.87500000	0.88281250	0.88495575	1.58530448
7	0.87500000	0.87890625	0.44444444	0.69420531
8	0.87500000	0.87695313	0.22271715	0.24865573
9	0.87597656	0.87695313	0.11148272	0.02588094
10	0.87646484	0.87695313	0.05571031	0.08550645
11	0.87670898	0.87695313	0.02784740	0.02981276
12	0.87670898	0.87683105	0.01392176	0.00196591
13	0.87670898	0.87677002	0.00696136	0.01195752
14	0.87670898	0.87673950	0.00348080	0.00499580
15	0.87672424	0.87673950	0.00174043	0.00151495
16	0.87672424	0.87673187	0.00087021	0.00022548
17	0.87672424	0.87672806	0.00043511	0.00064473
18	0.87672615	0.87672806	0.00021755	0.00020963
19	0.87672615	0.87672710	0.00010878	0.00000793
E_a,d가 1.75*10^(-6)일 때 interation 횟수는
25	0.87672621	0.87672623	0.00000170 에서
25 이다.

b) False-Position Method

출력데이터 (iteration number, xl, xu, ea, et)

0	0.50000000	1.00000000
1	0.79568610	1.00000000	100.00000000	100.00000000
2	0.86490596	1.00000000	8.00316564	9.24349214
3	0.87512417	1.00000000	1.16762956	1.34822737
4	0.87651133	1.00000000	0.15826004	0.18273143
5	0.87669743	1.00000000	0.02122750	0.02451018
6	0.87672236	1.00000000	0.00284324	0.00328338
7	0.87672570	1.00000000	0.00038076	0.00044015
8	0.87672615	1.00000000	0.00005099	0.00005940

False Position Method 가 더 빨리 수렴함을 볼 수 있다.