Linear convergence k 2 k a1+ k k k < xk+1-x|≤c|xk-x,C∈[0,1) myshow Bisection m Showbisect m
k k k a b a b 2 | | | | 0 0 1 0 0 * 2 | | , | | 2 ( ) k k k k k a b x x a b x Linear convergence | | | |, [0,1) xk1 x* c xk x* c myshowBisection.m Showbisect.m
4. 0000000000000000 3. 0000000000000000 3. 5000000000000000 0. 5000000000000000 3.12500000oooo 3. 2500000000000000 0. 1250000000000000 45B789g 3. 1250000000000000 3. 1875000000000000 0. 0625000000000000 3. 1250000000000000 3.156250 00000000o 0.031250U00 3.1406250000o0000 3.156250000ooo00 0. 0156250000000000 0. 0078125000000000 4 Notice that a new digit of pi is acquired 0.0o3s062500o 0.0019531250000 i every three or so iterations. This kind of 0.0oogT656250o 0004882812500 13 00o2441406250000 0.0001220703125000 15 1 uniform acquisition of significant digits is 0.0000610351562500 16 1 the hallmark of methods that converge 0.000o30s1T5781250 0.0000152587890625 18 A linearly 0.D0oooT6293945313 0.00038146972656 0.0oooo19073486328 21 3.1415920257568359 3.1415929794311523 0. 0000009536743164 3.1415925025939941 3.1415929794311523 0.00o4T683T1582 23 3.1415925025939941 3.1415927410125732 0. 0000002384185791 3.141592621803283了 3.1415927410125732 0. 0000001192092896 3.141532621803283T 3.1415926814079285 0. 000000596046448 3.1415926516056D61 3.1415926814079285 0.0ooo298023224 3.14159265160s6061 3.1415926665067673 0.0o0o0149011612 3.14153285160s661 3.1415926590561867 0. 000000074505806 29 3.1415926516os661 3.1415926553308964 0. 0000000037252903 3.1415926534682512 3.1415926553308964 0. 0000000018626451 31 3.1415926534682512 3.1415926543995738 0. 0000000009313226 3.1415926534682512 3.1415926539339125 0.DUo46s6613 3.1415926534682512 3.141592653T010819 0.000oo0002328306 3.1415928535846666 3.141532853To10819 0. 0000000001164153 3.1415926535846666 3.1415928536428742 0. 00000000005820TT 3.1415928535846666 3.1415926536137T04 0.00oooo00291038 3.1415926535846666 3.1415926535992185 0. 00000000001 45519 3.141592653584666f 3.1415926535919425 0.0000000000072760 3.1415926535883045 3.1415926535919425 3.1415926535883045 3.1415926535901235 0. 00000000000181 90 1 3.1415926535892140 3.1415326535901235 0. 0000000000009095
Notice that a new digit of pi is acquired every three or so iterations. This kind of uniform acquisition of significant digits is the hallmark of methods that converge linearly
The Newton Method Idea L(x)=f(x)+(x-x)f(x2) f(x X xc=5.268433f=2857e+000
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 -2 0 2 4 6 8 10 12 14 xc = 5.268433 fc = 2.857e+000 The Newton Method Idea ( ) ( ) ( ) '( ) c c c c L x f x x x f x '( ) ( ) c c c f x f x x x
f(x=tan(x/4) k Xk 03.1415926535897931 0.0000000000000000 3.7963140465723391 0.6547213929825460 23.2594354361754703 0.1178427825856772 33.1451315542075164 0.0035389006177233 43.1415957863900608 0.0000031328002676 53.1415926535922467 0.0000000000024536 63.1415926535897931 0.0000000000000000 M ytestnewtonm
k xk xk-pi ------------------------------------------------------------------- 0 3.1415926535897931 0.0000000000000000 1 3.7963140465723391 0.6547213929825460 2 3.2594354361754703 0.1178427825856772 3 3.1451315542075164 0.0035389006177233 4 3.1415957863900608 0.0000031328002676 5 3.1415926535922467 0.0000000000024536 6 3.1415926535897931 0.0000000000000000 f (x) tan(x / 4) 1 Mytestnewton.m
quadratical convergence k+1-X ≤C k No guarantee that x, is closer to a root than x f(x)=x2-a2,t=2,a=100,10,10°,10-20 A small value for f(x y works against rapid convergence f(x=arctan(x),f'(x) 1+x 2 计fx2>1.3917,x4|x divergence
quadratical convergence | | | | , 2 1 * * x x c x x k k No guarantee that x+ is closer to a root than xc A small value for f’(x *) works against rapid convergence ( ) , 2, 10 ,10 ,10 ,10 ,0 2 2 0 4 8 12 f x x a t a | | 1.3917,| | | | , 1 1 ( ) arctan( ), '( ) 2 c c if x x x x f x x f x divergence