復(fù)旦大學(xué)Matlab課件—— (1)

1、Matlab MathCleve Morler著陳文斌（)零點和根1Bisection 我們知道 在1和2之間，先試x=3/2，因為x的平方大于2，那么 在1,3/2 M=2;a=1;b=3;k=0; while b-aeps x=(a+b) /2; if x2 M b=x else a = x end k=k+1;end2Interval bisection觀察一下a,b變化的情況： 程序結(jié)束需要多少次？ 最后a,b是多少？ 用format hex表示呢？為什么是這樣的？3Interval bisectionf(x) Find a very small interval, perhaps t

2、wo successive floating point numbers, on which the function change sign.k=0; while abs(b-a)eps*abs(b) x=(a+b) /2; if sign(f(x)=sign(f(b) b=x; else a = x; end k = k+1; end f = inline(x2-2); a=1; b=2;計算sqrt(2)4Interval bisection 二分法是慢的，用上面的代碼，對任何函數(shù)她都需要運行52步，但是她是完全可靠的。這與我們后面介紹的方法不同。5Newtons Method 對于求f

3、(x) = 0的根，Newton法要求過y=f(x)上任一點的切線，并求切線與x-軸的交。迭代過程是k=0; while abs(x-xprev)eps*abs(x) xprev = x; x = x-f(x)/fprime(x)end6Newtons Method對于平方根問題，Newton法是優(yōu)雅而有效的Newton法k=0; while abs(x-xprev)eps*abs(x) xprev = x; x = 0.5*(x+M/x)end 1.500 1.41666666666667 1.41421568627451 1.41421356237469 1.41421356237309

4、1.414213562373096步7Newtons Method缺點： 函數(shù)f(x)必須是光滑的，要容易計算導(dǎo)數(shù);初始猜想要靠近零點。優(yōu)點：收斂速度快，二次收斂。如果函數(shù)的一階和二階導(dǎo)數(shù)連續(xù)，且初始值靠近零點，我們可以證明8Newtons Method下面構(gòu)造一個讓Newton方法無限進(jìn)行而不收斂的例子Newton方法不收斂，原因？910Secant Method用有限差分來代替導(dǎo)數(shù)的計算k=0; while abs(b-a)eps*abs(b) c=a;a=b; b=b+(b-c)/(f(c)/f(b)-1); k=k+1;end 1.33333333333333 1.400 1.4146

5、3414634146 1.41421143847487 1.41421356205732 1.41421356237310 1.414213562373107步11Secant Method如果f(x)的一階導(dǎo)數(shù)和二階導(dǎo)數(shù)連續(xù)Superline convergence即每作一次迭代有效位將增加1.6位再考慮一下 12Inverse Quadratic InterpolationSecant方法用前兩個點來確定第三個點，能不能用三個點呢？IQIk=0; while abs(c-b)eps*abs(c) x=polyinterp(f(a),f(b),f(c),a,b,c,0); a=b;b=c;c

6、=x; k=k+1;end1.666666666666671.451.4061.414213788335491.414213562373331.414213562373101.41421356237309a,b,c=0,1,213Inverse Quadratic Interpolation a,b,c = -2,0,2 a,b,c = -2.00001,0,1.99999看一下下面兩組數(shù)據(jù)，IQI方法會發(fā)生什么情況？14Zeroin Start with a and b so that f(a) and f(b) have opposite signs Use a secant step t

7、o give c between a and bRepeat the following steps until |b-a|eps|b| or f(b)=0 Arrange a,b and c so that f(a) and f(b) have opposite signs |f(b)|=|f(a)| c is the previous value of b If c=a, consider an IQI stepIf c=a, consider a secant stopIf the IQI or secant step is in the interval a,b,take it If

8、the step is not in the interval, use bisection.15fzerotxMatlab中zeroin的算法實現(xiàn)是fzero. x = fzero(FUN,x0) %x0可以是數(shù)，或區(qū)間 x = fzero(besselj(0,x),0) %? x = fzero(besselj(0,x),0,pi) x = fzerotx(besselj(0,x),0,pi)16 ezplot(besselj(0,x),-30,30)17函數(shù)調(diào)用 字符串表達(dá)式 cos(pi*t) besselj(0,x) Inline函數(shù)Function handle bessj0 z = fzerotx(bessj0,0,pi) function y=bessj0(x) y=besselj(0,x)符號表示 syms x F=besselj(0,x) z=f