1.牛顿法求方程根
假设f(x)在x=a处取到0，即f(a)=0。已知c在a的附近，有a=c+Δx。 根据泰勒公式展开， 省略高次项，得
2.简单代码实现
newt_cauchy.R
cauchy_mle=function(x,theta){sum = 0;for(i in 1:length(x)){sum = sum+(x[i]-theta)/(1+(x[i]-theta)^2);}sum;}cauchy_mle_der=function(x,theta){sum = 0;tmp = 0;for(i in 1:length(x)){tmp = (x[i]-theta)^2;sum = sum+(tmp-1)/(1+tmp)^2;}sum;}
test.R
#随机生成100个Cauchy样本，中位数为10source("A:\\studyproc\\陈希儒\\牛顿法解MLE\\newt_cauchy.R")sample=rcauchy(100,10,1)theta=median(sample)#用样本中值作为初始估计值std_diff=0.01;#阈值diff = cauchy_mle(sample,theta)diffs = c(abs(diff))while(abs(diff)>=std_diff && length(diffs)<1000){theta = theta - diff/cauchy_mle_der(sample,theta);diff = cauchy_mle(sample,theta)diffs = c(diffs,abs(diff))}print(diff)print(theta)
