MATLAB牛顿迭代法
-
牛顿迭代法（用于搜索零点）：
通过函数
f(
x
)
=
0
\ f(x) =0
f
(
x
)
=
0
的泰勒级数展开
f(
x
)
=
f
(
x
0
)
+
f
′
(
x
0
)
1
!
(
x
−
x
0
)
+
f
′
′
(
x
0
)
2
!
(
x
−
x
0
)
2
+
.
.
.
+
f
(
n
)
(
x
0
)
n
!
(
x
−
x
0
)
n
\ f(x) =f(x_0) +\frac {f'(x_0)} {1!}(x-x_0)+\frac {f”(x_0)} {2!}(x-x_0)^2+…+\frac {f^{(n)}(x_0)} {n!}(x-x_0)^n
f
(
x
)
=
f
(
x
0
)
+
1
!
f
′
(
x
0
)
(
x
−
x
0
)
+
2
!
f
′
′
(
x
0
)
(
x
−
x
0
)
2
+
.
.
.
+
n
!
f
(
n
)
(
x
0
)
(
x
−
x
0
)
n
取级数展开式的常数项和一次项：
f(
x
0
)
+
f
′
(
x
0
)
(
x
−
x
0
)
=
0
\ f(x_0) +f'(x_0)(x-x_0)=0
f
(
x
0
)
+
f
′
(
x
0
)
(
x
−
x
0
)
=
0
得到迭代函数：
x=
x
0
−
f
(
x
0
)
f
′
(
x
0
)
\ x=x_0 -\frac {f(x_0)} {f'(x_0)}
x
=
x
0
−
f
′
(
x
0
)
f
(
x
0
)
这里是一个一元函数的简单代码：
%牛顿迭代法
function [x,n,Xn,Yn] = newiteration(fun,dfun,x0,EPS)
% fun为目标函数，dfun为目标函数的一阶导数，x0为起始点，EPS为精度
a=feval(fun,x0);
b=a+1;
n=0;
%建立画图的点
Xn = zeros(5,1);
Yn = zeros(5,1);
while(abs(a-b) >= EPS)
a = feval(fun,x0) ;
df = feval(dfun,x0);
Xn(n+1,1) = x0;
Yn(n+1,1) = a;
if (feval(dfun,x0) == 0)
break
else
x0 = x0 - a/df;
end
b = feval(fun,x0);
n = n + 1;
end
x = x0;
调用函数：
%% 调用函数
clear all
clc
% syms x
fun=inline('x^2 - 9','x');
dfun = inline('2*x','x');
Xn = zeros(5,1);
Yn = zeros(5,1);
x0 = 1;
EPS = 0.001;
[x,n,Xn,Yn] = newiteration(fun,dfun,x0,EPS)
figure
plot(Xn,Yn,'k-','color','red')
hold on
x = 0:0.1:7;
y = x.^2 - 9;
y2 = zeros(1,71);
plot(x,y,'color','b')
plot(x,y2,'color','black')
运行结果：
x =3.000000001396984
n =5
迭代过程：
如果需要减少计算，可以通过将
f
′
(
x
0
)
\ f'(x_0)
f
′
(
x
0
)
设置为定值，收敛速度会减慢。