1）原理：零点定理：
f
∈
C
[
a
,
b
]
,
f
(
a
)
f
(
b
)
<
0
⟹
∃
c
∈
[
a
,
b
]
,
f
(
c
)
=
0
f\in C[a,b],f(a)f(b)<0\implies \exist c\in[a,b],f(c) = 0
f ∈ C [ a , b ] , f ( a ) f ( b ) < 0 ⟹ ∃ c ∈ [ a , b ] , f ( c ) = 0
2）实现：
% 二分法计算f(x)近似解
% 输入：
% f：函数句柄
% a，b：区间端点
% tol：解精度
% 输出：近似解
function xc=binsect(f, a, b, tol)
if sign(f(a))*sign(f(b)) >= 0
error('f(a)f(b) greater than zero ') % 错误
end
fa = sign(f(a));
fb = sign(f(b));
while ((b-a) >> 1) > tol
c = (a+b) >> 1
fc = sign(f(c))
if fc == 0
break
end
if fc^fa < 0 % 解落在[a,c]
b = c;
else
a = c;
end
end
xc = (a+b) >> 1
3）精度：在
n
n
n
b
−
a
2
n
+
1
\frac{b-a}{2^{n+1}}
2 n + 1 b − a
精确位数：若误差小于
0.5
∗
1
0
−
p
0.5*10^{-p}
0 . 5 ∗ 1 0 − p
1）原理：求解
f
(
x
)
+
x
f(x)+x
f ( x ) + x
x
−
f
(
x
)
x-f(x)
x − f ( x )
x
=
g
(
x
)
x = g(x)
x = g ( x )
不动点：
g
(
x
)
=
x
g(x) = x
g ( x ) = x 不动点迭代：
x
n
+
1
=
g
(
x
n
)
x_{n+1} = g(x_n)
x n + 1 = g ( x n )
x
n
x_n
x n
2）实现：
% 不动点计算g(x) = x的近似解
% 输入：
% g：函数句柄
% x0：初始估计
% k: 迭代次数
% 输出：近似解
function xc=fpi(g, x0, k)
x(1) = x0;
for i=1:k
x(i+1) = g(x(i))l
end
xc = x(k+1);
3）线性收敛：
lim
k
→
+
∞
∣
x
k
+
1
−
r
x
k
−
r
∣
=
S
<
1
\lim\limits_{k \to +\infty} |\frac{x_{k+1}-r}{x_k-r}| = S \lt 1
k → + ∞ lim ∣ x k − r x k + 1 − r ∣ = S < 1
S
S
S
g
(
x
)
g(x)
g ( x )
∣
g
′
(
r
)
∣
<
1
|g'(r) |<1
∣ g ′ ( r ) ∣ < 1 局部收敛 )
4）终止条件：不动点迭代的步数是难以确定的，且不一定收敛
绝对终止条件：
∣
x
k
+
1
−
x
k
∣
<
T
O
L
|x_{k+1} - x_k|<TOL
∣ x k + 1 − x k ∣ < T O L
相对终止条件：
∣
x
k
+
1
−
x
k
∣
∣
x
k
+
1
∣
<
T
O
L
\frac{|x_{k+1} - x_k|}{|x_{k+1}|}<TOL
∣ x k + 1 ∣ ∣ x k + 1 − x k ∣ < T O L
混合终止条件：
∣
x
k
+
1
−
x
k
∣
max
{
θ
,
∣
x
k
+
1
∣
}
<
T
O
L
\frac{|x_{k+1} - x_k|}{\max\{\theta,|x_{k+1}|\}}<TOL
max { θ , ∣ x k + 1 ∣ } ∣ x k + 1 − x k ∣ < T O L
1）后向误差：
∣
f
(
x
^
)
∣
|f(\hat{x})|
∣ f ( x ^ ) ∣ 前向误差：
∣
x
^
−
x
∣
|\hat{x}-x|
∣ x ^ − x ∣
2）计算机误差
在有限精度的计算机当中，在根的周围会出现多个假根，以及符号不稳定现象
多重根领域切线水平，一般更容易造成精度丢失
3）根的敏感公式：
f
(
r
)
=
0
,
f
(
r
+
Δ
r
)
+
ϵ
g
(
r
+
Δ
r
)
=
0
⟹
ϵ
≪
f
′
(
r
)
,
Δ
r
=
−
ϵ
g
(
r
)
f
′
(
r
)
f(r) = 0,f(r+\Delta r)+\epsilon g(r+\Delta r)=0\implies \epsilon \ll f'(r),\Delta r = -\frac{\epsilon g(r)}{f'(r)}
f ( r ) = 0 , f ( r + Δ r ) + ϵ g ( r + Δ r ) = 0 ⟹ ϵ ≪ f ′ ( r ) , Δ r = − f ′ ( r ) ϵ g ( r )
4）误差放大因子：相对前向误差与相对后向误差之比。
∣
g
(
r
)
r
f
′
(
r
)
∣
|\frac{g(r)}{rf'(r)}|
∣ r f ′ ( r ) g ( r ) ∣
1）原理：
x
k
+
1
=
x
k
−
f
(
x
k
)
f
′
(
x
k
)
x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}
x k + 1 = x k − f ′ ( x k ) f ( x k )
2）二次收敛：
lim
k
→
+
∞
∣
x
k
+
1
−
r
(
x
k
−
r
)
2
∣
=
S
<
+
∞
\lim\limits_{k \to +\infty} |\frac{x_{k+1}-r}{(x_k-r)^2}| = S \lt +\infty
k → + ∞ lim ∣ ( x k − r ) 2 x k + 1 − r ∣ = S < + ∞
f
f
f
f
(
r
)
=
0
,
f
′
(
r
)
≠
0
f(r) = 0,f'(r)\neq 0
f ( r ) = 0 , f ′ ( r ) = 0
r
r
r
lim
k
→
+
∞
∣
e
k
+
1
e
k
2
∣
=
f
′
′
(
r
)
2
f
′
(
r
)
\lim\limits_{k\to+\infty} |\frac{e_{k+1}}{e_k^2}| = \frac{f''(r)}{2f'(r)}
k → + ∞ lim ∣ e k 2 e k + 1 ∣ = 2 f ′ ( r ) f ′ ′ ( r )
f
f
f
[
a
,
b
]
[a,b]
[ a , b ]
m
+
1
m+1
m + 1
r
r
r
m
m
m
r
r
r
lim
k
→
+
∞
∣
e
k
+
1
e
k
∣
=
m
−
1
m
\lim\limits_{k\to+\infty} |\frac{e_{k+1}}{e_k}| = \frac{m-1}{m}
k → + ∞ lim ∣ e k e k + 1 ∣ = m m − 1
3）重根的改进迭代：
x
k
+
1
=
x
k
−
m
f
(
x
k
)
f
′
(
x
k
)
x_{k+1} = x_k - \frac{mf(x_k)}{f'(x_k)}
x k + 1 = x k − f ′ ( x k ) m f ( x k )
4）迭代失败：迭代过程中出现
f
′
(
x
k
)
=
0
f'(x_k) = 0
f ′ ( x k ) = 0
1）割线替代法：使用差商
f
(
x
i
)
−
f
(
x
i
+
1
)
x
i
−
x
i
+
1
\frac{f(x_i) - f(x_{i+1})}{x_i-x_{i+1}}
x i − x i + 1 f ( x i ) − f ( x i + 1 )
f
′
(
x
i
)
f'(x_i)
f ′ ( x i )
x
i
+
1
=
x
i
−
f
(
x
i
)
(
x
i
−
x
i
+
1
)
f
(
x
i
)
−
f
(
x
i
+
1
)
x_{i+1} = x_i - \frac{f(x_i)(x_i-x_{i+1})}{f(x_i) - f(x_{i+1})}
x i + 1 = x i − f ( x i ) − f ( x i + 1 ) f ( x i ) ( x i − x i + 1 ) 收敛速度介于线性和非线性之间（超线性）
2）试位方法：同二分法类似，属于插值方法
f
(
a
)
f
(
b
)
<
0
,
c
=
a
+
f
(
a
)
f
(
a
)
−
f
(
b
)
(
b
−
a
)
=
b
f
(
a
)
−
a
f
(
b
)
f
(
a
)
−
f
(
b
)
f(a)f(b) \lt 0,c = a + \frac{f(a)}{f(a)-f(b)}(b-a) = \frac{bf(a) - af(b)}{f(a) - f(b)}
f ( a ) f ( b ) < 0 , c = a + f ( a ) − f ( b ) f ( a ) ( b − a ) = f ( a ) − f ( b ) b f ( a ) − a f ( b ) 收敛速度不稳定
3）穆勒方法：计算过
(
x
k
−
2
,
f
(
x
k
−
2
)
)
,
(
x
k
−
1
,
f
(
x
k
−
1
)
)
,
(
x
k
,
f
(
x
k
)
)
(x_{k-2},f(x_{k-2})),(x_{k-1},f(x_{k-1})),(x_{k},f(x_{k}))
( x k − 2 , f ( x k − 2 ) ) , ( x k − 1 , f ( x k − 1 ) ) , ( x k , f ( x k ) )
y
=
p
(
x
)
y=p(x)
y = p ( x )
x
k
x_k
x k
x
k
+
1
x_{k+1}
x k + 1
4）逆二次插值
I
Q
I
IQI
I Q I
x
k
−
2
,
x
k
−
1
,
x
k
x_{k-2},x_{k-1},x_k
x k − 2 , x k − 1 , x k
x
=
p
(
y
)
x=p(y)
x = p ( y )
x
k
+
1
=
x
k
−
r
(
r
−
1
)
(
x
k
−
x
k
−
1
)
+
(
1
−
r
)
s
(
x
k
−
x
k
−
2
)
(
q
−
1
)
(
r
−
1
)
(
s
−
1
)
,
p
=
f
(
x
k
−
2
)
f
(
x
k
−
1
)
,
q
=
f
(
x
k
)
f
(
x
k
−
1
)
,
r
=
f
(
x
k
)
f
(
x
k
−
2
)
x_{k+1} = x_k - \frac{r(r-1)(x_k - x_{k-1})+(1-r)s(x_k - x_{k-2})}{(q-1)(r-1)(s-1)},p = \frac{f(x_{k-2})}{f(x_{k-1})},q = \frac{f(x_{k})}{f(x_{k-1})},r = \frac{f(x_{k})}{f(x_{k-2})}
x k + 1 = x k − ( q − 1 ) ( r − 1 ) ( s − 1 ) r ( r − 1 ) ( x k − x k − 1 ) + ( 1 − r ) s ( x k − x k − 2 ) , p = f ( x k − 1 ) f ( x k − 2 ) , q = f ( x k − 1 ) f ( x k ) , r = f ( x k − 2 ) f ( x k )
5）Brent方法：在有根区间依次选择使用逆二f次插值，割线方法，二分法，以确保至少使误差减少一半
稳定快速收敛
fzero()函数调用使用的就是Brent
fzero(f, [0 1]) % 寻找[0,1]上的根
fzero(f, 1) % 寻找1附近的根
fzero(f) % 自动寻根
fzero(f, optimset('Display', 'iter')) % 显示迭代情况