Lecture3Newton迭代法和弦截法

pk?1 ?
f ( pk?1) f '( pk?1 )
k ? 1,2,...
will converge to p for any initial approximation p0 ? [ p ? ? , p ? ? ] .
Newton's Method
证明 g(x) ? x ? f (x)
f '(x)
n ??
??ln
p?
pn?1
?
R ln
p?
pn
? ln A?? ?
0
lim
n ??
??ln
En?1
?
R ln
En
? ln
A?? ?
0
在直角坐标系下，点 ?ln En ,ln En?1 ? 逐渐接近直线
y ? Rx ? ln A ? 0
用最小二乘法可以求出 -R和-lnA，继而求出 R、A。
Speed of Convergence K=1时，则额外要求|g'(p)|<1。
exists a ? ? 0 such that
g '(x) ? k ? 1
for all x ? [ p ? ? , p ? ?.]
Newton's Method
此时，g(x)是 [ p ? ? , p ? ? ] 上的压缩映射。
g(x) ? p ? g(x) ? g( p) ? g '(? ) ? x ? p ? x ? p
?
0
pk ? A ? 0
pk ? A
Speed of Convergence
Definition Assume that
?pk
?? k
?
0
converges
to
p
and set En ? p ? pn for n ? 0 . If two positive
constants A? 0 and R ? 0 exist, and
f(xi ) ? 0 xi ? xi?1
f (xi)
xi?1 ? xi ?
f(xi ) f ?(xi )
slope= f' (xi)
[xi, f(xi)]
f (xi)-0
root
xi+1
xi
x
xi- xi+1
Newton's Method
Theorem 1.5 Assume that f ? C2[a,b]and there
exists a number p ? [a,b], where f(p)=0.
If f '( p) ? 0 , then there exists a ? ? 0 such
that the sequence
? ? p ? k k?0
define by the
iteration
pk
?
g ( pk?1) ?
Convergence Rate for NewtonRaphson Iteration
Theorem Assume that the Newton-Raphson
iteration produces a sequence
? ? p ? k k?0
that
converges to the root p of the function f(x).
g
'(x)
?
1?
?f
'(x)?2 ? f (x) ?f '(x)?2
f
(
x) ?
f (x) f ( x)
?f '(x)?2
由于 g'(x) 连续，且
g '( p) ?
f ( p)
?f '(
f '(p)
p)?2
?
0
then for any given number k in (0, 1), there
If R=1, the convergence is called linear.
If R=2, the convergence is called quadratic.
Speed of Convergence
说明 等价于
lim
n ??
p? p?
pn?1
R
pn
?
lim
n ??
En?1
R
?
A
En
lim
g '(x) ? k ? 1
根据压缩映射定理对于任意选定的初值 p0 ? [ p ? ? , p ? ? ] ，迭代序列收敛于 g(x)的不动点p.
Newton's Method
Corollary 1.2 Assume that A>0 is a real number
and let p0 ? 0 be an initial approximation
lim
n ??
p? p?
pn?1 pn R
?
lim
n ??
En?1 En R
?
A
then the sequence is said to converge to p with order of convergence R. The number A is called the asymptotic error constant.
? If p is a simple root, convergence is
quadratic and
pk ?
A?
pk?1 ?
A pk?1 ?
2
A?
pk?1 ?
A ?2 pk?1
A
2
? ?2
? pk2?1 ? 2 Ap k?1 ? A ? pk?1 ? A
2 pk?1
2 pk?1
pk ? pk ?
A A
?
? ???
pk?1 pk?1
? ?
A ?2 A ???
?
? ???
p0 p0
? ?
A ?2k A ???
?
A
Newton's Iteration for Finding
Square Roots
证明
f ( x) ? x2 ? A
g (x) ? x ?
f (x)
?
x?
x2 ?
A?
x?
A x
f '(x)
2x
2
可以看出，序列
? ? p ? k k?0
是Newton
迭代序列。
以下证明序列收敛于 A 。
Newton's Iteration for Finding Square Roots
to A .Define the sequence
? ? p ? k k?0
using
the recursive rule
pk ?
pk?1 ?
A pk?1
2
k ? 1,2,...
Then the sequence
? ? p ? k k?0
converges
to A .that is
lim
k ??
pk
Numerical Methods
Nonlinear Equations
Newton's Method
? Extend tangent百度文库line from current approximation [xi, f(xi)] to where it crosses the x axis
f(x)
f ?(xi ) ?