Lecture5： Bisection methods , Newton/Raphson, introduction to lists二分法，牛顿，拉复生方法，列表

Bisection methods 二分法

注意：
# bug: when x < 1, sqrt(x) > x = high eg.x=0.25 sqrt(x) = 0.5
# fix bug: high = max(x, 1.0)

def squareRootBi(x, epsilon):"""Assumes x >= 0 and epsilon > 0Return y s.t. y*y is within epsilon of x"""assert x >= 0, 'x must be non-negative, not' + str(x)assert epsilon > 0, 'epsilon must be positive, not' + str(epsilon)low = 0# bug: when x < 1, sqrt(x) > x = high eg.x=0.25 sqrt(x) = 0.5# fix bug: high = max(x, 1.0)high = xguess = (low + high) / 2.0ctr = 1while abs(guess ** 2 - x) > epsilon and ctr <= 100:# print 'low:', low, 'high:', high, 'guess:', guessif guess**2 < x:low = guesselse:high = guessguess = (low + high) / 2.0ctr += 1assert ctr <= 100, 'Iteration count exceeded'print 'Bi method. Num. iterations:', ctr, 'Estimate:', guessreturn guess

Newton Raphson 牛顿-拉夫森方法

Bisection methods 二分法:

Δ(guess)=guess2−x

\Delta(guess) = guess ^ 2 -x

Δ(guess)=0

\Delta(guess) = 0

Newton 求导

guess点的导数 = guess 点的斜率

Δ′(guess)=2guessi=k

\Delta'(guess)= 2guess_i = k

k=Δ(guessi)guessi+1−guessi

k = {\Delta(guess_i) \over guess_{i+1} - guess_i}

Δ(guessi)=guess2i−x

\Delta(guess_i) = guess_i ^2- x

由上面3个式子得到：

guessi+1=guessi−Δ(guessi)2guessi

guess_{i+1} = guess_i - {\Delta(guess_i) \over 2guess_i}

例子：

Δ(3)=32−16=−7

\Delta(3) = 3 ^2- 16 = -7

guessi+1=3−Δ(3)2×3=4.1666

guess_{i+1} = 3 - {\Delta(3) \over 2\times3} = 4.1666

def squareRootNR(x, epsilon):"""Assumes x >= 0 and epsilon > 0Return y s.t. y*y is within epsilon of x"""assert x >= 0, 'x must be non-negative, not' + str(x)assert epsilon > 0, 'epsilon must be positive, not' + str(epsilon)x = float(x)guess = x / 2.0guess = 0.001diff = guess ** 2 - xctr = 1while abs(diff) > epsilon and ctr <= 100:# print 'Error:', diff, 'guess:', guessguess = guess - diff/(2.0*guess)diff = guess ** 2 - xctr += 1assert ctr <= 100, 'Iteration count exceeded'print 'NR method. Num. iterations:', ctr, 'Estimate:', guessreturn guess

Lists 列表

3wschool Python 列表

non-scalar types 非基本类型

Tuple 元组（immutable 不可改变的）
String 字符串（immutable 不可改变的）
List 列表（mutable 可改变的）

>>> Techs = ['MIT', 'Cal Tech']
>>> print Techs
['MIT', 'Cal Tech']
>>> Ivys = ['Harvard', 'Yale', 'Brown']
>>> print Ivys
['Harvard', 'Yale', 'Brown']
>>> Univs = []
>>> Univs.append(Techs)
>>> print Univs
[['MIT', 'Cal Tech']]
>>> Univs.append(Ivys)
>>> print Univs
[['MIT', 'Cal Tech'], ['Harvard', 'Yale', 'Brown']]

for e in Univs:print efor c in e: print c['MIT', 'Cal Tech']
MIT
Cal Tech
['Harvard', 'Yale', 'Brown']
Harvard
Yale
Brown

>>> Univs = Techs + Ivys
>>> print Univs
['MIT', 'Cal Tech', 'Harvard', 'Yale', 'Brown']

>>> Ivys.remove('Harvard')
>>> print Ivys
['Yale', 'Brown']

>>> Ivys[1] = -1
>>> print Ivys
['Yale', -1]

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MIT公开课： Python 笔记6 二分法，牛顿-拉夫森方法，列表

Lecture5： Bisection methods , Newton/Raphson, introduction to lists二分法，牛顿，拉复生方法，列表

Bisection methods 二分法

Newton Raphson 牛顿-拉夫森方法

Lists 列表

MIT公开课： Python 笔记6 二分法，牛顿-拉夫森方法，列表相关推荐

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