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 > 0
    Return 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 = x
    guess = (low + high) / 2.0
    ctr = 1
    while abs(guess ** 2 - x) > epsilon and ctr <= 100:
        # print ‘low:‘, low, ‘high:‘, high, ‘guess:‘, guess
        if guess**2 < x:
            low = guess
        else:
            high = guess
        guess = (low + high) / 2.0
        ctr += 1
    assert ctr <= 100, ‘Iteration count exceeded‘
    print ‘Bi method. Num. iterations:‘, ctr, ‘Estimate:‘, guess
    return guess

Newton Raphson 牛顿-拉夫森方法

Bisection methods 二分法:

$$\Delta(guess) = guess ^ 2 -x$$
$$\Delta(guess) = 0$$

Newton 求导

guess点的导数 = guess 点的斜率

$$\Delta‘(guess)= 2guess_i = k$$
$$k = {\Delta(guess_i) \over guess_{i+1} - guess_i}$$
$$\Delta(guess_i) = guess_i ^2- x$$

由上面3个式子得到：

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

例子：

$$\Delta(3) = 3 ^2- 16 = -7$$
$$guess_{i+1} = 3 - {\Delta(3) \over 2\times3} = 4.1666$$

def squareRootNR(x, epsilon):
    """Assumes x >= 0 and epsilon > 0
    Return 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 / 2.0
    guess = 0.001
    diff = guess ** 2 - x
    ctr = 1
    while abs(diff) > epsilon and ctr <= 100:
        # print ‘Error:‘, diff, ‘guess:‘, guess
        guess = guess - diff/(2.0*guess)
        diff = guess ** 2 - x
        ctr += 1
    assert ctr <= 100, ‘Iteration count exceeded‘
    print ‘NR method. Num. iterations:‘, ctr, ‘Estimate:‘, guess
    return 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 e
    for 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]