开方实现

一种是二分查找的使用

二分法，在很多排序，查找等问题中经常使用

def squareRootBi(x, epsilon):
    """Assume x >= 0 and epsilon > 0
       Return y s.t. y*y is within epsilon of x"""
    #假设x>=0且ε>0，返回y，使得y*y在x的ε内
    assert x >= 0, 'x必须为非负数，而不是' + str(x)
    assert epsilon > 0, 'ε必须为正数，而不是' + str(epsilon)
    low = 0
    high = x
    #high = max(x,1)
    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, '循环计数次数超出范围'
    print 'Bi方法，循环数', ctr, '估值', guess
    return guess

另外一种，是求切点的值的思路，使用了牛顿迭代法

def squareRootNR(x, epsilon):
    """Assume x >= 0 and epsilon > 0
       Return y s.t. y*y is within epsilon of x"""
    #假设x>=0且ε>0，返回y，使得y*y在x的ε内
    assert x >= 0, 'x必须为非负数，而不是' + str(x)
    assert epsilon > 0, 'ε必须为正数，而不是' + 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 '差值:', diff, '猜想值:', guess
        guess = guess - diff/(2.0*guess)
        diff = guess**2 - x
        ctr += 1
    assert ctr <=100, '循环计数次数超出范围'
    print 'NR方法，循环数', ctr, '估值', guess
    return guess