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Glossary

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Definition

Drawing

drawing:mytest

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A graphical illustration of $\lim_{x -> 0} sin(x)/x =1$

x=var('x')
@interact
def _(x = slider(-7/10,7/10,1/20,1/2)):
    html('<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>')
    html('Below is a unit circle, so the length of the <font color=red>red line</font> is |sin(x)|')
    html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.') 
    html('is |sin(x)| and |tan(x)|, respectively where x is length of the arc.')
    html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|')
    html('and it follows easily from this that')
    html('cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.')
    html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.')
    if not (x == 0):
        pretty_print("sin(x)/x = "+str(sin(float(x))/float(x)))
    elif x == 0:
        pretty_print("The limit of sin(x)/x as x tends to 0 is 1")
    C=circle((0,0),1, rgbcolor='black')
    mvp = (cos(x),sin(x));tpt = (1, tan(x))
    p1 = point(mvp, pointsize=30, rgbcolor='red'); p2 = point((1,0), pointsize=30, rgbcolor='red')
    line1 = line([(0,0),tpt], rgbcolor='black'); line2 = line([(cos(x),0),mvp], rgbcolor='red') 
    line3 = line([(0,0),(1,0)], rgbcolor='black'); line4 = line([(1,0),tpt], rgbcolor='blue')
    result = C+p1+p2+line1+line2+line3+line4
    result.show(aspect_ratio=1, figsize=[3,3], axes=False)

Root Finding Using Bisection

by William Stein

def bisect_method(f, a, b, eps):
    try:
        f = f._fast_float_(f.variables()[0])
    except AttributeError:
        pass
    intervals = [(a,b)]
    two = float(2); eps = float(eps)
    while True:
        c = (a+b)/two
        fa = f(a); fb = f(b); fc = f(c)
        if abs(fc) < eps: return c, intervals
        if fa*fc < 0:
            a, b = a, c
        elif fc*fb < 0:
            a, b = c, b
        else:
            raise ValueError, "f must have a sign change in the interval (%s,%s)"%(a,b)
        intervals.append((a,b))
html("<h1>Double Precision Root Finding Using Bisection</h1>")
@interact
def _(f = cos(x) - x, a = float(0), b = float(1), eps=(-3,(-16..-1))):
     eps = 10^eps
     print "eps = %s"%float(eps)
     try:
         time c, intervals = bisect_method(f, a, b, eps)
     except ValueError:
         print "f must have opposite sign at the endpoints of the interval"
         show(plot(f, a, b, color='red'), xmin=a, xmax=b)
     else:
         print "root =", c
         print "f(c) = %r"%f(c)
         print "iterations =", len(intervals)
         P = plot(f, a, b, color='red')
         h = (P.ymax() - P.ymin())/ (1.5*len(intervals))
         L = sum(line([(c,h*i), (d,h*i)]) for i, (c,d) in enumerate(intervals) )
         L += sum(line([(c,h*i-h/4), (c,h*i+h/4)]) for i, (c,d) in enumerate(intervals) )
         L += sum(line([(d,h*i-h/4), (d,h*i+h/4)]) for i, (c,d) in enumerate(intervals) )
         show(P + L, xmin=a, xmax=b)

attachment:bisect.png

Newton's Method

Note that there is a more complicated Newton's method below.

by William Stein

def newton_method(f, c, eps, maxiter=100):
    x = f.variables()[0]
    fprime = f.derivative(x)
    try:
        g = f._fast_float_(x)
        gprime = fprime._fast_float_(x)
    except AttributeError:
        g = f; gprime = fprime
    iterates = [c]
    for i in xrange(maxiter):
       fc = g(c)
       if abs(fc) < eps: return c, iterates
       c = c - fc/gprime(c)
       iterates.append(c)
    return c, iterates
html("<h1>Double Precision Root Finding Using Newton's Method</h1>")
@interact
def _(f = x^2 - 2, c = float(0.5), eps=(-3,(-16..-1)), interval=float(0.5)):
     eps = 10^(eps)
     print "eps = %s"%float(eps)
     time z, iterates = newton_method(f, c, eps)
     print "root =", z
     print "f(c) = %r"%f(z)
     n = len(iterates)
     print "iterations =", n
     html(iterates)
     P = plot(f, z-interval, z+interval, rgbcolor='blue')
     h = P.ymax(); j = P.ymin()
     L = sum(point((w,(n-1-float(i))/n*h), rgbcolor=(float(i)/n,0.2,0.3), pointsize=10) + \
             line([(w,h),(w,j)],rgbcolor='black',thickness=0.2) for i,w in enumerate(iterates))
     show(P + L, xmin=z-interval, xmax=z+interval)