3229-Double Precision Root Finding Using Newton's Method.sagews

plot(1 / (1 + 25 * x^2),-1,1)

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
    var('x')    
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(x=z)
     n = len(iterates)
     print "iterations =", n
     html(iterates)
     P = plot(f, (x,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)

Double Precision Root Finding Using Newton's Method

f  c  eps  interval  [removed] [removed] [removed] [removed]

@interact
def _(q1=(-1,(-3,3)), q2=(-2,(-3,3)),
      cmap=['autumn', 'bone', 'cool', 'copper', 'gray', 'hot', 'hsv',
           'jet', 'pink', 'prism', 'spring', 'summer', 'winter']):
     x,y = var('x,y')
     f = q1/sqrt((x+1)^2 + y^2) + q2/sqrt((x-1)^2+(y+0.5)^2)
     C = contour_plot(f, (x,-2,2), (y,-2,2), plot_points=30, contours=15, cmap=cmap)
     show(C, figsize=3, aspect_ratio=1)
     show(plot3d(f, (x,-2,2), (y,-2,2)), figsize=5, viewer='tachyon')

q1  q2  cmap  [removed] [removed] [removed] [removed]

var('x')
@interact
def midpoint(n = slider(1,100,1,4), f = input_box(default = "x^2", type = str), start = input_box(default = "0", type = str), end = input_box(default = "1", type = str)):
    a = N(start)
    b = N(end)
    func = sage_eval(f, locals={'x':x})
    dx = (b-a)/n
    midxs = [q*dx+dx/2 + a for q in range(n)]
    midys = [func(x_val) for x_val in midxs]
    rects = Graphics()
    for q in range(n):
        xm = midxs[q]
        ym = midys[q]
        rects = rects + line([[xm-dx/2,0],[xm-dx/2,ym],[xm+dx/2,ym],[xm+dx/2,0]], rgbcolor = (1,0,0)) + point((xm,ym), rgbcolor = (1,0,0))
    min_y = find_minimum_on_interval(func,a,b)[0]
    max_y = find_maximum_on_interval(func,a,b)[0]
    html('<h3>Numerical integrals with the midpoint rule</h3>')
    html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')
    print "\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0])
    print "Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)])))
    show(plot(func,a,b) + rects, xmin = a, xmax = b, ymin = min_y, ymax = max_y)

n  f  start  end  [removed] [removed] [removed] [removed]

# by Nick Alexander (based on the work of Marshall Hampton)

var('x')
@interact
def midpoint(f = input_box(default = sin(x^2) + 2, type = SR),
    interval=range_slider(0, 10, 1, default=(0, 4), label="Interval"),
    number_of_subdivisions = slider(1, 20, 1, default=4, label="Number of boxes"),
    endpoint_rule = selector(['Midpoint', 'Left', 'Right', 'Upper', 'Lower'], nrows=1, label="Endpoint rule")):

    a, b = map(QQ, interval)
    t = sage.calculus.calculus.var('t')
    func = fast_callable(f(x=t), RDF, vars=[t])
    dx = ZZ(b-a)/ZZ(number_of_subdivisions)
   
    xs = []
    ys = []
    for q in range(number_of_subdivisions):
        if endpoint_rule == 'Left':
            xs.append(q*dx + a)
        elif endpoint_rule == 'Midpoint':
            xs.append(q*dx + a + dx/2)
        elif endpoint_rule == 'Right':
            xs.append(q*dx + a + dx)
        elif endpoint_rule == 'Upper':
            x = find_maximum_on_interval(func, q*dx + a, q*dx + dx + a)[1]
            xs.append(x)
        elif endpoint_rule == 'Lower':
            x = find_minimum_on_interval(func, q*dx + a, q*dx + dx + a)[1]
            xs.append(x)
    ys = [ func(x) for x in xs ]
         
    rects = Graphics()
    for q in range(number_of_subdivisions):
        xm = q*dx + dx/2 + a
        x = xs[q]
        y = ys[q]
        rects += line([[xm-dx/2,0],[xm-dx/2,y],[xm+dx/2,y],[xm+dx/2,0]], rgbcolor = (1,0,0))
        rects += point((x, y), rgbcolor = (1,0,0))
    min_y = min(0, find_minimum_on_interval(func,a,b)[0])
    max_y = max(0, find_maximum_on_interval(func,a,b)[0])

    # html('<h3>Numerical integrals with the midpoint rule</h3>')
    show(plot(func,a,b) + rects, xmin = a, xmax = b, ymin = min_y, ymax = max_y)
    
    def cap(x):
        # print only a few digits of precision
        if x < 1e-4:
            return 0
        return RealField(20)(x)
    sum_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ "f(%s)" % cap(i) for i in xs ]))
    num_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ str(cap(i)) for i in ys ]))
    
    numerical_answer = integral_numerical(func,a,b,max_points = 200)[0]
    estimated_answer = dx * sum([ ys[q] for q in range(number_of_subdivisions)])

    html(r'''
    <div class="math">
    \begin{align*}
      \int_{a}^{b} {f(x) \, dx} & = %s \\\
      \sum_{i=1}^{%s} {f(x_i) \, \Delta x}
      & = %s \\\
      & = %s \\\
      & = %s .
    \end{align*}
    </div>
    ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer))

f  Interval  Number of boxes  Endpoint rule  [removed] [removed] [removed] [removed]