Fall 2023: Numerical Analysis (Math 436/565)


Instructor Dr. Armin Straub
MSPB 313
(251) 460-7262 (please use e-mail whenever possible)
Office hours MW 10:00am-1:00pm, or by appointment
Lecture MW, 2:35-3:50pm, in MSPB 235
Midterm exams The tentative dates for our two midterm exams are:
Wednesday, October 4
Wednesday, November 15
Final exam Monday, December 11 — 3:30-5:30pm
Online grades Homework Scores
Exams: USAonline (Canvas)
Syllabus syllabus.pdf

Lecture sketches and homework

To help you study for this class, I am posting lecture sketches. These are not a substitute for your personal lecture notes or coming to class (for instance, lots of details and motivation are not included in the sketches). I hope that they are useful to you for revisiting the material and for preparing for exams.

Date Sketch Homework
08/23 lecture01.pdf Homework Set 1: Problems 1-4 (due 9/6)
08/28 lecture02.pdf Homework Set 1: Problems 5-8 (due 9/6)
08/30 lecture03.pdf Homework Set 2: Problems 1-3 (due 9/13)
09/06 lecture04.pdf Homework Set 2: Problems 4-7 (due 9/13)
09/11 lecture05.pdf Homework Set 3: Problems 1-5 (due 9/20)
09/13 lecture06.pdf Homework Set 3: Problem 6 (due 9/20)
09/18 lecture07.pdf Homework Set 4: Problems 1-3 (due 9/27)
09/20 lecture08.pdf Homework Set 4: Problems 4-6 (due 9/27)
09/25 lecture09.pdf Homework Set 5: Problems 1-4 (due 10/4)
09/27 lecture10.pdf Homework Set 5: Problems 5-6 (due 10/4)
exam practice problems (as well as solutions) are posted below
10/02 review get ready for the midterm exam on 10/4 (Wednesday)
10/09 lecture11.pdf Homework Set 6: Problems 1-3 (due 10/23)
10/11 lecture12.pdf Homework Set 6: Problem 4 (due 10/23)
10/16 lecture13.pdf Homework Set 6: Problems 5-6 (due 10/23)
10/18 lecture14.pdf Homework Set 7: Problem 1 (due 11/6)
10/23 lecture15.pdf Homework Set 7: Problem 2 (due 11/6)
10/25 lecture16.pdf Homework Set 7: Problem 3 (due 11/6)
10/30 lecture17.pdf Homework Set 7: Problem 4 (due 11/6)
11/01 lecture18.pdf Homework Set 8: Problems 1-2 (due 11/15)
11/06 lecture19.pdf Homework Set 8: Problem 3 (due 11/15)
11/08 lecture20.pdf Homework Set 8: Problem 4 (due 11/15)
exam practice problems (as well as solutions) are posted below
11/13 review get ready for the midterm exam on 11/15 (Wednesday)
11/20 lecture21.pdf Homework Set 9: Problem 1 (due 12/4)
11/27 lecture22.pdf Homework Set 9: Problem 2 (due 12/4)
11/29 lecture23.pdf complete all outstanding homework
12/04 lecture24.pdf final exam practice problems (as well as solutions) are posted below
12/06 review get ready for the final exam on 12/11 (Monday)
lectures-all.pdf (all lecture sketches in one big file)
Overview of all homework problems

About the homework

  • Homework problems are posted for each unit. Homework is submitted online, and you have an unlimited number of attempts. Only the best score is used for your grade.

    Most problems have a random component (which allows you to continue practicing throughout the semester without putting your scores at risk).

  • Aim to complete the problems well before the posted due date.

    A 15% penalty applies if homework is submitted late.

  • Collect a bonus point for each mathematical typo you find in the lecture notes (that is not yet fixed online), or by reporting mistakes in the homework system. Each bonus point is worth 1% towards a midterm exam.

    The homework system is written by myself in the hope that you find it beneficial. Please help make it as useful as possible by letting me know about any issues!

Exams and practice material

The following material will help you prepare for the exams.


If you just want to run a handful quick computations (without saving your work), you can use the text box below. However, keep in mind that the code entered in this box is run as a Python script. That means that we need to use print(...) if we want to see any output.

One convenient option for running Python code in the cloud is Colab by Google Research. You can sign in with any Google account (since Google is managing university email, you can use your university account).

Perform your first computation by entering something like "1+2" and pressing Shift+Enter.

Here are some other things to try:

  • The following code plots sine versus a quadratic interpolation:
    from numpy import linspace, pi, sin
    from scipy import interpolate
    import matplotlib.pyplot as plt
    xpoints = [0, pi/2, pi]
    ypoints = [sin(x) for x in xpoints]
    poly = interpolate.lagrange(xpoints, ypoints)
    xplot = linspace(0, pi, 100)
    plt.plot(xpoints, ypoints, 'o', xplot, sin(xplot), '-', xplot, poly(xplot), ':')
  • Interpolation using equally spaced points versus Chebyshev nodes:
    from numpy import linspace, pi, cos
    from scipy import interpolate
    import matplotlib.pyplot as plt
    def f(x):
        return 1/(1+25*x**2)
    n = 5
    xpoints = linspace(-1, 1, n)
    ypoints = [f(x) for x in xpoints]
    poly = interpolate.lagrange(xpoints, ypoints)
    xpoints2 = [cos((2*j+1)*pi/(2*n)) for j in range(n)]
    ypoints2 = [f(x) for x in xpoints2]
    poly2 = interpolate.lagrange(xpoints2, ypoints2)
    xplot = linspace(-1, 1, 100)
    plt.plot(xpoints, ypoints, 'o', xplot, poly(xplot), ':')
    plt.plot(xpoints2, ypoints2, 'o', xplot, poly2(xplot), ':')
    plt.plot(xplot, f(xplot), '-')
  • We can construct a (natural) cubic spline and plot it together with the knots:
    from numpy import linspace
    from scipy import interpolate
    import matplotlib.pyplot as plt
    xpoints = [1, 2, 4, 5, 7]
    ypoints = [2, 1, 4, 3, 2]
    spline = interpolate.CubicSpline(xpoints, ypoints, bc_type='natural')
    xplot = linspace(1, 7, 100)
    plt.plot(xplot, spline(xplot), '-', label='spline (natural)')
    plt.plot(xpoints, ypoints, 'o', label='knots')
    Other standard choices for the boundary conditions bc_type include 'not-a-knot' (the default) as well as 'clamped' and 'periodic' (this one requires the first and last point to have the same y-coordinates).
  • The following compares forward differences (order 1), central differences (order 2) and the Richardson extrapolation of the latter (order 4) for approximating the derivative of $ f'(1) = 2 \ln(2) $ where $ f(x) = 2^x $. Observe how all approximations turn sour if $ h = 10^{-n} $ is too small but how we are able to get a better "best approximation" using methods of higher order:
    from math import log
    def forward_difference(f, x, h):
        return (f(x+h)-f(x))/h
    def central_difference(f, x, h):
        return (f(x+h)-f(x-h))/(2*h)
    def central_difference_richardson(f, x, h):
        return (-f(x+2*h)+8*f(x+h)-8*f(x-h)+f(x-2*h))/(12*h)
    def f(x):
        return 2**x
    print([forward_difference(f, 1, 10**-n) - 2*log(2) for n in range(12)])
    print([central_difference(f, 1, 10**-n) - 2*log(2) for n in range(12)])
    print([central_difference_richardson(f, 1, 10**-n) - 2*log(2) for n in range(12)])
  • The following implements the trapezoidal rule and applies it for approximating the integral of $ 1/x $ from $ 1 $ to $ 3 $ (which is $ \log(3) $). Observe how the errors confirm that the trapezoidal rule has order 2.
    from math import log
    def trapezoidal_rule(f, a, b, n):
        h = (b - a) / n
        integral = f(a) + f(b)
        for i in range(1,n):
            integral += 2*f(a + i*h)
        return h/2*integral
    def f(x):
        return 1/x
    print([trapezoidal_rule(f, 1, 3, 10**n) - log(3) for n in range(1,6)])
  • In the following, we apply a Taylor method of order 3 to solve the differential equation $ y' = \cos(x) y $ with $ y(0) = 1 $. For comparison, we also plot the exact solution $ y(x) = e^{\sin(x)} $.
    from numpy import linspace, exp, cos, sin
    import matplotlib.pyplot as plt
    def taylor_3_cosy(x0, y0, xmax, n):
        h = (xmax - x0) / n
        ypoints = [y0]
        for i in range(n):
            y0 = y0 + cos(x0)*y0*h + 1/2*(cos(x0)**2-sin(x0))*y0*h**2 + \
            x0 = x0 + h
        return ypoints
    xpoints = linspace(0, 2, 100)
    plt.plot(xpoints, exp(sin(xpoints)), '-', label='y(x)')
    nn = [2, 4, 8]
    for n in nn:
        xpoints = linspace(0, 2, n+1)
        ypoints = taylor_3_cosy(0, 1, 2, n)
        plt.plot(xpoints, ypoints, ':o', label='n = %s' % n)
  • The following illustrates that the midpoint method converges with a global error of 2:
    from math import e, cos, sin
    def midpoint(f, x0, y0, xmax, n):
        h = (xmax - x0) / n
        ypoints = [y0]
        for i in range(n):
            y0 = y0 + f(x0+h/2,y0+f(x0,y0)*h/2)*h
            x0 = x0 + h
        return ypoints
    def f_cosx_y(x, y):
        return cos(x)*y
    print([midpoint(f_cosx_y, 0, 1, 2, 10**n)[-1] - e**sin(2) for n in range(6)])


If you take this class for graduate credit, you need to complete a project. The idea is to gain additional insight into a topic that you are particularly interested in. Some suggestions for projects are listed further below.

  • The outcome of the project should be a short paper (about 5 pages)
    • in which you introduce the topic, and then
    • describe how you explored the topic.
    Here, exploring can mean (but is not limited to)
    • computations or visualizations you did in, preferably, Python,
    • working out representative examples, or
    • combining different sources to get an overall picture.

Each project should have either a computational part (this is a great chance to learn more Python!) or have a more mathematical component. Here are some ideas:

  • Read the recent article on Newton's Method Without Division by Jeffrey D. Blanchard and Marc Chamberland. Explain the main result and implement several examples to illustrate it.
  • Describe and implement the Illinois method for computing roots.
  • Introduce and explore Chebyshev polynomials. For instance, prove that their roots provide the optimal nodes for minimizing the generic error bound for polynomial interpolation.
  • Introduce orthogonal polynomials and explore how they are used in solving differential equations or for approximating other functions.
  • Explore the discrete Fourier transform.
  • Describe and implement a fast multiplication algorithm such as the Karatsuba algorithm.
You are also very welcome to come up with your own ideas for a project. Please talk to me early with about idea, so I can give you a green light.