Math 702: Numerical Analaysis and Approximation II
Instructor Course Section Time Room

Eric Olson Math 702 Numerical Analysis II TR 910:15am AB206
Course Information
 Instructor:
 Eric Olson
 email:
 ejolson at unr edu
 Office:
 Tuesday and Thursday 12 noon DMS 238 and by appointment.
 Homepage:
 http://fractal.math.unr.edu/~ejolson/702
 Supplemental Texts:
 Justin Solomon, Numerical Algorithms: Methods for Computer
Vision, Machine Learning and Graphics, CRC Press, 2015.
 J.W. Thomas, Numerical Partial Differential Equations: Finite
Difference Methods, Springer Verlag, 2010.
 David Kincaid, Ward Cheney, Numerical Analysis: Mathematics of
Scientific Computing, 3rd Edition, American Mathematical Society, 2002.
 J Stoer, R Burlisch, Introduction to Numerical Analysis,
3rd Edition, Springer, 2002.
 William Press, Saul Teukolsky, William Vetterling, Brian Flannery,
Numerical Recipes, 3rd Edition, Cambridge University Press, 2007.

Roger Peyret, Spectral Methods for Incompressible Viscous Flow,
Springer, 2002.

Evans, Blackledge and Yardley, Numerical Methods for Partial
Differential Equations, Springer, 2000.

Evans, Blackledge and Yardley, Analytic Methods for Partial
Differential Equations, Springer, 1999.
Announcements
[19May2019] Answers for Programming and Homework 2
I have created a solution key
for the combined programming and homework assignment number 2.
Even if you received full credit
for your work, please compare my solutions to yours
to note any differences in approach or presentation.
[15May2019] Solution Key for Homework 1
I have created a solution key
for homework one. Even if you received full credit
for your work, please compare my solutions to yours
to note any differences in approach or presentation.
[14May2019] Final Exam
Our final exam will be held in the usual classroom AB206
from 7:30 to 9:30am on Tuesday, May 14.
[13May2019] Solution Key for Project 1
I have created a solution key
for programming project one. Even if you received full credit
for your work, please compare my solutions to yours to note
any differences in approach or presentation.
[07May2019] NavierStokes Part III
Here is the Fortran file continuing
our code for solving
a liddriven flow for the
NavierStokes equations in a twodimensional domain.
[02May2019] Exam 1
There will be an exam in class on May 2 to prepare you for
the final exam.
I've added a few topics to the list we discussed in class.
Tentatively, the following topics are under consideration:
 A proof of the orthogonality lemma for the discrete
Fourier transform.
 A proof of the inversion theorem for the discrete
Fourier transform
 A derivation of the conqueranddivide step for constructing
a fast Fourier transform of size N=2^{n}.
 An analysis showing the time complexity of the fast
Fourier transform is O(n logn).
 An algorithmic description of the Gauss–Newton method
for solving a nonlinear optimization problem.
 A proof of the Poincare inequality
∫_{0}^{L} f(x) − V^{2}dx
≤ L^{2}
∫_{0}^{L} f '(x)^{2}dx
where
V= L^{−1}
∫_{0}^{L} f(x) dx.
 A proof that the velocity profiles of the viscous Burgers
equations exponentially flatten out as
∫_{0}^{L} u − V^{2}
≤ M e^{−αt}
where
V= L^{−1}
∫_{0}^{L} u_{0}.
 A proof of Taylor's theorem which says
f(x+h) = f(x) + h f '(x) + (h^{2}/2!) f ''(x) +
··· + (h^{n}/n!) f^{(n)}(x)
+ R_{n}
where
R_{n} = ∫_{x}^{x+h}
((x+ht)^{n}/n!) f^{(n+1)}(t) dt.
 An explanation with proof how to simplify the geometric series
α^{p} +
α^{p+1} +
··· +
α^{q}
where p<q.
 A proof using Taylor's theorem that the finite difference
approximations have the following orders:
f '(x) ≈ (f(x+h)f(xh))/(2h) + O(h^{2})
and
f ''(x) ≈ (f(x+h)2f(x)+f(xh))/h^{2} + O(h^{2}).
[30Apr2019] NavierStokes Part II
Here is the Fortran file continuing
our code for solving
a liddriven flow for the
NavierStokes equations in a twodimensional domain.
[25Apr2019] NavierStokes Part I
Here is the Fortran file beginning
our code for solving
a liddriven flow for the
NavierStokes equations in a twodimensional domain.
[25Apr2019] Programming and Homework 2
Programming and Homework Project 2 on solving the viscous Burgers equation
is is now available. This combined assignment will be worth 40 points.
The data files and project
description may be found here.
[20Apr2019] Homework 1
Homework 1 covers the fast Fourier transform.
Please work the problems at the end of the handout and turn these
in by April 20. Here are the supporting files.
[18Apr2019] Successive Overrelaxation Method
Here are the Fortran files used
in class to test the SOR method for solving the linear
system Ax=b.
[08Apr2019] Fluid Flow
We will be working through this material
using finitedifferences to approximate solutions
to the Navier–Stokes equations.
[04Apr2019] Aliasing
Here are the Matlab files to show
the effect of aliasing and a simple way to dealias the
convolution used to compute the nonlinear term in the
the viscous Burgers equations.
[02Apr2019] The Viscous Burgers Equation
Here are the Matlab files using
Euler's and the RK2 method to solve the viscous Burgers equations.
[26Mar2019] Euler's Method
Here are the Matlab files using
Euler's method to solve the dissipative wave equation.
[12Mar2019] Solving the Linear Wave Equation
Here are the Matlab files using
the Fourier transform to solve
the dissipative wave equation.
[07Mar2019] Fourier Transform in Fortran
Here are the Fortran files used
in class for the Parallel Fourier Transform.
[08Mar2019] Programming Project 1
Programming Project 1 on solving nonlinear least squares
is is now available. The data files and project
description may be found here.
[28Feb2019] Fourier Transform in Fortran
Here are the Fortan files used
in class to demonstrate the fast Fourier translform.
[26Feb2019] Fourier Transform in Matlab
Here are the Matlab/Octave files used
in class to demonstrate the fast Fourier translform.
[07Feb2019] Nonlinear Optimization
Here are the Matlab/Octave and Maple files
used in class to perform nonlinear optimization using the
GaussNewton least squares method.
[05Feb2019] Frequency Modulation
Here are the Matlab/Octave files used
in class to create data sets resulting from nonlinear
frequency modulation with random noise.
Additional Resources
Grading
2 Homework Assignments 20 points each
2 Programming Projects 20 points each
1 Midterm 30 points
1 Final Exam 50 points

160 points total
Homework and Exams
Final Exam
The final exam will be held on
Tuesday, May 14 from 7:309:30am in AB206.
Equal Opportunity Statement
The Mathematics Department is committed to equal opportunity in education
for all students, including those with documented physical disabilities
or documented learning disabilities. University policy states that it is
the responsibility of students with documented disabilities to contact
instructors during the first week of each semester to discuss appropriate
accommodations to ensure equity in grading, classroom experiences and
outside assignments.
Academic Conduct
Bring your student identification to all exams. Work independently on
all exams and quizzes. Behaviors inappropriate to test taking may disturb
other students and will be considered cheating. Don't talk or pass notes
with other students during an exam. Don't read notes or books while taking
exams given in the classroom.
You may work on the programming assignments in groups of two if desired.
Homework may be discussed freely. If you
are unclear as to what constitutes cheating, please consult with me.
Last updated:
Thu Jan 31 08:55:25 PST 2019