## Fall 17: Numerical Methods I @ NYU Courant

## Instructor and organization:

Georg Stadler, Warner Weaver Hall Office #1111

Course numbers: MATH-GA 2010.001/CSCI-GA 2420.001

Lectures: Thursday 5:10-7:00pm; If you are interested in the class and are not registed, please email me.

Location: Warren Weaver Hall #101

Office Hours: To be announded or by appointment-please send an email.

If you email me about the class, please add [num1] in your subject
line, or use
**this
link**.

We will use **Piazza** for as email list and
discussion forum for this class. If you are registered for this class
you will receive an invitation to join the course on
Piazza at the beginning of the semester. Otherwise please email me and I will add you.

## Literature

Note that all Springer books are available from NYU’s campus and you can order a MyCopy softcover print for 25$.

Deuflhard, P. & Hohmann, A. (2003). **Numerical Analysis in Modern
Scientific
Computing**. Texts
in Applied Mathematiks [Series, Bk. 43]. New York, NY:
Springer-Verlag.

# Further Reading (available on reserve at the Courant Library.):

Quarteroni, A., Sacco, R., & Saleri, F. (2006). **Numerical
Mathematics (2nd ed.)**
Texts in Applied Mathematics [Series, Bk. 37]. New York, NY:
Springer-Verlag.

Bau III, D., & Trefethen, L.N. (1997). **Numerical Linear
Algebra**. Philadelphia,
PA: Society for Industrial & Applied Mathematics.

M. Overton (2004): **Numerical Computing with IEEE Floating Point
Arithmetic**, SIAM.

# If you need to brush up your MATLAB:

Gander, W., Gander, M.J., & Kwok, F. (2014). **Scientific Computing -
An Introduction Using Maple and
MATLAB**. Texts in Computation Science and Engineering [Series, Vol. 11].
New York, NY: Springer-Verlag.

Moler, C: (2004) **Numerical Computing with
Matlab**, SIAM.

## Class list

1) Sep 7: Organization; Overview of research on numerical methods at
CIMS; Motivational examples; Conditioning of
problems; Sources of errors; **Intro slides**,
**iPad notes on conditioning**

2) Sep 14: Stability of algorithms, measure of convergence speed;
Computer representation of numbers;
**Slides**, **iPad notes on floating point
representation**

3) Sep 21: Solving linear systems; forward/backward substitution, LU
decomposition, Choleski, solver libraries in Matlab;
**Slides**.

4) Sep 28: Data fitting and linear least squares problems, normal
equations, QR factorization.
**Slides**. Reading: Deuflhard/Hohmann,
Sec 3.1

5) Oct 5: Givens and Householder (D/H, Sec 3.2); Solution of nonlinear
equations (D/H, Sec 4.1-4.2): Fixed points and Newton’s method for
system of equations **Slides**; Fixed point
MATLAB **example file** from class.

6) Oct 12: Convergence and examples of Newton’s methods, Nonlinear
least squares (D/H, Sec
4.3). **Slides**;
MATLAB **example file** for
Newton method in 2D from class.

7) Oct 19: Data fitting with different objectives;
Nonlinear optimization, optimality conditions, convexity, descent
methods. **Slides**;

8) Oct 26: Newton method in optimization, linesearch, convergence of
descent methods. Introduction to eigenvalues. For slides see link
from previous class. **Summary** of quadratic
forms and convergence of steepest descent method. Descent with line
search **Matlab file**.

9) Nov 2: Eigenvalues and eigenvectors, power method and variants, QR
algorithm. **Slides**

10) Nov 9: SVD, orthongonal polynomials. **Slides**

11) Nov 16: Polynomial bases, interpolation, divided differences,
convergence. **Slides**

12) Nov 30: Trigonometric interpolation, Quadrature. **Slides on
interpolation**, and **Slides on quadrature**

13) Dec 7: Quadrature, Solution of large linear systems, Jacobi,
Gauss-Seidel. Updated quadrature slides see above. **Slides on
iterative solvers**

14) Dec 14: Solution of large linear systems; Relaxation of stationary iterative solvers, matrix-free methods, conjugate gradients. See (updated) slides from previous lecture.

15) Dec 21: Class final

## Homework assignments

*) **Homework #1 (as
PDF)**; You can also get the **TEX
file** to use as starting point if you want
to type your solutions in LaTeX.

*) **Homework #2 (as
PDF)** and the **LaTeX
file**.

*) **Homework #3 (as
PDF)** and the **LaTeX
file**.

*) **Homework #4 (as
PDF)** and the **LaTeX
file**.

*) **Homework #5 (as
PDF)** and the **LaTeX
file**.

*) **Homework #6 (as
PDF)** and the **LaTeX
file**.

*) **Homework #7 (as
PDF)** and the **LaTeX
file**.

## FAQs:

1) *I am a master student and am undecided if I should take your
class or the one-term Scientific Computing class. What’s the
difference?* In Numerical Methods I and II, more mathematical details
are presented as they are mainly intended for PhD students. The
Scientific Computing class covers most parts of Numerical Methods I
and some parts of Numerical Methods II. If you do not intend to take
the second part of Numerical Methods I, it makes more sense to take
Scientific Computing instead.

2) *Can you give a rough outline of the content for Numerical Methods
I and II?* The first part focuses on many aspects of numerical
mathematics (sources of errors, solution of linear and nonlinear
systems, least-squares problems, interpolation and quadrature) but
does not include the numerical solution of differential equations
(ODEs and PDEs), which is the main topic of part II in the spring
semester.

3) *Can I use Octave/Python instead of Matlab?* Sure.

4) *What is the work load in this class?* There will be a long-ish
homework assignment every two weeks, which involves a mix of
theoretical and numerical and programming exercises. These will expose
you to the material covered in class and I consider them critical to
understanding the methods and algorithms we discuss in class.

5) *Will there be exams?* There will be a final that will count for
about 50% of our grade. The homeworks you hand in amount to the
remaining 50% of our grade.

6) *I don’t have any programming experience in Matlab/Python. Can I
take your class?* I will discuss coding aspects, but this is not a
programming class. You are expected to have basic Matlab/Python
knowledge by the beginning of the class. If you are not sure about
your background, go through the first chapters of the above
recommended Matlab books (e.g., C. Moler’s book), and talk to
me. About half of the homework assignments will require basic
Matlab/Python plotting, computing or scripting. This is an important
part of the class.

7) *Do I need to know a compiled language (C,C++,Fortran) for your
class?* No. However, I will encourage you to experiment with
compiled languages, most likely through a few (simple) extra credit
homework problems.

8) *Can we collaborate for the homework assignments?* You are welcome
to discuss problems and talk to your colleagues, but you must write
every line of code and of your homework solutions yourself. See also
**NYU’s policy on Academic
Integrity**.

9) *How can I get access to Matlab?* CIMS has computer rooms you can
use. You can also purchase a student license for Matlab from the
computer store. There are also free alternatives to Matlab, e.g.,
Octave or Python.