# Numerical Methods I

Warren Weaver Hall, room 101, Thursdays, 5:10 - 7 pm
Courant Institute of Mathematical Sciences
New York University
Fall Semester, 2010

## Lecture materials

Before even the first class, I recommend that you read chapter I of the textbook on your own and make sure you can follow it (otherwise you should refresh your linear algebra background). A more accessible review of linear algebra can be found in these notes by Jonathan Goodman.
Also get a CIMS computer account and start playing with MATLAB, Linux, etc.

### 1. (Sept. 9th) Numerical computing

Read chapter 2 (Principles of Numerical Mathematics) of the textbook. My lecture slides cover basics only. If the book is hard to follow try these nice notes by my colleague Jonathan Goodman.

Here are the MATLAB codes for computing the harmonic sum in double and single precision.

You can see the proof of convergence of the Babylonian fixed-point iteration for computing square roots on Wikipedia.

### 2. (Sept. 16) Solving square linear systems: GEM and LU factorization

Read chapter 3 (Direct Methods for the Solution of Linear Systems) of the textbook. My lecture slides summarize the important points. Here is the MATLAB code MyLU.m.

### 3. (Sept. 23) Linear systems: Sparse matrices, iterative methods, and non-square systems

Finish reading chapter 3 (Direct Methods for the Solution of Linear Systems) of the textbook. We will only cover the first two sections of chapter 4 (Iterative Methods), and the basics of preconditioning. My lecture slides summarize the important points.

### 4. (Sept. 30) Eigenvalue problems

This chapter 5 in the textbook (Approximation of Eigenvalues and Eigenvectors) has lots of details: read at least the sections on conditioning, the power method, and the basic QR iteration. My lecture slides focus on the important points.

### 5. (Oct. 7th) Singular value problems

The SVD is not treated separately in the textbook, rather, it is mentioned in several places. My lecture slides focus on the important points.

### 6. (Oct. 14th) Solving nonlinear equations

This covers chapter 6 in the textbook (Rootfinding for Nonlinear Equations) and the first part of chapter 7 (Nonlinear Systems). My lecture slides focus on the important points.

### 7. (Oct. 21st) Mathematical Programming (Optimization)

This covers the second half of chapter 7 in the textbook (Numerical Optimization). My lecture slides focus on the important points.

### 8. (Oct. 28th) Polynomial Interpolation

This covers chapter 8 in the textbook (Polynomial Interpolation). My lecture slides focus on the important points.

### 9. (Nov 4th and Nov 11th) Orthogonal Polynomials

This covers the first half of chapter 10 in the textbook (Orthogonal Polynomials in Approximation Theory), including the basics of Fourier Transforms. My lecture slides focus on the important points. We will come back to chapter 9, Numerical Integration, later on.

### 10. (Nov 18th) Fourier and Wavelet Transforms

This continues discussing Fourier Transforms, including the FFT algorithm, and briefly introduces wavelets (not covered well in the book).

### 11. (Dec. 2nd) Numerical Integration

This covers chapter 9 in the textbook (Numerical Integration). My lecture slides focus on the important points.

### 12. (Dec. 9th) Monte Carlo Methods

This topic is not covered (much) in the textbook. Instead, take a look at these notes by Jonathan Goodman.