### Location & Time

MWF 1.25 AM-2.15 PM

218A Hosler Building

### Instructor

Kamesh Madduri

*Office hours*:

MWF 2.30 PM-3.30 PM (343E IST,

Skype: madduri.psu)

Appointment (

cal)

### TA

Diman Tootaghaj

*Office hours*:

Tue 8.30 AM-9.30 AM,

Th 11.30 AM-12.30 PM (338E IST)

### Important Dates

Aug 27. First class

Sep 07. HW1 posted

Sep 14. HW1 due

Sep 19. HW2 posted

Sep 26. HW2 due

Sep 28. Midterm 1

Oct 17. HW3 posted

Oct 24. HW3 due

Nov 02. HW4 posted

Nov 05. Midterm 2

Nov 12. HW4 due

Dec 03. HW5 posted

Dec 10. HW5 due

Dec 19. Final exam

### Overview

This class introduces students to key topics in numerical analysis, with emphasis on the computational mathematics behind numerical methods and on proving theorems. CMPSC/MATH 451 is a related course that has a significant overlap with topics studied in CMPSC/MATH 455 and CMPSC/MATH 456. Students may obtain credit for either this class or CMPSC/MATH 451, but not both.

### Topics

Floating Point Arithmetic, Approximations in Scientific Computing, Non-linear equations in one variable, Linear Systems: Direct Methods, Polynomial Interpolation, Piecewise Polynomial Interpolation, Numerical Differentiation, Numerical Integration.

### Syllabus

Last updated Oct 11.

### Schedule

- Lecture 1, Aug 27
- Course Logistics
- Introduction to Scientific Computing

- Lecture 2, Aug 29
- Taylor's theorem
- Types of error

- Lecture 3, Aug 31
- Analyzing algorithms
- Problem conditioning

- Lecture 4, Sep 5
- Floating-point systems

- Lecture 5, Sep 7
- Rounding unit
- Roundoff error accumulation
- Octave tutorial

- Lecture 6, Sep 10
- IEEE standard
- Cancellation error
- More Octave examples

- Lecture 7, Sep 12
- Non-linear equations: Introduction
- Interval bisection

- Lecture 8, Sep 14
- Fixed point iteration

- Lecture 9, Sep 17
- Order of convergence
- Newton's method
- Examples

- Lecture 10, Sep 19
- Secant method
- Octave examples

- Lecture 11, Sep 21
- Secant method order of convergence

- Lecture 12, Sep 24
- HW1 discussion
- Multiple roots

- Lecture 13, Sep 26
- Function minimization using Newton's method
- Linear Systems Introduction

- Lecture 14, Oct 1
- Matrices Review
- Vector and Matrix Norms

- Lecture 15, Oct 3
- Midterm 1 discussion

- Lecture 16, Oct 5
- Midterm 1 discussion (continued)
- Special classes of matrices
- Operation counts

- Lecture 17, Oct 8
- Forward and Backward substitution
- Gaussian Elimination

- Lecture 18, Oct 10
- Gaussian Elimination (continued)
- Examples

- Lecture 19, Oct 12
- GE with Pivoting
- Octave Linear Systems tutorial

- Lecture 20, Oct 15
- Octave code for Gaussian Elimation
- GE with Complete pivoting
- Uniqueness of GE

- Lecture 21, Oct 17
- Cholesky decomposition
- Example

- Lecture 22, Oct 19
- Sparse matrices, banded systems
- Error analysis, Matrix conditioning

- Lecture 23, Oct 22
- Linear systems exercises

- Lecture 24, Oct 24
- Linear systems exercises

- Lecture 25, Oct 26
- Interpolation: Introduction
- Monomial Basis

- Lecture 26, Oct 31
- Lagrange Interpolation
- Newton's basis

- Lecture 27, Nov 2
- Divided Differences

- Lecture 28, Nov 5
- Review for Midterm exam 2

- Lecture 29, Nov 7
- Midterm exam 2 solutions

- Lecture 30, Nov 9
- Error in polynomial interpolation
- Chebyshev interpolation

- Lecture 31, Nov 12
- Piecewise linear interpolation
- Piecewise cubic interpolation

- Lecture 32, Nov 26
- Review of piecewise interpolation
- Cubic spline interpolation

- Lecture 33, Nov 28
- Numerical Differentiation, finite difference schemes
- Richardson extrapolation

- Lecture 34, Nov 30
- Numerical Integration, Quadrature rules
- Newton-Cotes quadrature rules

- Lecture 35, Dec 3
- Precision of Newton-Cotes rules
- Composite quadrature
- Octave tutorial

- Lecture 36, Dec 5
- Clenshaw-Curtis quadrature
- Gaussian quadrature

- Lecture 37, Dec 7
- Adaptive quadrature
- Romberg integration
- Octave examples

- Lecture 38, Dec 10
- Review lecture

- Lecture 39, Dec 12
- Review lecture

- Lecture 40, Dec 14
- Review lecture

### Textbook

Uri M. Ascher and Chen Greif,

A First Course in Numerical Methods, SIAM, 2011.

### Prerequisites

MATH 220; MATH 230 or MATH 231; Three credits of programming (CMPSC 201 or CMPSC 202 or CMPSC 121).

### Class material

Presentations, lecture notes, and homework assignments will be posted on Angel.