### Location & Time

Section 1: MWF 11.15AM-12.05PM

Section 2: MWF 12.20PM-01.10PM

110 Walker Building

### Instructor

Kamesh Madduri

*Office hours*:

MW 1.30PM-2.30PM (343E IST)

F 1.30PM-2.30PM (Skype,

ID madduri.451.officehours)

Appointment (

cal)

### TA

Ping Chi

*Office hours*:

TTh 1.00PM-2.00PM (338E IST)

### Important Dates

Aug 22. First class

Sep 02. HW1 given out

Sep 02. HW1 due

Sep 16. HW2 given out

Sep 23. HW2 due

Sep 26. Midterm exam 1

Oct 07. HW3 given out

Oct 14. HW3 due

Oct 26. HW4 given out

Oct 31. Midterm exam 2

Nov 04. HW4 due

Nov 14. Extra-credit HW given out

Nov 21. HW5 given out

Nov 30. Extra-credit HW due

Dec 05. HW5 due

### Overview

This class introduces students to key ideas behind numerical computation, with
emphasis on the implementation of common numerical methods. CMPSC/MATH 455 is a
related course that covers a subset of the topics that we will study in this class.
Students may obtain credit for either this class or CMPSC/MATH 455, but not both.

### Topics

Approximations in Scientific Computing, Floating Point Arithmetic, Systems of Linear
Equations, Linear Least Squares, Non-linear Equations, Interpolation, Numerical Integration,
Numerical Differentiation, Initial Value Problems for ODEs.

### Syllabus

Last updated Sep 19.

### Schedule

- Lecture 1, Aug 22
- Course Logistics
- Introduction to Scientific Computing
- Error Analysis
- Lecture 2, Aug 24
- Error Analysis
- Sensitivity and Conditioning
- Review
- Lecture 3, Aug 26
- Floating-point systems
- IEEE-754 standard
- Lecture 4, Aug 29
- Floating-point computations
- Octave tutorial
- Lecture 5, Aug 31
- Octave tutorial (continued)
- Linear systems introduction
- Vector Norms
- Lecture 6, Sep 2
- Matrix Norms and Condition Number
- Error Bounds
- HW 1 online
- Lecture 7, Sep 7
- Transformations
- Triangular systems
- Forward and Back substitution
- Lecture 8, Sep 9
- Gaussian Elimination
- HW 1 due
- Lecture 9, Sep 12
- LU Factorization with Partial Pivoting
- Exercises
- Lecture 10, Sep 14
- HW1 discussion
- Linear systems with Octave
- Lecture 11, Sep 16
- Octave examples continued
- Gauss-Jordan elimination
- HW 2 online
- Lecture 12, Sep 19
- Solving modified systems
- Special types of linear systems
- Lecture 13, Sep 21
- Linear systems software
- Non-linear systems: Introduction
- Lecture 14, Sep 23
- Review of floating-point arithmetic
- Lecture 15, Sep 26
- Review of linear systems
- Midterm Exam 1
- Lecture 16, Sep 28
- Midterm Exam solutions discussion
- Non-linear systems: conditioning
- Lecture 17, Sep 30
- Convergence rate
- Intermediate value theorem
- Interval Bisection
- Lecture 18, Oct 3
- Fixed-point iterations
- Newton's method
- Lecture 19, Oct 5
- Newton's method convergence theorems
- Secant method
- Lecture 20, Oct 7
- Inverse Quadratic Interpolation
- Linear Fractional Interpolation
- Dekker's method, Brent's method
- HW3 online
- Lecture 21, Oct 10
- Non-linear equations: exercises, Halley's method
- Octave code
- Lecture 22, Oct 12
- Non-linear equations, exercises
- Non-linear systems: Fixed-point, Newton method
- Lecture 23, Oct 14
- Secant updating methods: Broyden method
- Lecture 24, Oct 17
- Interpolation: Introduction
- Basis functions, monomial basis, Horner evaluation
- Lecture 25, Oct 19
- Lagrange Interpolation
- Newton Interpolation
- Lecture 26, Oct 21
- Divided differences
- Orthonormal polynomials
- Legendre polynomials
- Lecture 27, Oct 24
- Chebyshev points
- Piecewise polynomial interpolation
- Lecture 28, Oct 26
- Hermite cubic, cubic splines
- Octave examples for polynomial interpolation
- Lecture 29, Oct 28
- Octave examples for polynomial, spline interpolation
- Review of non-linear systems
- Lecture 30, Oct 31
- Review of Interpolation
- Midterm exam 2
- Lecture 31, Nov 2
- Numerical Quadrature: Introduction
- Method of Undetermined Coefficients
- Lecture 32, Nov 4
- Newton-Cotes Quadrature
- Clenshaw-Curtis Quadrature
- Lecture 33, Nov 7
- Midterm exam 2 discussion
- Gaussian Quadrature
- Lecture 34, Nov 9
- Progressive Gaussian Quadrature
- Composite Quadrature
- Adaptive Quadrature
- Octave code: Newton-Cotes rules
- Lecture 35, Nov 11
- Octave code: Composite and Adaptive Quadrature
- Other Integration Problems
- Lecture 36, Nov 14
- Numerical Differentiation
- Richardson Extrapolation
- Lecture 37, Nov 28
- ODEs: Introduction
- Lecture 38, Nov 30
- ODEs: Existence, Uniqueness
- Lecture 39, Dec 2
- ODEs: Stability
- Euler's method
- Lecture 40, Dec 5
- Backward Euler method
- Stiff ODEs
- Lecture 41, Dec 7
- Runge-Kutta methods
- Numerical Integration: Review
- Lecture 42, Dec 9
- Review for final exam

### Textbook

Michael Heath,

Scientific Computing: An Introductory Survey, second edition, Mc-Graw Hill Higher Education, 2002.

### Prerequisites

Three credits of programming; MATH 230 or MATH 231.

### Class material

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