Computer Science @ UC Davis | Course Descriptions

ECS 130 SCIENTIFIC COMPUTATION (4) III

Lecture: 3 hours

Discussion: 1 hour

Prerequisite: Course 30 or Engineering 6; Mathematics 22A or Mathematics 67

Grading: Letter; 6-8 homework (40%), midterm (20%), and final (40%)

Catalog Description:
Matrix-vector approach using MATLAB for floating point arithmetics, error analysis, interpolations, numerical integration, matrix computations, nonlinear equations and optimization. Parallel computing for matrix multiplication and the Cholesky factorization.

Expanded Course Description:

Power Tools of the Trade
1. Vector and plotting
2. Building exploratory environments
3. Floating point arithmetic
4. Error
5. Designing functions
Polynomial Interpolation
1. The Vandermonde approach
2. The Newton Representation
3. Properties
Piecewise Polynomial Interpolation
1. Piecewise linear interpolation
2. Piecewise cubic Hermite interpolations
3. Cubic splines
Numerical Integration
1. The Newton-Cotes rules
2. Composite rules
3. Adaptive quadrature
Matrix Computations
1. Setting up matrix problems
2. B. Matrix operations
3. Recursive matrix operations
4. Shared and distributed memory matrix multiplications
Linear systems
1. Triangular problems
2. Banded problems
3. Full problems
4. Error analysis
The QR and Cholesky Factorizations
1. Least squares fitting
2. The QR factorization
3. The Cholesky factorization
4. High-performance Cholesky decomposition
Nonlinear Equations and Optimization
1. Finding roots
2. Minimization a function of one variable
3. Minimizing multivariate functions
4. Solving systems of nonlinear equations

Textbook:
Moler, Numerical Computing with Matlab, SIAM, 2004

Computer Usage/Homework:
Each homework includes problems related to the basic mathematical and algorithmic concepts and techniques discussed in class. Students use MATLAB as a problem-solving environment to practice numerical techniques, to learn and further develop appreciation for the connection between continuous mathematics and discrete computing. The programming projects and homework enhance the learning of the many features of MATLAB which will be useful in a computational career.

Discussion:
Students, individually or in small groups, will work on programming assignment problems.

Engineering Design Statement:
The course material includes practical design and implementation issues in scientific computing for reliable and efficient solutions of the most common mathematical problems in science and engineering. Lectures discuss various mathematical and numerical techniques, and practical implementation issues. The homework including the programming problems follow these themes too. Homework grades are based in part on these design issues. Final examination questions are based on algorithmic design and analysis issues discussed in lecture and from the homework.

ABET Category Content:
Engineering Science: 2 units
Engineering Design: 1 unit

Goals:
Students will:

learn fundamental techniques in scientific computation
"play" with the mathematics that stands behind each and every new method that they learn
use graphics to appreciate convergence and error
use MATLAB's matrix-vector programming language to solidify the understanding of linear algebra
prepare for advanced array-level computing

Program Outcomes:

The ability to work independently
The ability to communicate well via oral presentations
The ability to effectively express ideas through written communication
The ability to apply knowledge of basic science, mathematics and engineering principles to solve computing and information processing problems
The ability to write correct and good programs
The ability to construct appropriate abstractions to manage complexity and to think creatively about new problems

Instructors: Z. Bai, B. Hamann, K. Joy

Prepared by: Z. Bai (September 2008)

Overlap Statement:
Some of mathematical topics of ECS 130 overlap with MAT 128A/B/C, EAD 115 and ENG 180, such as interpolation and integration. However, the overlap is only very limited. Some of the ECS 130 topics are normally not treated in these courses, such as recursive and parallel matrix operations, and parallel Cholesky factorization.

ECS 130 is a one-quarter course, as opposed to the two-quarter EAD or three-quarter MAT sequences. ECS 130 is especially designed for undergraduate computer science majors who can benefit from a substantial knowledge of numerical computing, but allot only a limited time for this purpose due to many other complementary course requirements. MAT 128A/B/C, EAD 115/116 and ENG 180 cover wide areas of mathematical problems and numerical methods. In general, the focus of these courses is on the theoretical and mathematical aspects of numerical methods.

ECS 130 will only cover fundamental parts of mathematical problems and numerical methods in scientific computation. Moreover, ECS 130 uses a matrix-vector approach with MATLAB as a problem-solving environment. Students who have taken ENG 6 will have additional background in using MATLAB language, but it is not required for ECS 130. The overlap between ECS 130 and ENG 6 is only on the introduction of MATLAB, thus the overlap is minimal. In ECS 130, numerical algorithms and analysis, finite precision arithmetic, graphics and matrix-vector manipulation are folded into the course in a way that gets students to appreciate the connection between continuous mathematics and numerical computing, and the subtleties of numerical computing.

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