## MATH3230B Numerical Analysis (MATH3230B), 2012/2013

### Announcement

• Some one has left a pair of eyeglasses in LT3 after tutorial 13. Please come to LSB 228 to take it back!

### General Information

#### Lecturer

Professor Jun Zou
Office: LSB, Rm 210    Tel: 3943-7985    Email:

#### Teaching Assistant

Ka Chun Lam
Office: LSB, Rm 228    Tel: 3943 7957    Email:
Chengfeng Wen
Office: LSB, Rm 228    Tel: 3943 7957    Email:

#### Time and Venue

Lecture: Wed. 4:30-6:15pm (LSB LT4); Thu. 4:30-5:15pm (LSB LT2)
Tutorial: Wed. 6:30-7:15pm (LSB LT3)

### Course Description

#### Course outline:

Numerical Analysis is an important branch in Applied Mathematics. It aims at numerically solving all kinds of mathematical problems which arise from practical applications and can be modeled by different mathematical equations or inequalities, for example, linear or nonlinear differential equations and integral equations. For a systematic understanding of the field Numerical Analysis, we will offer two basic courses: Numerical Analysis (MATH3230) and Numerical Methods for Differential Equations (MATH3240), which were previously called Numerical Analysis I and Numerical Analysis II.

The first course Numerical Analysis (MATH3230) introduces the fundamental concepts and methods, and basic numerical analysis tools in the field. The course starts with several traditional but most influential numerical methods for solving nonlinear equations of one variable, e.g., bisection method, Newton's method, quasi-Newton's method and fixed-point iterative method. Then several basic iterative methods are introduced for solving systems of nonlinear equations of multiple variables, including Newton's method, Broyden's method and steepest descent method.

This course emphasizes not only numerical methods, but also the analysis of their convergence and convergence rates. So for nearly every numerical method we study in this course, we shall analyze their convergence and how fast they converge.

We know that if the concerned problem is not so small, numerical methods must be realized through computers. So the course will then discuss how the computers handle the numbers and carry out arithmetic operations with reasonably acceptable accuracies.

Solving linear systems of equations is an inevitable stage among all the numerical solutions of differential and integral equations. So matrix and vector must be two important components involved in the stage, that is why the course continues with the basic properties and operations for matrices and vectors, then presents some most fundamental numerical algorithms for linear systems, e.g., Gauss elimination, LU, LDU and Cholesky factorization.

Interpolation is a simple and often efficient methodology to extract a good approximation to some given function or data. The course will introduce several basic interpolation methods, e.g., Vandermonde interpolation, Lagrange interpolation and Newton interpolation, and analyze their approximation accuracies.

Finally, the course comes closer to our aforementioned aim, with the introduction to numerical integration and differentiation, which, as the name indicates, studies how to approximate given integrals and derivatives. This knowledge will be our basic tool and frequently needed in the course Numerical Methods for Differential Equations (MATH3240). Numerical integration will be addressed in detail in this course, including the introduction of a variety of most commonly used quadrature rules and the analysis of their numerical accuracies, while numerical differentiation will be discussed only briefly since it will be one of the major focuses of the course Numerical Methods for Differential Equations.

#### Course prerequisite:

Advanced calculus, linear algebra, programming using MATLAB

Tutorial work & programming or top 20% in the mid- and final exams: 10%; Mid-Exam: 35%; Final Exam: 55%.

#### Mid-exam date:

March 13, 2013, Wednesday (4:30-6:30pm)

Attention: The Mid-Exam venue may be different from the currently used classroom.

### Textbooks

• D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, Brooks/Cole Publishing Co., New York, 1996. (QA297.K563)
• G.W. Stewart, Afternotes on Numerical Analysis, SIAM, 1996. (QA297.S785)

### References

• G.H. Golub and C.F. van Loan, Matrix computations, Johns Hopkins University Press, MD, 1989
Call No.: QA188.G65.
• R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole-Thomson Learning, 2001
• C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Person Addison Wesley, 2003