Instructor:

Kyriacos Zygourakis
AL B217
Phone: 713-348-5208
Email: kyzy at rice.edu

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Syllabus:

Complex Variables and Applications

• Definittion of complex numbers. Operations.
• Analytic functions. Cauchy Riemann equations. Harmonic functions. Laplace's equation.
• Conformal mapping. Definition, examples and important special functions. Mobius transform. Schwartz-Cristoffel transform.
• Steady state heat flow. Irrotational flow.
  

Linear Algebra and Applications

This part of the course will first introduce the formal mathematical language of functional analysis. Familiar concepts of matrix algebra will be reformulated using the abstract framework of vector spaces. We will then show that many seemingly different mathematical systems (matrices, sets of functions, polynomials, solutions of integral and differential equations etc.) are vector spaces.

Thus, the theory of vector spaces can be used to unify all these diverse phenomena into a single study. At the same time, we will establish that these results simplify the solution of many significant and complicated engineering problems.

Vector Spaces and Linear Transformations.

• Overview of the problem of solving large systems of linear equations. Which applications give rise to such systems ? Which are the theoretical problems that must be answered ?
• Vector spaces and subspaces. Linear dependence, basis and dimension. Linear transformations between finite-dimensional spaces and their matrix representation. Rank and nullity of linear transformations. Elementary matrices and the computation of the rank.
• The theory of simultaneous linear equations. Homogeneous and nonhomogeneous systems. The Fredholm alternative.

Solution of Systems of Linear Equations Ax = b

• Gauss elimination and the LU-decomposition. Pivoting and operation counts. Rudimentary error analysis. Ill-conditioned matrices.
• Band matrices and how they arise in practice. Finite difference solution of partial differential equations.
• Brief overview of iterative methods for solving linear systems. Comparison of the various numerical algorithms.

The Eigenvalue Problem.

• Determinants and their properties.
• Inner products, norms, orthogonality.
• Eigenvalues and eigenvectors of matrices. Diagonalization and similarity transformations. Systems of difference equations. Functions of matrices.
• Solution of systems of ordinary differential equations. Stability. Unitary transformations, normal matrices and the spectral decomposition of operators.

Quadratic Forms and Variational Principles.

• Positive definite quadratic forms. Minimization problems. Least squares method. Rayleigh quotient. Maximin and minimax principles.
• Numerical computation of eigenvalues and eigenvectors.
• Overview of the finite elements method.

Applications of Differential Equations

• Stability of linear systems. Phase-plane. Phase portraits of linear systems.
• Predator-prey systems. Population biology. Epidemiology.

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Textbook:

A set of notes (available at the bookstore) will be used for the linear algebra part of the course. The students are strongly encouraged to develop their own complete set of notes using the lecture material, class handouts and additional references listed below.

Additional references:

1. Brown, J.W and R. V. Churchill "Complex Variables and Applications," 6th Edition, McGraw-Hill (1995).
2. Strang, Gilbert "Linear Algebra and Its Applications", 3rd Edition, HBJ College & School Division (1988).
3. Amundson, N.R. "Mathematical Methods in Chemical Engineering: Matrices and Their Application," Prentice-Hall (1966).
4. Dahlquist, G., A. Bjorck and N. Anderson "Numerical Methods," Prentice-Hall (1974).
5. Hirsch, M.W. and S. Smale "Differential Equations, Dynamical Systems and Linear Algebra", Academic Press (1974).
6. Noble, B. and J.W. Daniel "Applied Linear Algebra", 3rd Edn, Prentice-Hall (1988).

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Course Organization:

Homework assignments

One set of problems will be assigned approximately every week. They will be due on the same day of the following week. No late homework will be accepted, unless prior arrangements have been made with the instructor.

Computer assignments

Some assignments given during the latter part of the semester will require use of the computer. An interactive computer program (MATLAB) will be used for some assignments. MATLAB will serve as a convenient "laboratory" for computations involving matrices since it provides easy access to matrix software developed by the LINPACK and EISPACK projects. In another project, various linear equation solvers will be compared. The LINPACK subroutine package will be used for this purpose.

The students must first demonstrate that they can formulate the assigned problems correctly. Following that, MATLAB or LINPACK routines will be used to derive the results. Thus, the emphasis will be on analyzing the results and the programming effort required will be minimal.

No prior knowledge of computer operating systems will be assumed. However, the students should be able to write simple programs in FORTRAN. If necessary, tutoring sessions will be scheduled to demonstrate the use of MATLAB and LINPACK.

Exams

There will be a mid-semester and a final exam for this course. Both will be of the limited-time check-out type and will consist of open- and closed-book parts.

Honor Code Policy Students are encouraged to talk to each other, the teaching assistants, the instructors, or anyone else about any assignment in the course that is not specifically designated as pledged. This assistance is limited to the discussion of the problem and perhaps sketching of a solution. Consulting another student's solution (even from a previous CENG 672 class) is prohibited, and submitted solutions to assignments may not be copied from any source.

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