Title

Applied Numerical Linear Algebra

Effective Term

2016 Spring Quarter

Learning Activities

Discussion/Laboratory - 3.0 hours

Discussion - 1.0 hours

Description

Numerical linear algebra (NLA) with emphasis on applications in engineered systems; matrix factorizations; perturbation and rounding error analyses of fundamental NLA algorithms.

Prerequisites

ECS 130 or EAD 209 or MAT 167

Enrollment Restrictions

Pass One and Pass Two open to Graduate Students in Computer Science only.

**Summary of Course Content**

I. Review of Finite Arithmetic

A. Approximating real numbers -- what can go wrong?

B. The IEEE floating-point arithmetic standard

II. Conditioning and Numerical Stability

A. Perturbation theory and conditioning

B. Numerical stability, backward stability

C. Accuracy of computed solutions

III. Introduction to Rounding Analysis

IV. Numerical Matrix Algebra

A. Elementary complexity analysis

B. Software details

C. Memory management

D. Structured matrices

V. Solution of Linear Systems

A. Elementary transformations

B. Gaussian elimination

C. Triangular matrix factorizations

D. Scaling

E. Perturbation theory and conditioning

F. A posteriori bounds and iterative improvement

G. LINPACK and LAPACK implementations

VI. Solution of Linear Least Squares Problems

A. Perturbation theory for pseudoinverses and linear least squares problems

B. Householder transformations

C. QR factorization

D. Other algorithms (normal equations, Gram-Schmidt, Givens transformations, SVD, etc.)

E. LINPACK and LAPACK implementations

VII. Computing Eigenvalues and Eigenvectors

A. Perturbation theory and conditioning

B. Reduction to Hessenberg and tridiagonal form

C. The QR algorithm

D. EISPACK and LAPACK implementations

VIII. Other QR-type Algorithms

A. Golub-Reinsch SVD algorithm

B. QZ algorithm for generalized eigenvalue problems

C. VZ algorithm

IX. Computation of Functions of Matrices

A. The matrix exponential

B. The matrix logarithm

C. Condition estimation for matrix functions

X. Solution of Special Matrix Equations

A. Lyapunov equations

B. Sylvester equations

C. Algebraic Riccati equations

**Illustrative Reading**

A.J. Laub, Computational Matrix Analysis (not published), LaTeX notes

**Potential Course Overlap**

This course does not have a significant overlap with any other course. It covers some of the same general topics as Math 229A, but does so at a more applied and software-related level.

Applications in computational science and engineering are emphasized throughout. This course is suitable for graduate students in any department in the College of Engineering or the Division of Mathematical and Physical Sciences.

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