Course Overview
This course provides a rigorous, graduate level introduction to the fundamental theoretical and algorithmic principles of numerical linear algebra and scientific computing. Topics include computer arithmetic, stability analysis, matrix decompositions, spectral theory, iterative solvers for big sparse linear systems, and eigenvalue problems. The course emphasizes the development of efficient and stable numerical algorithms with a focus on stable computational implementations and algorithm development.
Syllabus & Lecture Topics
| Sec. | Topic & Detailed Content |
|---|---|
| 1.1 | Introduction to Computational Science and Engineering |
| 1.2 |
Number Representation in Computing Systems
|
| 1.3 | Rounding Errors and Wilkinson's Error Analysis Principles |
| 1.4 | Condition Number Analysis ($\kappa$) |
| 2.1 | Vector and Matrix Norms |
| 2.2 | Taylor Series Expansion and Truncation Errors |
| 2.3 | Numerical Stability: Stable and Unstable Algorithms |
| 3.1 | Linearity and Linear Combinations |
| 3.2 | Linear Independence and Matrix Multiplication |
| 3.3 | The Four Fundamental Subspaces of Linear Algebra |
| 3.4 | Elementary Row Operations and Permutation Matrices |
| 3.5 |
Methods for Solving Linear Systems $A ec{x} = ec{b}$
|
| 4.1 |
Gaussian Elimination
|
| 4.2 | LU Decomposition and Matrix Factorization |
| 4.3 | Symmetric and Positive Definite (SPD) Matrices |
| 4.4 | Cholesky Decomposition ($A = RR^T$) |
| 4.5 | Elimination via Givens Rotation Matrices |
| 4.6 | Stability and Condition Number Analysis for Linear Systems |
| 5.1 |
The Eigenvalue Problem
|
| 5.2 |
Spectral Decomposition (Eigen-Decomposition)
|
| 5.3 | Matrix Exponential $\exp(A)$ |
| 5.4 | Dimensionality Reduction: Perspectives on PCA and Spectral Decomposition |
| 5.5 | Inner Products, Orthogonality, and Correlation |
| 5.6 | Similarity Transformations and Symmetric Matrices |
| 5.7 | Gerschgorin Disk Theorem and Eigenvalue Bounds |
| 5.8 |
Numerical Methods for Solving the Eigenvalue Problem
|
| 6.1 | Singular Value Decomposition (SVD) and Applications |
Primary Textbooks & References
An Introduction to Computational Science
Numerical Linear Algebra
Elements of Scientific Computing
Fundamentals of Scientific Computing
Scientific Computing