Numberical Linear Algebra
AM 213A
This course focuses on numerical solutions to classic problems of linear algebra, providing students with comprehensive training in fundamental computational techniques and their practical implementation. Topics covered include matrix factorizations such as LU, Cholesky, and QR decompositions; iterative methods for solving linear equation systems; least squares methods, power methods, and QR algorithms for eigenvalue problems; and analysis of conditioning and stability in numerical algorithms. Students will gain hands-on experience implementing numerical algorithms from scratch to solve engineering and scientific problems, with emphasis on developing and coding their own algorithmic solutions rather than relying on existing linear algebra libraries. The course aims to master essential numerical methods for numerical linear algebra while building practical programming skills through algorithm development and implementation. Basic knowledge of mathematical linear algebra is assumed as a prerequisite. This class will use Fortran 90 (or above) or C as the programming language.
Course Notes
| Title | Description | Date | Author |
|---|---|---|---|
| Lecture Notes | Numerical Linear Algebra Course Notes | Mar 3, 2025 |
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Course Work
My solutions to the homework assignments and exams presented to us in this course.
| Title | Description | Date | Author |
|---|---|---|---|
| Homework 2 | This homework assignment requires students to prove various linear algebra theorems about eigenvalues, eigenvectors, and matrix properties, analyze the numerical conditioning of functions including plotting comparisons of mathematically equivalent but computationally different expressions, and optionally prove the equivalence between positive definiteness and positive eigenvalues for Hermitian matrices. | Feb 4, 2025 | |
| Homework 3 | This homework assignment requires students to implement Gaussian elimination and LU decomposition algorithms with partial pivoting in Fortran/C/Python, apply them to solve linear systems and find a plane equation through three 3D points, and complete three theory problems covering numerical stability, block matrix operations, and complex-valued linear systems. | Feb 19, 2025 | |
| Homework 4 | This homework assignment requires students to implement and compare Cholesky decomposition and QR decomposition methods for solving least-squares polynomial fitting problems using both single and double precision arithmetic, along with solving theoretical problems about orthogonal projectors and Householder reflectors. | Mar 4, 2025 | |
| Homework 5 | This homework assignment requires students to implement three numerical algorithms (Householder reduction to tridiagonal form, QR algorithm with and without shift, and inverse iteration for eigenvectors) in Fortran/C/Python and solve five theoretical problems related to matrix eigenvalue computations, Householder transformations, and matrix properties. | Mar 12, 2025 | |
| Midterm Exam | This midterm exam covers seven main areas: definitions of fundamental linear algebra concepts like matrix rank, orthogonality, SVD, eigenvalues, and numerical stability; proving that real symmetric matrices are diagonalizable using Schur decomposition; analyzing the Cholesky decomposition method for solving normal equations and its numerical issues with ill-conditioned matrices; proving eigenvalue bounds for stochastic matrices and invertibility conditions; examining properties of symmetric positive definite matrices including eigenvalue positivity; analyzing Gaussian elimination without pivoting, counting its operations, identifying numerical issues, and describing partial pivoting as a solution; and studying an iterative algorithm involving Cholesky decomposition, proving invariant properties, similarity transformations, and computing explicit iterations. | Feb 12, 2025 | |
| Final Coding Project | This final coding project requires students to implement two main problems: SVD-based image compression using LAPACK routines to compress a black-and-white dog image at various singular value levels and visualize the results, and iterative linear algebra methods including Gauss-Jacobi and Gauss-Seidel algorithms to solve systems of equations, with an optional conjugate gradient implementation for extra credit. | Mar 20, 2025 |
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