FDU_numerical_algorithm_24fall

Assignments and Project

• HW1. Numerical error analysis, relative error estimation, avoiding numerical cancellation, LU factorization of banded matrices, and implementation of Gaussian elimination with time complexity analysis.

• HW2. Cholesky factorization (two methods), partial and complete pivoting in Gaussian elimination, and performance testing.

• HW3. Pivoting strategies in Gaussian elimination and analysis of ill-conditioned matrices.

• HW4. QR decomposition using Householder transformations and Cholesky-based methods.

• HW5. QR decomposition via Classical Gram-Schmidt (CGS) and Modified Gram-Schmidt (MGS) methods, time complexity testing of QR factorization, and least squares problems.

• HW6. Arnoldi process with four variants (CGS/CGS2/MGS/MGS2), and QR decomposition for rank-deficient least squares problems.

• HW7. Power method for computing the largest eigenvalue, eigenvalue problems with Hilbert matrices, and eigenvector computation for upper triangular matrices.

• HW8. Francis double-shift QR algorithm, Hessenberg reduction, and comparison between Householder transformations and the Arnoldi process.

• HW9. Jacobi method for computing eigenvalues of symmetric matrices, including full pivoting, cyclic Jacobi, and analysis of incorrect rotation angle selection.

• HW10. Matrix exponential computation using scaling and squaring method, and comparison between truncated Taylor expansion and Padé approximation.

• HW11. Jacobi iterative method for solving Poisson's equation.

• HW12. GMRES (Generalized Minimal Residual) method and Steepest Descent method.

• HW13. Direct methods for solving Poisson's equation using Jacobi iteration, and the Lanczos algorithm.

• Final Project. Implementation of Francis QR algorithm for real matrices, including Hessenberg reduction and double-shift implicit QR iteration for eigenvalue decomposition.