Module 1 (12 Hours)
Fundamentals: an overview of matrix computations, norms of vectors and matrices, singular value decomposition (SVD), IEEE floating point arithmetic, analysis of roundoff errors, stability, and ill-conditioning.

Module 2 (12 Hours)
Linear systems: LU factorization, Gaussian eliminations, Cholesky factorization, stability, and sensitivity analysis Jacobi, Gauss-Seidel, and successive overrelaxation methods Linear least-squares: Gram-Schmidt orthonormal process, rotators and reflectors, QR factorization, stability of QR factorization QR method linear least-squares problems, normal equations, Moore-Penrose inverse, rank-deficient least-squares problems, sensitivity analysis.

Module 3 (12 Hours)
Eigenvalues and singular values - Schur's decomposition, reduction of matrices to Hessenberg and tridiagonal forms Power, inverse power, and Rayleigh quotient iterations QR algorithm, implementation of implicit QR algorithm Sensitivity analysis of eigenvalues Reduction of matrices to bidiagonal forms, QR algorithm for SVD.

Learn the basic matrix factorization methods for solving systems of linear equations and linear least squares problems.

Learn the basic computer arithmetic and the concepts of conditioning and stability of a numerical method.

Learn how to implement and use these numerical methods.

Learn about eigenvalues and singular values.

CO1: Understand and implement common iterative algorithms.
CO2: Students will leave the course with a comprehensive understanding of matrix factorization methods, an ability to apply these methods to a variety of domains, and hands-on experience with the tools and algorithms used in matrix factorization.
CO3: By the end of the course, students should be able to identify and apply appropriate sensitivity analysis techniques to various models across different domains.
CO4: Have a solid grasp of the numerical challenges involved in computing SVD and the strategies to address them.
CO5: Be equipped to apply these techniques to real-world problems in data science, engineering, and machine learning.

L. N. Trefethen and David Bau, Numerical Linear Algebra, SIAM, 1997

D. S. Watkins, Fundamentals of Matrix Computation, 2nd Edn., Wiley, 2002

J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997

B. N. Datta, Numerical Linear Algebra and Applications, 2nd Edn., SIAM, 2010