Numerical Methods

About This Course

Numerical methods are a set of tools that allow us to transform real-world problems into mathematical problems, and then into numerical solutions. Fundamental problems often involve matrices, vectors, and linear systems; in fields such as engineering, physics, data science, and applied mathematics, the ability to solve problems efficiently and reliably is essential.

This course is structured into two complementary chapters. The first chapter, Matrix Analysis, establishes the algebraic foundations necessary for the rest of the course. We begin by exploring vector spaces, including the concepts of sub-spaces, linear combinations, linear dependence, bases, and dimension. This leads naturally to the study of matrices, where we cover matrix operations, special matrix types, trace, determinant, inverse, eigenvalues, eigenvectors, and the critical link between linear maps and matrices. Finally, we introduce norms and inner products, which provide the essential tools for measuring vector and matrix magnitudes, quantifying distances, and assessing the accuracy of numerical computations.

The second chapter, Direct Methods for Solving Linear Systems, applies this theoretical foundation to one of the most fundamental problems in numerical computing: solving \( Ax=b \). We begin with triangular systems, which serve as the building blocks for more advanced techniques. From there, we develop Gaussian elimination, examining both the basic and pivoting elimination algorithms. Finally, we conclude with the LU factorization.

By the end of this course, you will not only be able to apply these algorithms, but you will also understand the underlying mathematical principles that drive them.