Numerical Methods

This course provides an in-depth exploration of numerical methods, emphasizing their theoretical foundations and practical applications in solving mathematical problems. Students will engage with key concepts in floating point arithmetic, linear system solving, and the computation of eigenvalues and eigenvectors.

Key Topics Include:

1. Floating Point Arithmetic:

• Understanding the representation of real numbers in computers, including precision and accuracy issues.
• Analysis of rounding errors, truncation errors, and their impact on numerical computations.

2. Solving Linear Systems:

• Examination of direct and iterative methods for solving linear equations.
• Implementation of algorithms such as Gaussian elimination, LU decomposition, and the Jacobi and Gauss-Seidel methods.
• Exploration of matrix factorization techniques and their computational efficiency.

3. Eigenvalues and Eigenvectors:

• Introduction to the concepts of eigenvalues and eigenvectors in linear algebra.
• Methods for computing eigenvalues and eigenvectors, including the Power Method and QR algorithm.

Learning Outcomes:

By the end of the course, students will be able to:

• Understand and apply floating point arithmetic in computational settings.
• Solve linear systems using both direct and iterative numerical methods.
• Compute and interpret eigenvalues and eigenvectors.

This course is ideal for students seeking a comprehensive understanding of numerical methods and their relevance in scientific computing, engineering, and applied mathematics. Practical programming assignments will reinforce theoretical knowledge, enabling students to implement numerical algorithms effectively.