Applied Numerical Computing

• 🦉 [Slides] Introduction to the course & logistics
• 🦉 [Slides] Mathematical modeling & engineering problem solving
• 🦉 [Slides] Introduction to Python programming: Part I
• 🦉 [Slides] Introduction to Python programming: Part II
• 🦉 [Slides] Modeling and errors
• 🦉 [Slides] Nonlinear equations
• 🦉 [Slides] Introduction to applied linear algebra: Part I
• 🦉 [Slides] Introduction to applied linear algebra: Part II
• 🦉 [Slides] Linear algebraic equations
• 🦉 [Slides] Ordinary differential equations
• 🦉 [Slides] Optimization techniques: Part I
• 🦉 [Slides] Optimization techniques: Part II
• 🦉 [Slides] Curve fitting

• [Slides] Linear programming for beginners
• [Slides] Quadratic programming for beginners
• [Slides] Mixed-integer quadratic programming for beginners
• [Slides] The geometry of quadratic functions

Here are some important topics covered in this course:

• The simplest numerical technique "Euler's method" used to solve ordinary differential equations (ODEs). Application to engineering problems, e.g., bungee jumping velocity and cantilever beam deflection.

• Introduction to Python programming. How to perform numerical computations in Python with NumPy?

• Higher-order Taylor series approximation: Any smooth function can be approximated by a polynomial.

• Newton-Raphson iteration formula from a Taylor series. The rate of convergence in Newton-Raphson method is quadratic.

• Intuitive understanding of linear, superlinear, and quadratic convergence.

• Intuitive understanding of linear transformation, determinants, and inverse of matrix ($2\times 2$ and $3\times 3$ matrices). Cramer's rule and Gauss elimination for solving a system of linear equations (e.g., $2\times 2$ and $3\times 3$ matrices).
• Understanding orthogonal matrix and its properties. How to use LU decomposition to solve linear systems?
• What are ill-conditioned or well-conditioned systems? Condition number is a measure of how sensitive a linear system is to these small errors.
• Numerical methods: Solving linear system $Ax=b$ iteratively with Gauss-Seidel method and conjugate gradient method.

• Linear programming: Linear objective function & linear equality/inequality constraints.

• For any questions and feedback, please contact Dr. Xinyu Chen (chenxy346@gmail.com).
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