Computational Methods

You already took Calculus classes and learned how to find the perfect derivative, the elegant area under a curve using pen and paper, optimize problems, and solve differential equations. It is clean and precise, but in the real world, it falls short. Most high-level problems in physics, engineering, and biology do not have a neat "back-of-the-book" answer. Instead, they are messy, chaotic, and mathematically impossible to solve by hand. This is where we stop being mathematicians and start being engineers. In Computational Methods, we use tools you learned in Calculus to build the engine that takes over when analytical math hits a wall. We start learning how machines actually represent numbers and why those tiny rounding errors can occasionally crash a rocket. From there, we move into the art of the approximation. You will learn how to use Taylor polynomials to implement industry-standard methods to analyze complex problems with no analytical solution. Using Python and the powerful NumPy/SciPy ecosystem, you will transform abstract theories into high-performance simulations. Whether we are solving differential equations, fitting data with non-linear models, or potentially diving into advanced applied math topics, you will learn how to make a computer do the heavy lifting without it falling into a pit of numerical instability.

Computational Methods Potential Course Topics

1. Numeric Representation and Computer Architecture 

a. Binary, Octal, and Hexadecimal Representations and Arithmetic 

b. Integer and Floating Point Representations and Arithmetic 

c. Machine Epsilon

2. Programming in Python 

a. Review of basic concepts 

b. Special packages: Matplotlib, Numpy, Scipy

3. Finding Polynomial Roots 

a. Bisection Method 

b. Newton-Raphson Method 

c. Fixed-Point Method 

d. Secant Method 

e. Convergence analysis

4. Numerical Integration 

a. Risch’s Algorithm

b. Trapezoidal Rule 

c. Simpson’s Rule

5. Numerical Differentiation 

a. Taylor expansion 

b. First, second, and fourth-order schemes 

c. Truncation error 

d. Roundoff error

6. Solving Ordinary Differential Equations (ODEs) 

a. Analytical methods for first-order ODEs 

b. Computational Methods for first-order ODEs - Initial Value Problem (IVP): 

i. Euler’s Method 

ii. Predictor-Corrector Methods 

iii. Heun’s Method iv. Runge-Kutta Methods v. Error analysis 

c. Analytical methods for second-order ODEs 

d. Computational Methods for second-order ODEs - Initial Value Problems (IVP): 

i. Second-order ODE as a system of two First-order ODEs 

ii. Euler’s Method 

iii. Predictor-Corrector Methods 

iv. Heun’s Method v. Euler-Cromer Method 

vi. Error analysis 

e. Computational Methods for second-order ODEs - Boundary Value Problem (BVP): 

i. Finite Difference Method 

ii. Error analysis

7. Special Topics 

a. Interpolation

i. Linear and quadratic interpolations 

ii. Cubic splines

b. Image Processing 

c. Radiomics analysis 

d. Matrix Algebra (these topics are taught in the Linear Algebra class) 

i. Introduction to Matrix Algebra 

ii. Gaussian Elimination Method

iii. LU Decomposition Method 

e. Regression (these topics are taught in the Linear Algebra class) 

i. Linear Regression 

ii. Non-linear regression

While the core of the course consists of Units 1–6, some special topics will be covered depending on available instructional days and students’ interests.