Computation for Mathematics
Using computers to solve mathematical problems with algorithms, programming, and visualization.
Programming/Scripting for Computational Mathematics
• Automate math tasks with code (Python is common).
• Learn variables, loops, conditionals, functions.
• Example: writing scripts to solve equations or plot graphs.
Mathematical Libraries and Functions
• NumPy → arrays, linear algebra.
• SciPy → integration, optimization, differential equations.
• Matplotlib → plotting.
• SymPy → symbolic algebra and calculus.
Analytical & Coordinate Geometry
• Represent shapes and relationships algebraically.
• Distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2).
• Midpoint: ((x1+x2)/2, (y1+y2)/2).
• Circle: (x-a)^2 + (y-b)^2 = r^2.
Matrices
• Rectangular arrays of numbers.
• Operations: addition, multiplication, transpose, inverse.
• Determinant: tells invertibility.
• Solve Ax = b → Gaussian elimination or numpy.linalg.solve(A,b).
Computational Calculus
• Use computers to approximate derivatives/integrals.
• Differentiation → rate of change.
• Differential equations → equations involving derivatives.
• Integration → area under a curve.
Numerical Methods for Calculus
• Derivative approximation: (f(x+h) - f(x))/h.
• Integration: trapezoidal rule, Simpson’s rule.
• Solving ODEs: Euler’s method, Runge-Kutta.
• Python: scipy.integrate.odeint, quad.
Gradients
• Gradient = vector of partial derivatives.
• Example: ∇f(x,y) = (∂f/∂x, ∂f/∂y).
• Direction of steepest increase.
• Useful in machine learning for optimization.
Optimisation
• Goal: find min/max of functions.
• Gradient Descent: iterative minimization.
• Update: x_new = x_old - η∇f(x_old).
• η = learning rate.
• Core of ML model training.
Python Examples
import numpy as np
import matplotlib.pyplot as plt
from scipy import integrate
# Matrix solve
A = np.array([[2,1],[1,3]])
b = np.array([1,2])
x = np.linalg.solve(A,b)
# Numerical integration
f = lambda x: np.sin(x)
res, _ = integrate.quad(f, 0, np.pi)
# Gradient Descent (simple)
x = 5
lr = 0.1
for _ in range(10):
grad = 2*x # derivative of f(x)=x^2
x -= lr * grad