Numerical Methods
Computational techniques for solving mathematical problems
Numerical Analysis Fundamentals
Numerical methods provide algorithms for solving mathematical problems that typically cannot be solved analytically, especially in real-world applications.
Core Numerical Methods
Root Finding
- Bisection Method
- Newton's Method
- Secant Method
Numerical Integration
- Trapezoidal Rule
- Simpson's Rule
- Gaussian Quadrature
Numerical Methods with Python
import numpy as np
from scipy import integrate
import matplotlib.pyplot as plt
# Define function
def f(x):
return np.sin(x) * np.exp(-x/3)
# Integration range
a, b = 0, 10
x = np.linspace(a, b, 100)
# Numerical integration methods
trap_result = integrate.trapz(f(x), x)
simps_result = integrate.simps(f(x), x)
quad_result, _ = integrate.quad(f, a, b)
# Visualization
plt.figure(figsize=(12, 6))
plt.plot(x, f(x), 'b-', label='f(x)')
plt.fill_between(x, f(x), alpha=0.3)
plt.grid(True, alpha=0.3)
plt.legend()
plt.title('Numerical Integration Example')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.show()
print("Integration Results:")
print(f"Trapezoidal Rule: {trap_result:.6f}")
print(f"Simpson's Rule: {simps_result:.6f}")
print(f"Gaussian Quadrature: {quad_result:.6f}")Advanced Numerical Methods
Differential Equations
- • Euler Method
- • Runge-Kutta Methods
- • Finite Differences
Linear Systems
- • Gaussian Elimination
- • LU Decomposition
- • Iterative Methods