Numerical Algorithms

Numerical algorithms are computational techniques for solving mathematical problems approximately when analytical solutions are unavailable.

They are widely used in scientific computing, engineering simulations, optimization, and physical modeling.

🔢 Major Categories of Numerical Algorithms

1. Linear Algebra

  • Gaussian elimination
  • LU/QR/Cholesky factorization
  • Iterative solvers (CG, BiCGStab)
  • Sparse matrix solvers

2. Nonlinear Systems

  • Newton–Raphson method
  • Fixed-point iteration

3. Optimization

  • Gradient descent
  • Quasi-Newton methods
  • Simplex algorithm

4. Numerical Integration

  • Simpson’s rule
  • Trapezoidal rule
  • Gauss quadrature

5. Differential Equations

  • Runge–Kutta (RK2–RK4)
  • Adams–Bashforth
  • Crank–Nicolson

6. Eigenvalue Problems

  • Power method
  • Lanczos method

🔧 Open-Source Libraries

NumPy

Matrix operations, FFTs, linear algebra.
🔗 https://numpy.org

SciPy

Scientific algorithms: ODE solvers, optimization, interpolation.
🔗 https://scipy.org

Eigen

High-performance C++ linear algebra.
🔗 https://gitlab.com/libeigen/eigen

PETSc

Scalable parallel solvers for PDEs.
🔗 https://petsc.org

SuiteSparse

Sparse matrix algorithms.
🔗 https://github.com/DrTimothyAldenDavis/SuiteSparse

🎥 YouTube Playlists

  • Numerical Methods for Engineers (NPTEL)
    🔗 https://www.youtube.com/c/nptel

  • Numerical Computing with Python
    🔗 https://www.youtube.com/results?search_query=numerical+python+methods

📘 Learning Resources

  • Numerical Recipes (open summaries)
    🔗 https://numerical.recipes

  • MIT OpenCourseWare — Numerical Methods
    🔗 https://ocw.mit.edu