About Numerical Methods
Numerical methods are computational algorithms used to solve mathematical problems that are difficult or impossible to solve analytically. In computational engineering, these methods form the backbone of simulation software, optimization tools, and data analysis frameworks. My expertise spans various categories of numerical methods, from basic root-finding algorithms to advanced integration techniques used in finite element analysis.
Methods & Techniques
Gaussian Quadrature
A numerical integration technique that approximates definite integrals using weighted sums at specific points (nodes). It achieves maximum accuracy for polynomials of degree 2n-1 using n points.
Applications:
Newton Interpolation
A polynomial interpolation method using divided differences to construct a polynomial that passes through given data points. Particularly useful for equally and unequally spaced data.
Applications:
LU Decomposition
Matrix factorization method that decomposes a matrix into lower and upper triangular matrices. Essential for solving systems of linear equations efficiently.
Applications:
Runge-Kutta Methods
Family of iterative methods for solving ordinary differential equations. The 4th-order method (RK4) provides excellent balance between accuracy and computational cost.
Applications:
Newton-Raphson Method
Root-finding algorithm that uses iterative approximation based on function derivatives. Provides quadratic convergence for well-behaved functions.
Applications:
Finite Difference Methods
Numerical techniques for approximating derivatives using discrete function values. Foundation for solving partial differential equations on grids.
Applications:
Interactive Numerical Methods
Explore implementations of key numerical methods. Click "Run" to execute the algorithms and see detailed results.
Root-finding algorithm with quadratic convergence for solving nonlinear equations.
Converged in 4 iterations Root found: x = 1.52137971 Verification: f(1.52137971) = -1.42e-08 Iteration details: 1: x = 1.52777778, error = 2.78e-02 2: x = 1.52140212, error = 6.38e-03 3: x = 1.52137971, error = 2.24e-05 4: x = 1.52137971, error = 2.81e-10
Implementation Experience
Languages & Tools
- • C++: High-performance numerical libraries and finite element solvers
- • Julia: Scientific computing and mathematical modeling
- • Python: NumPy, SciPy for rapid prototyping and data analysis
- • MATLAB: Algorithm development and visualization
Project Applications
- • Finite element structural analysis solvers
- • Computational fluid dynamics simulations
- • Heat transfer and diffusion modeling
- • Optimization algorithms for engineering design