🔢 Numerical Analysis Fundamentals

Learn the principles of numerical differentiation (forward difference, central difference, Richardson extrapolation) and numerical integration (trapezoidal rule, Simpson's rule, Gauss quadrature). Understand error evaluation and convergence, and implement using NumPy/SciPy. Applications to materials property temperature dependence analysis and thermal calculations are also covered.

Learn direct methods (Gaussian elimination, LU decomposition, Cholesky decomposition) and iterative methods (Jacobi method, Gauss-Seidel method, conjugate gradient method) for solving large-scale simultaneous linear equations. Efficient handling of sparse matrices and numerical stability evaluation using condition numbers are also implemented.

Learn numerical methods for finding eigenvalues and eigenvectors of matrices (power method, QR method, Jacobi method). Cover eigenvalue problems for symmetric matrices and applications in materials science (vibration analysis, quantum chemistry calculations), implementing with NumPy/SciPy.

Learn interpolation methods that pass through data points (Lagrange interpolation, spline interpolation) and approximation using least squares method. High-precision interpolation using Chebyshev polynomials and applications to materials data (XRD patterns, phase diagrams) are also implemented.

Learn root-finding algorithms for nonlinear equations (Newton's method, bisection method) and optimization methods (gradient descent, conjugate gradient method, constrained optimization). Solve practical optimization problems in materials exploration and alloy design using scipy.optimize.