Computational methods, algorithms, and numerical analysis for scientific problems
← Back to Computer ScienceMaster Python fundamentals and essential libraries for scientific computation.
Study efficient algorithms for matrix computations and linear systems.
Learn algorithms for solving nonlinear equations and optimization problems.
Study methods for data fitting and function approximation.
Learn quadrature rules and integration techniques for functions and data.
Study numerical methods for solving initial and boundary value problems.
Learn finite difference and finite element methods for PDEs.
Study stochastic simulation and sampling techniques.
Master frequency domain analysis and signal processing.
Learn computational aspects of machine learning and data science.
Study parallel computing and optimization for large-scale problems.
Learn best practices for developing robust scientific software.
Master Python fundamentals and essential libraries for scientific computation and data analysis.
Master Python syntax, control flow, functions, and built-in data structures (lists, tuples, dictionaries, sets). Learn list comprehensions and generators for efficient code.
Learn ndarray creation, indexing, slicing, and broadcasting. Master vectorized operations, array manipulation, and linear algebra functions for efficient numerical computing.
Explore SciPy modules: optimize, integrate, interpolate, linalg, stats, and special. Learn to solve scientific problems using high-level mathematical functions.
Create publication-quality plots with matplotlib. Master 2D/3D plotting, subplots, styling, animations, and interactive visualizations for data exploration.
Learn DataFrame and Series operations, data cleaning, grouping, merging, and time series analysis. Master data I/O and exploratory data analysis techniques.
Master interactive computing with Jupyter. Learn markdown, magic commands, widgets, and best practices for reproducible research and data science workflows.
Learn profiling tools, Cython, Numba JIT compilation, and algorithm optimization. Understand bottlenecks and techniques for speeding up Python code.
Understand Python memory model, garbage collection, and memory-efficient programming. Learn techniques for handling large datasets and memory profiling.
Study efficient algorithms for matrix computations, linear systems, and eigenvalue problems.
Learn LU factorization for solving linear systems Ax = b. Understand partial and complete pivoting for numerical stability and computational complexity O(n³).
Master QR decomposition using Householder reflections and Givens rotations. Apply to least squares problems and orthogonal transformations.
Understand SVD A = UΣV^T for matrix analysis, pseudoinverse computation, low-rank approximation, and principal component analysis applications.
Learn power method, QR algorithm, and Jacobi method for eigenvalue computation. Understand convergence properties and applications to matrix diagonalization.
Study Jacobi, Gauss-Seidel, and SOR methods for large sparse systems. Analyze convergence conditions and computational efficiency compared to direct methods.
Master Conjugate Gradient (CG) for symmetric positive definite systems and GMRES for general systems. Understand preconditioning for faster convergence.
Learn sparse matrix storage formats (CSR, CSC, COO) and specialized algorithms. Study graph-based orderings and fill-in reduction for sparse factorizations.
Understand condition numbers κ(A) = ||A|| ||A⁻¹||, error propagation, and numerical stability. Learn backward/forward error analysis for robust algorithms.
Learn algorithms for solving nonlinear equations and finding optimal solutions to mathematical problems.
Master bisection method for guaranteed convergence and Newton's method x_{n+1} = x_n - f(x_n)/f'(x_n) for quadratic convergence. Analyze convergence rates and requirements.
Learn secant method for derivative-free root finding and Brent's method combining bisection with faster methods for robust and efficient root finding.
Study fixed-point problems x = g(x) and iteration schemes. Understand contraction mapping theorem and conditions for convergence of iterative methods.
Learn steepest descent x_{k+1} = x_k - α∇f(x_k) and variants. Study step size selection, momentum methods, and adaptive learning rates for optimization.
Apply Newton's method x_{k+1} = x_k - H⁻¹∇f(x_k) using Hessian matrix for second-order optimization. Understand quadratic convergence and computational costs.
Learn BFGS and L-BFGS algorithms that approximate Hessian using gradient information. Study secant conditions and memory-efficient implementations.
Study Lagrange multipliers, KKT conditions, and penalty/barrier methods. Learn sequential quadratic programming (SQP) for nonlinear constrained problems.
Explore simulated annealing, genetic algorithms, and particle swarm optimization for finding global minima in non-convex optimization landscapes.
Study methods for data fitting, function approximation, and constructing smooth representations of discrete data.
Learn polynomial interpolation theory, existence and uniqueness theorems. Understand Runge's phenomenon and oscillatory behavior of high-degree polynomials.
Master Lagrange interpolation formula and Newton's divided differences. Compare computational efficiency and numerical stability of different representations.
Study piecewise polynomial interpolation with splines. Learn cubic splines, B-splines, and smoothing splines for flexible and stable curve fitting.