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Linear Algebra • Calculus • Optimization • Differential Equations • Scientific Computing

Master the art of computational mathematics — from solving linear systems and numerical integration to optimization algorithms and differential equation solvers.

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Numerical Analysis Concepts & Algorithms

Numerical Analysis Basics

Floating point, error analysis, stability, conditioning.

Rounding & Truncation Errors

Precision, accuracy, machine epsilon.

Numerical Stability

Well-conditioned vs ill-conditioned problems.

Bisection Method

Bracketing, convergence, iterations.

Newton-Raphson

Tangent method, quadratic convergence.

Secant Method

Derivative-free, convergence rate.

Fixed-Point Iteration

Contraction mapping, convergence criteria.

Gaussian Elimination

LU decomposition, back substitution.

LU Decomposition

Doolittle, Crout, pivoting.

Cholesky Decomposition

Positive definite matrices, LL^T.

QR Decomposition

Gram-Schmidt, Householder, Givens.

Eigenvalue Algorithms

Power iteration, QR algorithm, SVD.

Singular Value Decomposition

Applications, low-rank approximation.

Jacobi Iteration

Diagonal dominance, convergence.

Gauss-Seidel

Successive over-relaxation, convergence.

Conjugate Gradient

Krylov subspaces, linear systems.

Polynomial Interpolation

Lagrange, Newton, divided differences.

Spline Interpolation

Natural, cubic, B-splines.

Least Squares Approximation

Linear/nonlinear, polynomial fitting.

Trapezoidal Rule

Composite, error analysis.

Simpson's Rule

1/3, 3/8, composite, error.

Gaussian Quadrature

Legendre, Gauss-Legendre, nodes/weights.

Adaptive Quadrature

Error control, adaptive Simpson.

Finite Differences

Forward, backward, central, error.

Differentiation

Richardson Extrapolation

Error improvement, derivatives.

Differentiation

Euler Method

Forward Euler, local/global error.

Runge-Kutta Methods

RK4, adaptive step, stiff systems.

Multistep Methods

Adams-Bashforth, Adams-Moulton, BDF.

Boundary Value Problems

Shooting method, finite difference.

PDE Basics

Elliptic, parabolic, hyperbolic.

Finite Difference for PDEs

Implicit/explicit, stability, CFL.

Finite Element Method

Mesh, basis functions, assembly.

Gradient Descent

Steepest descent, learning rate, convergence.

Newton's Method for Optimization

Hessian, quadratic convergence.

Conjugate Gradient for Optimization

Nonlinear CG, Fletcher-Reeves.

Matrix Norms

Frobenius, spectral, condition number.

Sparse Matrix Methods

Storage, iterative solvers, preconditioning.

NumPy for Numerical Computing

Arrays, linear algebra, operations.

SciPy Scientific Computing

Optimization, integration, ODE solvers.

MATLAB for Numerical Analysis

Matrices, solving, visualization.

Signal Processing Applications

FFT, filtering, convolution.

Image Processing

Filters, transforms, restoration.

ML & Numerical Methods

Optimization, linear algebra, gradients.

Error Propagation

Absolute/relative error, sensitivity.

Condition Number

Matrix conditioning, stability.

📊 Numerical Analysis Mastery Roadmap

🔰 Foundation

Error analysis, root finding (bisection, Newton)

📶 Core

Linear systems, interpolation, integration, ODEs

🔥 Deep Dive

Eigenvalues, optimization, PDEs, FEM

🏆 Frontier

Advanced iterative methods, sparse systems, scientific computing

📊 Compute with Precision — Master Numerical Analysis

Hands-on implementations, algorithm analysis, and real-world scientific computing challenges.

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