Numerical Methods

Computing answers when formulas run out. Approximate computation for engineers: floating-point and error analysis, root finding, direct and iterative linear solvers, interpolation and splines, numerical differentiation and integration, ODE solvers (Euler to RK4 and stiff methods), and finite-difference PDE solvers — each with runnable Python.

• Numerical Foundations & Errors — Every numerical answer is approximate. Learn exact vs approximate solutions, floating-point representation and machine epsilon, round-off and truncation error, absolute and relative error, error propagation, subtractive cancellation, conditioning vs stability, and convergence order.
• Root Finding — Solve f(x) = 0 when algebra can't. Learn bisection, false position, fixed-point iteration, Newton–Raphson and the secant method — formulas, algorithms, preconditions, convergence rates, failure cases and stopping criteria — with Python implementations compared on the same equation.
• Numerical Linear Systems — Solving Ax = b efficiently and accurately. Learn Gaussian elimination cost and pivoting, LU decomposition for multiple right-hand sides, iterative methods (Jacobi, Gauss–Seidel, SOR), convergence conditions (diagonal dominance, spectral radius), conjugate gradient for SPD systems, sparsity and the condition number.
• Interpolation & Splines — Estimate values between known data points. Learn linear interpolation, polynomial interpolation (Lagrange form, Newton divided differences, forward/backward differences), interpolation error, Runge's phenomenon, and cubic splines used in CAD, CNC paths and lookup tables.
• Numerical Differentiation & Integration — Compute derivatives and integrals from data or formulas without antiderivatives. Learn forward, backward and central differences and their error orders, the trapezoidal rule, Simpson's 1/3 and 3/8 rules, error behaviour, Richardson extrapolation and Gaussian quadrature — with a Python comparison.
• Numerical ODE Solvers — Integrate differential equations step by step. Learn Euler's method, modified Euler/Heun (RK2), classical RK4, local and global error, stability regions and stiff equations, multistep (Adams) and predictor–corrector methods, adaptive step size, and solving systems — with Euler vs RK4 accuracy tests.
• Numerical PDEs (Finite Differences) — Solve heat, wave and Laplace equations on a grid. Learn grid discretisation, finite-difference stencils, explicit (FTCS) and implicit schemes, Crank–Nicolson, stability conditions (r = αΔt/Δx² ≤ ½, CFL), iterative solution of Laplace's equation, and verification against exact solutions — with a demonstration of instability.