Numerical Analysis

Binary fractions, IEEE 754, machine epsilon, rounding, overflow and underflow

Error of one operation, accumulation, catastrophic cancellation and compensated summation

Taylor's theorem with remainder, order of accuracy, the finite-difference sweet spot, and how libraries evaluate functions

Condition numbers, forward and backward error, stable and unstable algorithms, and chaos

The intermediate value theorem, bisection, false position, the Illinois fix and Brent's method

Linearisation, quadratic convergence, failure modes, the secant method and safeguarded Newton

x = g(x), the contraction mapping theorem, linear rate, Aitken and Steffensen acceleration

The Jacobian, Newton's method in n dimensions, finite-difference Jacobians, damping, Broyden and power flow

Row operations, triangular systems, A = LU, operation counts and many right-hand sides

Partial pivoting, vector and matrix norms, κ(A), residual versus error, backward stability and refinement

Symmetric positive definite matrices and energy, A = LLᵀ, band storage and the Thomas algorithm

CSR storage, assembly from elements, fill-in, reordering and sparse direct solvers

Splitting A = M − N, the Jacobi and Gauss–Seidel iterations, spectral radius and diagonal dominance

SOR, Young's optimal ω, cost versus grid size, red–black ordering and stopping criteria

Energy minimisation, steepest descent, conjugate directions, √κ convergence and GMRES

Changing the spectrum, Jacobi and incomplete Cholesky preconditioners, multigrid in one idea, seepage

Overdetermined systems, minimising ‖Ax − b‖, projection, normal equations and weighting

Orthogonal matrices, Gram–Schmidt and its instability, Householder reflections, Givens rotations

A = UΣVᵀ, singular values as gains, rank and pseudoinverse, low-rank approximation, regularisation

Gauss–Newton, the damped step, Levenberg–Marquardt, GNSS point positioning and residual analysis

Eigenpairs and their physical meaning, power iteration, the Rayleigh quotient, PageRank and Gershgorin discs

Inverse iteration, shifted inverse iteration, Rayleigh quotient iteration and deflation

Similarity transforms, QR iteration, Hessenberg reduction, shifts and deflation, eig versus eigh

Kφ = ω²Mφ, mass-orthogonal modes, modal superposition, shear buildings and footbridges

Existence and uniqueness, the Lagrange and Newton forms, the barycentric formula and the error term

Runge's phenomenon, the node polynomial, Chebyshev points, Lebesgue constants and spectral accuracy

Piecewise interpolation, cubic splines and their tridiagonal system, shape preservation, Bézier curves

The discrete Fourier transform, the fast Fourier transform, spectral analysis, leakage and aliasing

Difference formulas from Taylor series, the step-size trade-off, noisy data, Richardson extrapolation, complex step

Trapezoid and Simpson rules, degree of precision, composite rules and their error, total impulse

Romberg integration, adaptive Simpson, error estimation and scipy.integrate.quad

Free nodes, orthogonal polynomials, Gauss–Legendre rules, change of interval and finite elements

Expectations as integrals, 1/√N convergence, the curse of dimensionality, uncertainty propagation

Initial value problems, Euler's method, local and global error, order, consistency, stability and convergence

Midpoint and RK4, Butcher tableaux, embedded pairs, step-size control and solve_ivp

Adams–Bashforth and Adams–Moulton, predictor–corrector, starting values, zero-stability and the root condition

Stiffness and stability regions, backward Euler and the trapezoidal rule, A- and L-stability, BDF, Newton inside each step

First-order systems, events, Störmer–Verlet and symplectic integration, rigid-body attitude dynamics

x′ = Ax, modes and stability, computing e^{At}, exact discretisation for digital control, transient growth, the Lyapunov equation

Two-point BVPs, the shooting method, finite-difference discretisation, convergence and solve_bvp

The heat equation, explicit FTCS and its stability limit, implicit and Crank–Nicolson schemes, von Neumann analysis, consolidation

Advection and wave equations, upwind, Lax–Friedrichs and Lax–Wendroff, the CFL condition, numerical diffusion and dispersion

The five-point stencil, Dirichlet and Neumann boundaries, sparse assembly and solution, seepage under a dam

Energy and the Ritz method, the weak form and Galerkin, hat functions, element matrices and assembly, bars, beams and mass matrices

The weak form in the plane, linear triangles, assembly, Prandtl torsion, the patch test, corner singularities and mesh convergence

Optimality conditions, golden-section and Brent in one dimension, steepest descent, Armijo backtracking

Newton's method for minimisation, BFGS, scipy.optimize.minimize and derivative-free Nelder–Mead

Lagrange multipliers and KKT conditions, penalty and SQP methods, linear programming, minimum-mass truss layout

Pseudo-random generators and their flaws, sampling distributions, numpy.random.Generator, confidence intervals

Covariance and the multivariate normal, error ellipses, linearised propagation J P Jᵀ, the unscented transform, collision probability

Recursive least squares, predict and update, the Kalman gain, a tracking filter and inertial–GNSS fusion

The extended Kalman filter, the unscented Kalman filter, consistency, estimating what you cannot measure, the Riccati equation