Computational Mathematics | Xingyu Yang

Reading guide

Computational mathematics investigates how finite-precision computation represents, approximates, and analyzes continuous mathematical objects. Begin with numerical error and algorithm quality, then progress through root-finding (bisection, fixed-point iteration, Newton’s method), algebraic polynomial interpolation, error analysis and Chebyshev nodes, Hermite interpolation with derivative constraints, and piecewise cubic splines. The sequence connects an algorithm’s structural hypotheses to its convergence rates and numerical stability.

These notes organize the core MTH2051 ideas; private applied exercises and assessment material remain in the course archive. The numerical differentiation and integration sequence covers Weeks 6–9, from difference formulas and Newton–Cotes to composite quadrature, Richardson/Romberg, Gauss–Legendre, and adaptive Simpson. Each chapter includes explicit hypotheses, proofs, themed illustrations, and interactive experiments. Multiple and improper integrals remain future topics.

Reading path

Numerical Error and Algorithm Quality

How a computed number inherits modelling choices, truncation, round-off, and the four properties that make an algorithm trustworthy.

Bisection and Fixed-Point Iteration

Roots versus fixed points, the bisection bracket, and contraction mapping on a closed interval.

Newton's Method

Newton as a locally contractive map, quadratic convergence, and the same idea for nonlinear systems.

Polynomial Interpolation

Constructing polynomials through discrete data points: Lagrange basis, existence and uniqueness, and Newton divided differences.

Interpolation Error and Chebyshev Nodes

Cauchy's error remainder formula, the Runge phenomenon, and how Chebyshev node clustering tames boundary oscillations.

Hermite Interpolation

Interpolating both values and derivatives: double-root uniqueness, decoupling via squared Lagrange basis, error osculation, and confluent divided differences.

Spline Interpolation

From global stiffness to local smoothness: piecewise linear error, cubic spline degrees of freedom, boundary conditions, and tridiagonal systems.

Numerical differentiation

Construct finite differences from Taylor and interpolation, prove error constants, and balance truncation with roundoff.

Newton–Cotes quadrature

Derive open and closed quadrature weights and prove their precision and signed errors with Peano kernels.

Composite quadrature

Assemble trapezoidal, midpoint and Simpson rules; prove global errors, mesh identities, and smoothness limits.

Richardson extrapolation and Romberg integration

Cancel known error powers, derive a finite Euler–Maclaurin expansion, and build a sample-reusing Romberg table.

Gauss–Legendre quadrature

Choose nodes by orthogonality; prove root properties, maximal precision, positive weights, and the Hermite error formula.

Adaptive Simpson integration

Derive the correction by fifteen, allocate tolerance budgets, and expose estimator failures with a smooth counterexample.