Build a numerical result another person can audit

Try

A numerical answer is a conclusion supported by a chain: a mathematical model, an algorithm, an implementation, a stopping rule and an error interpretation. Failure at any link can invalidate the digits. A reproducible report should let another person reconstruct that chain. Begin with a concrete question such as approximating a root to absolute error 10⁻⁶, fitting a noisy linear model, or integrating an ODE to a stated final time.

Predict

Predict whether printing twelve digits means twelve accurate digits. The output format does not measure conditioning, approximation error or arithmetic error. Likewise two agreeing programs may share the same formula or bug. Independent evidence changes the representation or mechanism: compare a polynomial quadrature case with an exact integral, a linear ODE with its closed recurrence and continuous solution, or a linear solve with both residual and condition information.

Experiment

Use this evidence map to avoid substituting one check for another.

Evidence	What it supports	What it does not establish alone
Small residual	Nearby equation satisfaction	Small forward error
Refinement ratio	Observed asymptotic behavior	Universal convergence theorem
Exact test case	Correct behavior on that case	All-input correctness
Theorem with checked hypotheses	A class-level guarantee	Correct implementation

Choose one method from this unit and fill in an actual example for every row.

Understand

When an error expansion Q_h=Q+Ch^p+o(h^p) holds with C≠0, successive-grid differences have ratio approaching 2^p. With an independent exact Q, observed order is log₂(|Q_h−Q|/|Q_(h/2)−Q|). Without Q, use three-grid differences, but recognize shared bias and cancellation. A plateau may indicate rounding, reference error or a failed smoothness assumption rather than the formal order being false. Certified absolute bounds can be added through the triangle inequality; estimated errors should not be relabeled certified merely because their sum is small. State units and norms so contributions are comparable.

Apply

Method selection follows structure. A valid sign bracket favors bisection or a safeguarded hybrid. Smooth local root information can justify Newton. Noisy data often favor least squares over exact interpolation; QR or SVD expose different rank and conditioning issues. Smooth integrands support high-order quadrature, while singular features may require transformed variables or interval splitting. Stiff ODEs may need implicit methods despite their solve cost. A method’s order, stability region and work per step answer different questions. Compare methods at a meaningful resource budget and include rejected steps, derivative evaluations, factorizations and reused solves in accounting.

Continue

For a cumulative implementation project, supply runnable code separately from this written lesson, fixed reproducible inputs, independent reference calculations, exact edge-case tests and a failure report. Include at least one case where the preferred method should refuse or qualify an answer. This unit covers core numerical mechanisms and selected proofs; full production-library engineering, advanced sparse/Krylov algorithms, PDE discretization and a complete derivation of high-order Runge–Kutta conditions remain further work. Completed reading or a filled report does not establish implementation proficiency without reviewing the code, tests and reasoning.

Review calculus and linear algebra when a prerequisite needs practice. Return to the numerical analysis unit to choose a targeted retry.