--- name: math-thumb-numerical-methods description: "Apply Chapter 20 (Numerical Methods: Approximation You Can Trust) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to budget a numerical computation and verify its accuracy with an independent check." --- # Numerical Methods: Approximation You Can Trust Use this chapter to help the reader make a checked mathematical decision. All 19 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Function or data; domain; root bracket or integration interval; smoothness; conditioning; precision and tolerance; coupled error sources. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Does a small residual support the claimed answer?** Normalize it, estimate conditioning, and check the quantity of interest before inferring forward accuracy. - **Is the algorithm itself stable?** Compute a backward error in a structure and scale that represent scientifically acceptable input perturbations. - **Is arithmetic representation losing information?** Use compensated reduction, Horner nesting, or barycentric interpolation before increasing precision. - **Is a numerical derivative needed?** Scale the input, choose forward or centered differences from domain access, sweep nearby steps, and use complex-step only through analytic code. - **Are interpolation samples selectable?** Use Chebyshev-like nodes for a global polynomial; otherwise prefer a representation suited to fixed, noisy, or nonsmooth data. - **Is an integral smooth and freely sampled?** Use an appropriate Gaussian family; split at known nonsmooth locations and audit adaptive error estimates. - **Is a convergence order advertised?** Run at least three resolutions with controlled coupled errors and estimate the observed rate before extrapolating. - **Is a scalar root bracketed?** Budget bisection first, then permit secant or Newton acceleration without surrendering containment. - **Can a root solver stop safely?** Require a meaningful residual plus location, correction, or bracket evidence, with condition checks near singular roots. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Convergence order, residual size, and solver success flags require interpretation. Establish continuity before treating a sign change as a root bracket. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 20.3.4, 20.3.5, 20.3.7 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C20-D01: Turn a bracket into a numerical certificate**: rule 20.3.5. - **C20-D02: Find where a smaller derivative step stops helping**: rule 20.2.2. - **C20-D03: Measure trapezoid convergence**: rule 20.3.1. - **C20-D04: Watch equal spacing blow up**: rule 20.2.4. - **C20-D05: Recover small terms with compensated summation**: rule 20.1.3. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.