Separate a solver's stopping signal from solution accuracy
When does a tiny residual or smaller step fail to imply a more accurate scientific result?
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What to compare
Use the known double-root relation and exact rational Simpson errors for the fixed polynomial. A finite-precision zero is not a certified real root.
Useful outputs
- Known-root/function examples with residual, location error and precision stated separately.
- Complete refinement grids, including roundoff-dominated steps.
- A source/version-pinned comparison plan for a library solver.
Inputs and prerequisites
- Four original examples use the qualified standard-library path.
- Library comparisons need qualified SciPy dependencies and frozen options.
What this work would not establish
- No general convergence or condition-number theorem is established by execution.
- Reference arithmetic and assumptions remain inspectable author-controlled material.
Start with these sources.
A bounded bisection solver with bracket controls
Provide a small bisection implementation, a valid cubic-root example, an invalid-bracket control, and a high-precision Newton comparison.
Residual size versus root-location error
Compare residuals near simple and double roots at the same candidate locations to illustrate why a small residual can have different interpretations.
Finite-difference step-size sensitivity
Sweep sixteen step sizes for forward and central differentiation of exp(x) at zero, exposing truncation and rounding behavior against the analytical derivative.
Exact-error comparison of trapezoid and Simpson quadrature
Use rational arithmetic to compare composite quadrature errors for x^4 on [0,1] across five nested grid resolutions.
Optimization (scipy.optimize)
A solver-selection guide spanning local and global optimization, roots, constrained problems, linear programming and assignment.
Integration (scipy.integrate)
An official guide to quadrature from functions or samples, multiple integration and initial-value ODE solving.
SciPy
Numerical routines for integration, differential equations, optimization, transforms, and statistical analysis.
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