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?
Make this plan your own ↓Read, edit and export without an account. This is preparation; no run or result is claimed.
Make the question your own.
7 sources · Exact preparation context
- A bounded bisection solver with bracket controls →
Original local preparation by the Executable Science seed collection; upstream API references remain separately attributed.
Record revision: 92ff71866fde5148f733525493fcb867ccc7220343917930db069205c837726e
- Residual size versus root-location error →
Original local preparation by the Executable Science seed collection; upstream API references remain separately attributed.
Record revision: c159002a54b69ac3afbce054a7ff420a55583f3aadf4cf1e4e1e2a392b54d7e4
- Finite-difference step-size sensitivity →
Original local preparation by the Executable Science seed collection; upstream API references remain separately attributed.
Record revision: 68269bbc6c0186e76544c0365bed30b78e7dcb860e579a63d8409ac65a82b00f
- Exact-error comparison of trapezoid and Simpson quadrature →
Original local preparation by the Executable Science seed collection; upstream API references remain separately attributed.
Record revision: 626fe39bdd80162055297ca68c8299bfb703ac3b4d4fd423ede102ceb39744ba
- Optimization (scipy.optimize) →
SciPy documentation. Optimization (scipy.optimize). https://docs.scipy.org/doc/scipy-1.18.0/tutorial/optimize.html; observed version 1.18.0.
Record revision: 246b8f15e986d5140a3c26ae2483b8ecaa2f49ef26f68b013c22548cd97dd3bc
- Integration (scipy.integrate) →
SciPy documentation. Integration (scipy.integrate). https://docs.scipy.org/doc/scipy-1.18.0/tutorial/integrate.html; observed version 1.18.0.
Record revision: bdf9b7f1f3a06dbdeeff688edab033d166a13914a2e67b508250eddf00d97a24
- SciPy →
Preserve the upstream project name, version, repository link, applicable notices, and contributor attribution when preparing an artifact for reuse.
Record revision: 64e7d2969d3c0d0a1f2738f9ba89149cf0caf457928df51942d138e7a8d2fa73
Source links open the current listing. This plan retains the revisions above; listing review does not grant permission to copy or adapt source files.
Before carrying out this plan
- Four original examples use the qualified standard-library path.
- Library comparisons need qualified SciPy dependencies and frozen options.
Limits to keep in view
- No general convergence or condition-number theorem is established by execution.
- Reference arithmetic and assumptions remain inspectable author-controlled material.
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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
Executable Science local pilot drafts
Provide a small bisection implementation, a valid cubic-root example, an invalid-bracket control, and a high-precision Newton comparison.
Investigate Does the solver preserve a shrinking root bracket and reject the specified invalid input?
Residual size versus root-location error
Executable Science local pilot drafts
Compare residuals near simple and double roots at the same candidate locations to illustrate why a small residual can have different interpretations.
Investigate How does a multiple root change the relationship between residual magnitude and distance to the known root?
Finite-difference step-size sensitivity
Executable Science local pilot drafts
Sweep sixteen step sizes for forward and central differentiation of exp(x) at zero, exposing truncation and rounding behavior against the analytical derivative.
Investigate How does derivative error change across a fixed grid of shrinking step sizes?
Exact-error comparison of trapezoid and Simpson quadrature
Executable Science local pilot drafts
Use rational arithmetic to compare composite quadrature errors for x^4 on [0,1] across five nested grid resolutions.
Investigate What convergence pattern appears without floating-point rounding for these two quadrature rules?
Optimization (scipy.optimize)
SciPy documentation
A solver-selection guide spanning local and global optimization, roots, constrained problems, linear programming and assignment.
Investigate Which solver classes satisfy a fixed accuracy and feasibility target on a small declared suite of smooth and constrained problems?
Integration (scipy.integrate)
SciPy documentation
An official guide to quadrature from functions or samples, multiple integration and initial-value ODE solving.
Investigate How do selected quadrature rules behave on smooth, oscillatory and endpoint-singular functions with independently known integrals?
SciPy
Upstream project repository
Numerical routines for integration, differential equations, optimization, transforms, and statistical analysis.
Investigate Which declared tolerances are attained by adaptive quadrature on a fixed family of endpoint-singular integrals?