Interval Verified Root Isolation and Nonlinear System Study
A completed mathematics study of outward-rounded interval arithmetic, certified polynomial root isolation and verified two-dimensional nonlinear systems.
Software compatibility
The retained release uses Python 3.14.6. The Docker workflow uses Python 3.12.14 and reproduces all certification counts and numerical widths within the declared tolerance.
Project definition
Problem statement
An ordinary numerical solver can return a plausible root approximation without proving that the root exists nearby, is unique, or is the only root in a declared domain.
The mathematical problem is to preserve rounding uncertainty, exclude root-free regions and certify simple roots with interval inclusion tests while reporting singular or unresolved cases honestly.
Project objectives
- Implement closed binary64 intervals with outward-rounded arithmetic operations.
- Reduce polynomial dependency with natural and midpoint-shifted Horner forms.
- Isolate simple polynomial roots with interval Newton contraction and adaptive subdivision.
- Certify two-dimensional nonlinear system roots with a Krawczyk operator.
- Audit interval containment across one thousand deterministic arithmetic cases.
- Measure tolerance sensitivity, dependency overestimation, floating-point residuals and degree scaling.
- Retain certified boxes, unresolved boxes, result tables, figures and checksums.
Project structure
Project components
Interval arithmetic core
Implements containment, intersection, directed endpoint expansion, arithmetic, powers and bisection.
Polynomial isolator
Combines two valid polynomial enclosure forms with interval Newton contraction and subdivision.
System isolator
Applies interval exclusion and a two-dimensional Krawczyk operator with widest-axis splitting.
Benchmark suite
Covers smooth, oscillatory, clustered, cancellation-prone and repeated-root polynomial cases plus three nonlinear systems.
Evidence builder
Writes root boxes, summaries, audits, sensitivity tables, checksums and seventeen labelled figures.
Methodology
Project workflow
- 01Load the study
The program reads tolerances, box budgets, precision and the deterministic audit seed.
- 02Partition domains
Finite polynomial and system domains are divided into candidate intervals or boxes.
- 03Exclude and contract
Interval evaluation removes impossible regions before Newton or Krawczyk images contract candidates.
- 04Certify or retain
Strict inclusion certifies one simple root, while singular or budget-limited candidates remain unresolved.
- 05Compare and audit
Known or high-precision references, arithmetic samples, tolerance sweeps and runtime cases are evaluated.
- 06Export evidence
CSV, JSON, checksums and figures are generated from the same versioned configuration.
Demonstration scenario
The production run certifies 34 simple roots across eight regular polynomial benchmarks and all nine expected roots across three nonlinear systems. Every arithmetic audit passes. A repeated root remains unresolved as required by the simple-root claim boundary.
Engineering
Tools and method
- Tools
- The project uses Python, NumPy, SciPy, mpmath, Pandas, Matplotlib, Jupyter for subject analysis, simulation, and results.
- Arithmetic model
- Closed binary64 intervals with each elementary endpoint expanded by one representable value.
- Polynomial range
- Intersection of natural Horner and interval-computed midpoint-shifted Horner enclosures.
- Univariate theorem
- Interval Newton exclusion, contraction and strict interior uniqueness for simple roots.
- System theorem
- Krawczyk fixed-point inclusion using a rounded numerical preconditioner and interval Jacobian.
- Verification
- Automated tests, coverage, lint, dependency audit, cross-version Docker reproduction and document validation.
Testing
Evaluation
Evaluation measures
- Certified and unresolved box counts by polynomial
- Containment of analytically known or high-precision reference roots
- Maximum certified interval and system-box width
- Exclusions, contractions, bisections and processed boxes
- Arithmetic audit sample containment across one thousand cases
- Requested tolerance against achieved width and computational work
- Natural dependency, ordinary floating-point residuals and degree runtime scaling
Project boundaries
- Certificates apply to the binary64 coefficients stored by the program and the declared finite domains.
- The release supports polynomial arithmetic and two-dimensional systems, not transcendental interval functions or arbitrary precision.
- The repeated-root control is intentionally unresolved by the simple-root method.
- The implementation is tested and theorem-based but is not a machine-checked proof of Python, NumPy or the hardware runtime.
Included
- 01Complete Python source code
- 02Outward-rounded binary64 interval arithmetic core
- 03Natural and midpoint-shifted polynomial enclosures
- 04Interval Newton simple-root isolator
- 05Two-dimensional Krawczyk system isolator
- 06Nine polynomial benchmarks and three nonlinear systems
- 07One thousand sampled interval-arithmetic audit cases
- 08Tolerance, dependency, residual and runtime studies
- 09Seventeen reproducible project figures and complete retained evidence
- 10102-page project report in PDF and editable Word formats
- 1112-page setup and usage guide in PDF and editable Word formats
- 1245 annotated references and two attributed 2026 literature figures
- 1310 automated tests with 93.86 percent branch-aware coverage
Project record
No information is collected on this page.
- Permanent project ID
- GP-MA-1BWR1UU
- Catalogued
- 21 Aug 2026
- Completed
- 28 Aug 2026
- Verified
- 28 Aug 2026
- Demonstration
- Included in repository
Handover
After purchase
- 01Payment is confirmed
The project is marked unavailable and cannot be purchased again.
- 02Repository access is granted
The buyer's submitted GitHub account receives access to the private repository.
- 03The purchase record is delivered
The certification sheet is prepared from the reviewed buyer details and sent privately by email.