Mathos Core Library: Fast, Reliable Math Routines for DevelopersMathos Core Library is a modern, lightweight collection of numerical routines designed for developers who need dependable mathematical building blocks without the overhead of large, domain-specific frameworks. Whether you’re writing a performance-critical backend service, a scientific prototype, or numerical components for game engines and embedded systems, Mathos aims to provide clear, well-tested, and efficient implementations of common and advanced math operations.
Why Mathos?
- Focused scope. Mathos concentrates on core numerical tools (linear algebra, special functions, numerical solvers, random number utilities, and basic statistics) rather than trying to be every package under the sun. That reduces API surface area and cognitive load.
- Performance-first mindset. Implementations emphasize predictable performance and minimal allocations. Critical routines expose both high-level convenience APIs and low-level primitives for zero-copy usage.
- Reliability and correctness. Core algorithms include robust edge-case handling and are accompanied by property-based tests and real-world datasets to validate results across platforms.
- Interoperability. Mathos is designed to play well with existing numeric ecosystems: arrays and buffers can be passed in without conversion where possible, and adapters are provided for common matrix/array types.
Key Components
Linear Algebra
Mathos provides a compact linear algebra module covering:
- Dense and sparse matrix representations.
- Basic operations: addition, multiplication, transposition, and scaling.
- Solvers: LU decomposition, Cholesky for positive-definite matrices, QR factorization, and iterative solvers (Conjugate Gradient, GMRES) for large sparse systems.
- Eigenvalue approximations and basic SVD utilities for moderate-sized problems.
Design choices prioritize numerical stability (pivoting strategies, condition checks) and memory efficiency (in-place operations, tiled algorithms).
Special Functions & Numerical Methods
- Accurate implementations of common special functions (gamma, beta, error function) with fallbacks for extreme arguments.
- Root-finding utilities: Brent’s method, Newton–Raphson with safeguarded steps, and secant methods.
- Integration routines: adaptive Simpson and Gauss–Kronrod quadrature for one-dimensional integrals.
- Interpolation: linear, cubic Hermite (PCHIP), and spline-based interpolators with monotonicity-preserving options.
Random Numbers & Statistics
- High-quality pseudorandom generators (Xoshiro256**, PCG) and convenience factories for normal, Poisson, exponential, and other distributions.
- Streaming statistical estimators (online mean, variance, quantile sketches) that work with limited memory.
- Sampling utilities for weighted sampling, reservoir sampling, and bootstrapping.
Utilities & Numeric Helpers
- Safe numeric conversions and clamping helpers.
- Units-agnostic angle utilities, numerically stable trig reductions.
- Bitwise and integer math helpers for hashing, combinatorics, and optimized modular arithmetic.
API Design & Ergonomics
Mathos exposes a layered API:
- Low-level primitives: in-place vector/matrix operations that avoid allocations, ideal for hot loops.
- Mid-level convenience functions: safe-allocating operations with ergonomic signatures for rapid development.
- High-level composables: solver pipelines and prebuilt workflows (e.g., least-squares fitters, smoothing filters).
APIs are intentionally small and consistent: function names and parameter orders follow predictable patterns, and error handling returns rich diagnostics (condition numbers, iteration counts, convergence flags) rather than opaque failures.
Performance Considerations
- In-place variants are provided for all major operations to reduce GC/allocator pressure.
- Memory layout defaults are chosen for cache-friendly access; explicit control over row-major vs column-major layouts is exposed.
- Parallelism: Mathos includes multi-threaded kernels where appropriate, using configurable thread pools and work-stealing strategies. Parallelism is opt-in and non-invasive to single-threaded behavior.
- Benchmarks are included in the repo and continuous integration to avoid regressions. Benchmarks compare against common alternatives and demonstrate where Mathos favors predictability over asymptotic micro-optimizations.
Example micro-optimizations:
- Blocking/tiled matrix multiplication for better L1/L2 cache utilization.
- Avoidance of temporary allocations in chained operations using expression templates or builder patterns (language-dependent).
- Numerically stable summation algorithms (Kahan, pairwise) where precision matters.
Testing, Verification & Documentation
- Extensive unit tests, property-based tests, and fuzz testing target numerical edge cases (NaNs, infinities, denormals, underflow/overflow).
- Regression suites include data from scientific use-cases and known tricky matrices (near-singular, ill-conditioned).
- API docs include examples, performance notes, and recommended idioms for typical workflows (e.g., choosing solvers by matrix properties).
- Cookbooks: concise recipes for common tasks like solving PDE discretizations, parameter estimation, and real-time filtering in games.
Integration Patterns & Examples
Use-case: Solving a large sparse system from finite-difference discretization
- Store the operator in compressed sparse row (CSR) form.
- Use Mathos’ Conjugate Gradient with Jacobi or incomplete Cholesky preconditioner.
- Monitor residual norms and iteration counts for adaptive tolerance.
Use-case: Robust curve fitting
- Build Jacobian via analytic or finite-difference helpers.
- Use Levenberg–Marquardt implementation with damping schedule and parameter bounds.
- Emit covariance estimates for parameter uncertainty.
Use-case: Real-time signal smoothing in an embedded system
- Use in-place low-pass filters and online variance estimators to maintain minimal memory and CPU footprint.
- Choose fixed-point-friendly algorithms when floating-point hardware is constrained.
Comparison with Alternatives
| Aspect | Mathos Core Library | Large Numerical Frameworks | Lightweight Utility Libraries |
|---|---|---|---|
| Scope | Focused core math routines | Broad, domain-specific features | Very small helpers |
| Performance | Predictable, allocation-aware | High-performance but heavier | Fast for small tasks |
| API surface | Small and consistent | Large and complex | Minimal, sometimes inconsistent |
| Use case fit | Embedded, backend, mid-size scientific apps | High-end scientific computing, ML stacks | Quick scripts, glue code |
Contributing & Community
Mathos welcomes contributors who can:
- Add optimized kernels or language bindings.
- Improve numerical robustness and add test cases.
- Write recipes and examples for real-world tasks.
Contribution guidelines emphasize reproducible benchmarks, performance tests, and clear documentation for any new algorithm added.
When to Choose Mathos
Choose Mathos when you want:
- A compact, well-documented set of numerical routines.
- Predictable performance and low-allocation behavior.
- Tools that are easy to integrate into larger systems without pulling heavy dependencies.
Avoid if you need specialized domain functionality (advanced PDE solvers, large-scale machine-learning primitives) that mature, larger ecosystems already provide.
Mathos Core Library aims to be the reliable, efficient toolbox for developers needing essential mathematical operations—balancing speed, accuracy, and simplicity so you can focus on solving domain problems rather than reinventing numeric primitives.
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