thales 0.4.2

A comprehensive Computer Algebra System (CAS) library for symbolic mathematics, equation solving, calculus, and linear algebra
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# Implementation Status

This document tracks the implementation status of Thales features against the PRD.

## Legend

- Implemented: Feature is complete and tested
- In Progress: Partially implemented
- Planned: Scheduled for a future release
- De-scoped: Will not be implemented

## Core Parsing

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Expression parsing | Yes | Yes | Yes | Implemented | `parse_expression()` |
| Equation parsing | Yes | Yes | Yes | Implemented | `parse_equation()` |
| Equation system parsing | Yes | Yes | Yes | Implemented | Via JSON array in FFI |
| Implicit multiplication | Yes | Yes | Yes | Implemented | `2x`, `3sin(x)` |
| LaTeX input (basic) | Yes | Yes | Yes | Implemented | Fractions, roots, greek letters, trig |
| LaTeX input (integrals) | Yes | No | No | In Progress | `\int`, `\int_{a}^{b}` — FFI/Swift pending |
| LaTeX input (limits) | Yes | No | No | In Progress | `\lim_{x \to a}` — FFI/Swift pending |
| LaTeX input (summation) | Yes | No | No | In Progress | `\sum_{i=a}^{b}` — FFI/Swift pending |
| LaTeX input (matrices) | No | No | No | Planned | `\begin{pmatrix}` |
| LaTeX input (partial) | No | No | No | Planned | `\partial` |
| LaTeX output | Yes | Yes | Yes | Implemented | `to_latex()` |
| MathML input | No | No | No | De-scoped | Use LaTeX instead |

## Algebra

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Linear equation solving | Yes | Yes | Yes | Implemented | |
| Quadratic equation solving | Yes | Yes | Yes | Implemented | |
| Cubic equation solving | Yes | Yes | Yes | Implemented | |
| Polynomial solving | Yes | Yes | Yes | Implemented | |
| Transcendental equations | Yes | Yes | Yes | Implemented | |
| Equation systems (linear) | Yes | Yes | Yes | Implemented | |
| Equation systems (nonlinear) | Yes | Yes | Yes | Implemented | Newton-Raphson, Broyden |
| Inequality solving | Yes | Yes | Yes | Implemented | |
| Partial fractions | Yes | Yes | Yes | Implemented | |
| Pattern matching simplification | Yes | Yes | Yes | Implemented | |

## Calculus

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Symbolic differentiation | Yes | Yes | Yes | Implemented | |
| Nth derivative | Yes | Yes | Yes | Implemented | |
| Gradient | Yes | Yes | Yes | Implemented | |
| Symbolic integration | Yes | Yes | Yes | Implemented | |
| Integration by parts | Yes | Yes | Yes | Implemented | |
| Integration by substitution | Yes | Yes | Yes | Implemented | |
| Definite integrals | Yes | Yes | Yes | Implemented | |
| Improper integrals | Yes | Yes | Yes | Implemented | |
| Numerical integration | Yes | Yes | Yes | Implemented | Adaptive Simpson |
| Limits | Yes | Yes | Yes | Implemented | |
| Limits to infinity | Yes | Yes | Yes | Implemented | |

## Series

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Taylor series | Yes | Yes | Yes | Implemented | |
| Maclaurin series | Yes | Yes | Yes | Implemented | |
| Laurent series | Yes | Yes | Yes | Implemented | |
| Asymptotic series | Yes | Yes | Yes | Implemented | |
| Fourier series | Yes | Yes | No | In Progress | Swift wrapper pending |
| Series composition | Yes | No | No | Implemented | Rust-only |
| Series reversion | Yes | No | No | Implemented | Rust-only |

## ODE Solvers

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| 1st-order separable | Yes | Yes | Yes | Implemented | |
| 1st-order linear | Yes | Yes | Yes | Implemented | |
| 1st-order IVP | Yes | Yes | Yes | Implemented | |
| 2nd-order homogeneous | Yes | Yes | No | In Progress | Swift wrapper pending |
| 2nd-order IVP | Yes | Yes | No | In Progress | Swift wrapper pending |
| Higher-order (constant coeff) | Yes | Yes | No | In Progress | Swift wrapper pending |
| Runge-Kutta (RK4) | Yes | Yes | No | In Progress | Swift wrapper pending |
| ODE systems (numerical) | Yes | No | No | In Progress | FFI pending |
| Variation of parameters | No | No | No | Planned | |

## Numerical Methods

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Newton-Raphson | Yes | Yes | Yes | Implemented | |
| Bisection method | Yes | Yes | Yes | Implemented | |
| Brent's method | Yes | Yes | Yes | Implemented | |
| Secant method | Yes | No | No | Implemented | Rust-only, tested |
| Smart solver (auto-select) | Yes | Yes | Yes | Implemented | |
| Gradient descent | Yes | No | No | Implemented | Rust-only |
| Levenberg-Marquardt | Yes | No | No | Implemented | Rust-only |

## Special Functions

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Gamma function | Yes | Yes | Yes | Implemented | |
| Beta function | Yes | Yes | Yes | Implemented | |
| Error function (erf) | Yes | Yes | Yes | Implemented | |
| Complementary erf (erfc) | Yes | Yes | Yes | Implemented | |

## Transforms and Geometry

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| 2D coordinate transforms | Yes | No | No | Implemented | Cartesian, Polar |
| 3D coordinate transforms | Yes | No | No | Implemented | Cartesian, Cylindrical, Spherical |
| Transform2D (translate/rotate/scale) | Yes | No | No | Implemented | |
| Complex number operations | Yes | Yes | Yes | Implemented | |
| Complex nth roots | Yes | No | No | Implemented | |
| Matrix operations | Yes | No | No | Implemented | |

## Dimensional Analysis

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Dimension arithmetic | Yes | No | No | Implemented | multiply, divide, power |
| Unit conversions | Yes | No | No | Implemented | |
| Quantity with units | Yes | No | No | Implemented | |

## Precision and Optimization

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Precision-controlled evaluation | Yes | Yes | No | In Progress | Swift wrapper pending |
| Manual computation optimization | Yes | Yes | No | In Progress | Swift wrapper pending |
| Small angle approximations | Yes | Yes | Yes | Implemented | |

## Expression Simplification

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Algebraic simplification | Yes | Yes | Yes | Implemented | |
| Trigonometric simplification | Yes | Yes | Yes | Implemented | |
| Pattern-based rewriting | Yes | Yes | Yes | Implemented | |

## Resolution Paths

| Feature | Rust | FFI | Swift | Status | Notes |
|---------|------|-----|-------|--------|-------|
| Step-by-step solutions | Yes | Yes | Yes | Implemented | |
| LaTeX formatted steps | Yes | Yes | Yes | Implemented | |
| Operation tracking | Yes | Yes | Yes | Implemented | |