# TODO
## Done
### Core
- [x] Expression AST, parser, evaluator with Context
- [x] Expression equality checking (canonical form with commutative sorting, constant folding)
### Symbolic algebra
- [x] Symbolic differentiation (product, quotient, chain rules)
- [x] **Multi-variable symbolic differentiation** (partial derivatives and gradients)
- [x] Algebraic simplification (constant folding, identities)
- [x] **Symbolic integration** for polynomial, exponential, trigonometric, and inverse-trigonometric primitives
### Numerical calculus
- [x] Numerical derivatives (high-order finite difference) and gradients
- [x] Trapezoidal, Simpson's, and adaptive quadrature
- [x] **Romberg with Richardson extrapolation**
- [x] **Fourier series** (numerical coefficient computation via Simpson's rule, evaluation)
- [x] **Monte Carlo integration** (1-D and N-D, reproducible LCG, standard error)
- [x] **Stochastic primitives** (Rng, uniform/normal/exponential sampling, normal/exp PDF/CDF)
- [x] **Moment / cumulant helpers** (skewness, excess kurtosis, cumulants up to order 4)
- [x] **Hilbert-matrix-aware solvers** (Tikhonov regularisation, Hilbert matrix construction, least-squares for rectangular systems)
### Equation solving
- [x] Bisection, Newton–Raphson, secant
- [x] Durand–Kerner polynomial root finding
- [x] **Polynomial root isolation** (VAS method with i128 exact arithmetic)
- [x] **Newton's method for nonlinear systems** (central-difference Jacobian)
### FFT
- [x] Cooley–Tukey radix-2 from scratch: forward, inverse, 2D, real-input
- [x] Magnitude / power spectra
- [x] Convolution and cross-correlation
- [x] Window functions (Hann, Hamming, Blackman, Rectangular)
### Matrix
- [x] Arithmetic (add, sub, mul, scalar), transpose, trace
- [x] Gaussian elimination: determinant, inverse, linear solve
- [x] **Rank** estimation
- [x] **LU decomposition** with partial pivoting
- [x] **Cholesky decomposition** `A = L · Lᵀ` for symmetric positive-definite matrices
- [x] **SVD** `A = U · Σ · Vᵀ` via one-sided Jacobi rotations
- [x] **Power iteration** for the dominant eigenvalue/eigenvector
- [x] **QR algorithm** for full symmetric eigenvalue decomposition (Householder tridiagonalisation + Wilkinson-shift QR iteration)
- [x] **Hessenberg decomposition** `A = Q·H·Qᵀ` via Householder reflections
- [x] **Real Schur decomposition** `A = Q·T·Qᵀ` via shifted QR on Hessenberg form
- [x] **`det` REPL command** for matrix determinant
### Statistics
- [x] Mean, median, variance, standard deviation
- [x] Quartiles, IQR
- [x] Pearson correlation
- [x] Linear regression
### Number theory
- [x] GCD, LCM, extended GCD, modular inverse, modular exponentiation
- [x] Primality (trial division + Miller–Rabin, deterministic for n < 3.3e24)
- [x] Prime factorization, sieve of Eratosthenes
- [x] Binomial coefficients, factorial, Fibonacci (fast doubling)
- [x] Euler's totient
- [x] **Jacobi symbol**
- [x] **Continued fractions** (rational and real-valued approximants)
- [x] **Linear Diophantine solver**
- [x] **Discrete logarithm** (baby-step giant-step)
- [x] Chinese Remainder Theorem
### ODE
- [x] Euler, RK4, RK4 systems
- [x] Adaptive RKF45
### Interpolation
- [x] Lagrange, Newton divided-difference, linear
- [x] **Natural / clamped cubic spline** (Thomas algorithm for second derivatives)
- [x] **Chebyshev polynomials** `T_n(x)`, Chebyshev nodes, series approximation, Clenshaw evaluation
- [x] **Legendre polynomials** `P_n(x)`, associated `P_n^m(x)`, **Gauss–Legendre quadrature**
### Special functions
- [x] Gamma, log-Gamma (Lanczos)
- [x] Beta function
- [x] erf, erfc (via incomplete gamma)
- [x] sinc, incomplete gamma P
- [x] **Bessel functions** `J_0(x)`, `J_1(x)`, integer-order `J_n(x)` (Maclaurin series + asymptotic + forward recurrence)
### Taylor series
- [x] Symbolic expansion around any point
- [x] **Laurent series** (expansion around poles, negative powers, principal + analytic parts)
### Rational arithmetic
- [x] **Rational number type** (exact arithmetic, GCD reduction, i128 intermediate, parsing, REPL)
### Web notebook
- [x] **Math notebook** (`.mnb` file format, JSON cells with TeX/math input + output)
- [x] **Web notebook server** (minimal HTTP server, Jupyter-like UI with KaTeX rendering, cell eval, save/load)
- [x] **Step-by-step solving** (`dispatch_steps` — shows intermediate steps for diff, solve, taylor, integrate, simplify, rat, laurent)
- [x] **Exact rational evaluation in notebook** (fraction expressions evaluated as `Rational`, returning exact fractions instead of decimals)
- [x] **KaTeX math rendering** (input preview + output rendering, plain-to-LaTeX converter, decimal-to-fraction display)
- [x] **Fallback to simplify on unbound variables** (expressions with variables that can't be evaluated are simplified instead of erroring)
- [x] **`det` command in notebook/REPL** (matrix determinant via `det <rows>`)
- [x] **Shared context across cells** (`let`/`fn` bindings persist across cells, `/api/reset` + `/api/context` endpoints, `dispatch_with_ctx` API)
- [x] **Cell types** (Math cells evaluated with KaTeX, Text cells for documentation — `CellType` enum, JSON `cell_type` field)
- [x] **Cell management** (move up/down, duplicate, toggle type, execution status indicators, context panel, Reset & Run All)
- [x] **Inline plots** (`plot` commands render PNG images directly in the notebook via base64, `plot_function_to_bytes`/`plot_multi_to_bytes`/`plot_scatter_to_bytes`)
- [x] **Markdown text cells** (text cells render Markdown via marked.js — headings, lists, code, blockquotes)
- [x] **Execution counters** (`In [n]:` indicators like Jupyter, Alt+Enter to run + add cell)
### Other
- [x] Complex number type with arithmetic, polar conversion, powers
- [x] PNG plotting (line, multi-series, scatter) via `plotters`
- [x] LaTeX / TeX input (`\frac`, `\sqrt`, `\sin`, `\pi`, `\left(\right)`, `^{...}`, `\Gamma`, `\log_2`, …; `$...$`, `$$...$$`, `\[...\]`, `\(...\)`)
- [x] Interactive REPL (rustyline-powered with history)
- [x] CLI subcommands and REPL dispatch for all features
- [x] **436 inline unit tests** + 165 integration tests — all passing
- [x] AGENTS.md, README.md, ARCHITECTURE.md, SPEC.md
### Fast math
- [x] **Chebyshev-based fast math library** — `ChebyshevApprox` struct, `fast_sin`/`fast_cos`/`fast_tan`/`fast_exp`/`fast_log`/`fast_sqrt`/`fast_pow` with argument reduction, `fast` REPL command
### Big integers
- [x] **Big integer support** — `bigint` module with arbitrary-precision primality (Miller–Rabin), factorization (trial division + Pollard's rho), GCD, LCM, factorial, Fibonacci (fast doubling), binomial, modular exponentiation, totient. REPL commands `fact`/`fib`/`binom` auto-upgrade to BigInt on u64 overflow (SymPy-style). `big` command for explicit big-integer ops on inputs > u64::MAX.
### Automatic differentiation
- [x] **Dual numbers** — `autodiff` module with `Dual` type (value + derivative), full arithmetic operator overloads, elementary functions (sin, cos, tan, exp, ln, log, sqrt, powf, pow_dual, asin, acos, atan, sinh, cosh, tanh, abs), `eval` for `Expr` AST, `derivative`, `gradient`, `jacobian`, `ad` REPL command. Naming follows AD literature conventions.
### Serialization
- [x] **Expression serialization** — `serialize` module with three interchangeable formats: S-expressions (`to_sexpr`/`from_sexpr`), JSON (`to_json`/`from_json`), and RPN (`to_rpn`/`from_rpn`). All round-trip via `Expr::equals`. Hand-rolled JSON parser (no serde dep). `serialize <fmt> <expr>` and `serialize <fmt> import <text>` REPL commands.
### Interval arithmetic
- [x] **Interval arithmetic** — `interval` module with `Interval` type (closed `[lo, hi]` bounds), full arithmetic (+, -, *, /, neg), `sqr`/`powi` with zero-crossing awareness, elementary functions (sin, cos, tan, exp, ln, log10, log2, sqrt, abs, asin, acos, atan, sinh, cosh, tanh, cbrt) with monotonicity and extrema tracking, set operations (intersect, hull, contains, overlaps), and `eval_interval` for rigorous bounds over the `Expr` AST. `interval <expr> with <var>=[lo,hi],...` REPL command. Pure Rust, no deps.
### Matrix decompositions
- [x] **QR decomposition** — Householder reflections for `A = Q·R` where Q is orthogonal and R is upper-triangular. Handles rectangular m×n matrices (tall and wide). Least-squares solve for overdetermined systems. `qr <rows>` REPL command.
## Brainstorming
### High Priority
(none currently)
### Medium Priority
(none currently)
### Low Priority
- [ ] Arbitrary-precision arithmetic (BigDecimal)
- [x] **Automatic differentiation (dual numbers)** — `autodiff` module with `Dual` type, `derivative`, `gradient`, `jacobian`, `ad` REPL command
- [x] **MathML support** — W3C Presentation MathML export (`to_mathml`, `to_mathml_doc`) and import (`from_mathml`). `mathml` REPL command for both directions. Supports `<mn>`, `<mi>`, `<mo>`, `<mrow>`, `<mfrac>`, `<msup>`, `<msub>`, `<msqrt>`, `<mroot>`, `<mfenced>`, `<mstyle>`, `<mtext>`.
- [x] **Expression serialization** — S-expressions, JSON, RPN (`serialize` module, `serialize` REPL command)
- [x] **Interval arithmetic** — rigorous bounds via `Interval` type + `eval_interval` over Expr AST (`interval` module, `interval` REPL command)
- [ ] GPU-accelerated FFT (via `wgpu`)
- [ ] 3D plotting (surface plots, contour plots)
- [ ] Animated plot output (GIF/WebM)