dyadic-examples

32 runnable examples for dyadic — the Six-Axiom 2-Adic Operator Calculus library.

cmake -B build && cmake --build build && cmake --build build --target run

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What This Is

dyadic is a header-only C++20 library for arithmetic and calculus over the 2-adic integers ℤ₂ and the ring of formal power series ℤ₂[[t]]. This companion project provides 32 runnable examples covering the full dyadic API, including its four extension headers (<dyadic/dynamic_polynomial.h>, <dyadic/pade.h>, <dyadic/continued_fractions.h>, <dyadic/matrix.h>) which live in the dyadic repository under include/dyadic/.

Extensions

Four extension headers shipped with the dyadic library, drop-in ready — just #include <dyadic/dynamic_polynomial.h> (or whichever) in your project:

Header	What It Adds	Depends On
<dyadic/dynamic_polynomial.h>	Heap-allocated polynomial with runtime-variable degree	dyadic.h
<dyadic/pade.h>	Padé approximant construction in ℤ₂[[t]]	dyadic.h
<dyadic/continued_fractions.h>	Continued fraction expansion and convergent evaluation in ℤ₂[[t]]	dyadic.h, dynamic_polynomial.h
<dyadic/matrix.h>	Generic M×N matrix over ℤ/2^Wℤ with linear algebra	dyadic.h

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dynamic_polynomial.h

Runtime-degree polynomial — the same eval, formal derivative, forward difference, and basis-conversion operations as the compile-time Polynomial<N,W,Basis>, but with heap-allocated storage so the degree is determined at runtime.

DynamicPolynomial<uint32_t> p({1, 2, 3}); // 1 + 2x + 3x²auto df = formal_derivative(p); // 2 + 6xauto taylor = change_basis<TaylorBasis>(p);
// Convert to/from compile-time Polynomial
Polynomial<4, uint32_t> static_p{{1, 2, 3, 4}};
auto dyn = to_dynamic(static_p);
auto back = to_static<4>(dyn);

Arithmetic (+, −, *) works between DynamicPolynomial values of any degree; mixed-degree operations auto-resize the result.

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pade.h

Constructs the [m/n] Padé approximant P(t)/Q(t) from the first m+n+1 terms of a formal power series A(t) = Σ a_i t^i, matching A(t) to order O(t^{m+n}). Uses Gaussian elimination over ℤ₂ to solve for Q's coefficients (requires a_m odd for a unique normal solution with Q(0)=1).

// Geometric series: 1 + t + t²
Polynomial<3, uint32_t> geom{{1, 1, 1}};
auto [P, Q] = pade_approximant<1, 1>(geom);
// P = 1, Q = 1 − t → 1/(1-t) recovered

Works at compile time — all operations are constexpr.

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continued_fractions.h

Expands a power series A(t) into the continued fraction form

A(t) = c₀ + t / (c₁ + t / (c₂ + t / (...)))

by repeated constant-term extraction and degree reduction. The k-th convergent P_k/Q_k is computed via the recurrence:

P₋₁ = 1, P₀ = c₀, Q₋₁ = 0, Q₀ = 1
P_k = c_k · P_{k-1} + t · P_{k-2}
Q_k = c_k · Q_{k-1} + t · Q_{k-2}

DynamicPolynomial<uint32_t> geom({1, 1, 1, 1, 1, 1, 1, 1});
auto c = cf_expand(geom, 6); // {1, 1, 1, 1, 1, 1}auto [P, Q] = cf_convergent(c, 3); // 3rd convergent
W val = P.eval(2) * modinv_odd(Q.eval(2)); // evaluate at t=2

Rational functions produce finitely terminating CF expansions; transcendental series give infinite ones.

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matrix.h

Generic Matrix<M, N, W> over ℤ/2^Wℤ with Gaussian elimination using odd-pivot selection (invertible elements in ℤ₂ are exactly the odd integers).

Matrix<3, 3, uint32_t> A{{{ {1, 2, 3}, {0, 1, 4}, {5, 6, 0} }}};
auto det = A.determinant();
auto inv = A.inverse(); // M×M, Gauss-Jordanauto x = A.solve({7, 11, 13}); // linear systemint r = A.rank(); // Gaussian elimination mod 2

All operations (+, −, *, determinant, rank, inverse, solve, transpose, trace, rref) are constexpr.

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The 32 Examples

Each example is a self-contained .cpp file in examples/. Build and run them all with cmake --build build --target run.

#	Example	What It Demonstrates
01	basis_conversion	Monomial ↔ FallingFactorial ↔ Taylor roundtrip
02	formal_derivative	D = d/dt, Dⁿ annihilates deg N-1 polynomials
03	forward_difference	Δ = e^D − I, identity ΔP = P(t+1) − P(t)
04	taylor_shift	P(t) → P(t + δ) in monomial and FF bases
05	indefinite_sum	Σ = Δ⁻¹, exact inverse in FF basis
06	witt_vectors	Witt addition/multiplication, ghost map, Frobenius, Verschiebung
07	adams_teichmuller	Adams operations ψⁿ, Teichmüller lifts
08	compose_reversion	Power series composition P(Q(t)), Lagrange inversion
09	carry_chain	C = (I−N)⁻¹, one-pass carry propagation
10	2adic_primitives	v₂, modinv_odd, exact_divide, Artin-Schreier ℘(x)
11	stirling_numbers	S₂(n,k), s₁(n,k),
12	polynomial_arithmetic	+, −, ×, eval in all three bases
13	log_exp_series	Binomial series (1+t)^α, geometric series
14	newton_iteration	Newton's method for sqrt, cuberoot in ℤ₂
15	pade_approximants	Rational evaluation, geometric series in ℤ₂[[t]]
16	bernoulli_numbers	Bₙ via generating function, Σ C(n+1,k)B_k = 0
17	eulerian_numbers	A(n,k) table, Worpitzky identity
18	bell_numbers	Bₙ via Stirling sum, recurrence B_{n+1} = Σ C(n,k)B_k
19	hensel_lifting	Root lifting mod 2^k for polynomials over ℤ₂
20	witt_division	Witt vector inverse a⁻¹, ghost inverse verification
21	resultant_discriminant	Resultant, discriminant via Sylvester matrix
22	continued_fractions	CF expansion, convergent P/Q, finite CF for rationals
23	benchmark_precision	Timing across word sizes (uint8–uint64), precision analysis
24	compiletime_benchmarks	Compile-time vs runtime operation timing
25	csv_output	Generates CSV files for Stirling/Witt/precision visualization
26	witt_log_exp	Witt inverse, exp/log homomorphisms, Adams compose
27	dynamic_polynomial	Runtime-degree polynomial, basis conversion, D, Δ
28	hardware_arithmetic	adc, add_overflow, mul_overflow, carry chain integration
29	pade_approximant	Padé [1/1], [2/2], rational recovery, constexpr usage
30	matrix	Matrix operations, determinant, inverse, rank, solve
31	discrete_hedging	Δ/Σ operators in finance: daily returns, discrete gamma, cumulative P&L, continuous vs discrete hedge ratios
32	extension_verify	Conformance tests for all four extension headers (49 checks)

Build

Dependencies

• C++20 compiler (GCC 12+, Clang 17+)
• dyadic — expected at ../dyadic relative to this repo (include path to ../dyadic and ../dyadic/include)

Quick start

cmake -B build
cmake --build build
# Run all 32 examples (returns 0 on pass, non-zero if any example fails)
cmake --build build --target run
# Run a single example
./build/01_basis_conversion

The run target runs every example in sequence and prints PASS/FAIL results. Example 32_extension_verify returns exit code 0 when all conformance checks pass, 1 otherwise.

Project integration

If you already have a CMake project, link against the dyadic interface target:

add_subdirectory(path/to/dyadic-examples)
target_link_libraries(my_appPRIVATEdyadic)

This gives you the -I paths for both <dyadic.h> and <dyadic/...> extension headers in a single step.

Project Structure

dyadic-examples/
├── CMakeLists.txt # Build system for all 32 examples
├── cmake/
│ └── run_all.sh # Script invoked by `--target run`
├── examples/
│ ├── 01_basis_conversion.cpp
│ ├── 02_formal_derivative.cpp
│ ├── ... # (32 .cpp files total)
│ ├── 30_matrix.cpp
│ ├── 31_discrete_hedging.cpp
│ └── 32_extension_verify.cpp
├── LICENSE # MIT
└── README.md

Design Notes

DynamicPolynomial Edge Cases

• degree() returns -1 for empty polynomials: An empty polynomial (default-constructed with no coefficients) has degree -1. Its eval() returns 0. Multiplying an empty polynomial with any other polynomial returns an empty polynomial.
• operator+= doesn't resize this to o's size: If o.size() > this->size(), *this is resized. But *this is never truncated if o is shorter — the extra coefficients remain.
• All-zero DynamicPolynomial is non-empty: Constructing with DynamicPolynomial<W>({0, 0, 0}) gives a degree-0 polynomial (the trailing zeros are not trimmed). Use coeff.resize(degree() + 1) to trim.
• Carry-chain operator*: Uses poly_mul (same as static Polynomial), NOT coefficient-wise multiplication. Use poly_mul_cw on the internal coeff array for coefficient-wise.
• to_static<N>(dyn) truncates: If dyn.degree() >= N, the top coefficients are silently dropped.

Continued Fractions

• All-zero series: cf_expand on an all-zero series yields all-zero CF coefficients (not an error).
• Single-term series: A series with only a constant term produces a single CF coefficient equal to that constant.
• cf_convergent(k) for k=0: Returns P_0 = c_0, Q_0 = 1.

Matrix

• Singular inputs: inverse() returns the zero matrix for singular inputs (no exception). solve() returns a zero vector for singular systems. This matches the "no exceptions" design of the core library.
• Rectangular matrices: rank(), transpose(), and operator ==/!= work on arbitrary M×N dimensions. Determinant and inverse are only defined for square matrices.

Related

• dyadic — the core library
• dyadic/verify.h — compile-time proofs

License

MIT — see LICENSE.