quantumquirkxyz Contract
- Input: a numerical or symbolic mathematical problem.
- Output: Markdown artifact with algorithm choice, error analysis, computation, and verification.
- Side effects: none.
- Dependencies: none.
- Stop condition: artifact complete with error bounds and verification.
- Risk: low — computation only, no system changes.
- Boundary: may execute code for numerical work but only in-memory, no file writes.
Computational Math
Solve a mathematical problem numerically or symbolically with an algorithm whose error, stability, and cost are explicit. Never ship a black-box answer.
When to use
- The user wants a numerical result, symbolic simplification, or to choose between algorithms.
- A formula or expression needs to be evaluated, integrated, differentiated, or solved.
- An engineering / physics / quant problem needs a numerical kernel.
Process
1. Frame the problem
Identify: input → output, the mathematical object (scalar / vector / function / matrix / ODE / PDE / integral), and the regime (smoothness, scale, boundary conditions). State whether an analytical or numerical answer is expected.
Completion criterion: problem restated as a clean mapping with regime noted.
2. Choose the algorithm
Pick the algorithm that matches the regime:
- Root finding: bisection (safe), Newton's (fast, needs derivative + good init), secant.
- Integration: Simpson / Gauss (smooth, 1D), Monte Carlo (high-D), adaptive quadrature (singularities).
- ODEs: explicit Euler (cheap, low order), RK4 (default), implicit (stiff), symplectic (Hamiltonian).
- Linear systems: direct (LU, Cholesky) vs iterative (CG, GMRES).
- Symbolic: CAS (SymPy / Mathematica) with manual simplification on top.
Name the algorithm and one reason it fits.
Completion criterion: algorithm named, with a one-line justification referencing the regime.
3. Estimate error and stability
For numerical work, state:
- Local truncation error and global error order.
- Conditioning — well-posed or ill-posed? Sensitivity to input noise.
- Stability — bounded-input-bounded-output? Stiffness concerns?
- Cost — O(n²)? O(n³)? Memory?
For symbolic work, state the simplification rules used and any non-canonical choices.
Completion criterion: error, stability, and cost all addressed for the chosen algorithm.
4. Implement / compute
Either run code (Python with NumPy/SciPy/SymPy) or hand-compute if tractable. For code, include:
- The algorithm in named functions, not one giant cell.
- Input validation (NaN, inf, dimension mismatch).
- Sanity asserts on outputs (sign, magnitude, symmetry).
Completion criterion: output produced and reproducible from the code shown.
5. Verify
Cross-check by at least one independent path:
- Different algorithm at lower fidelity.
- Asymptotic / limiting case (small/large parameter).
- Comparison against a known analytic solution or benchmark.
- Dimensional / sign / magnitude check.
Completion criterion: independent verification done; discrepancy explained or fixed.
6. Deliver
Markdown artifact with: problem, algorithm choice, error/stability/cost, code or computation, verification, and final answer with stated uncertainty (e.g. "to 6 significant digits").
Completion criterion: artifact contains algorithm, error bounds, verification, and a final answer whose precision is justified.