role-algorithms:numerical-methods
Implements numerical methods — floating-point arithmetic (IEEE 754, Kahan summation, catastrophic cancellation), matrix operations (LU/QR/SVD decomposition, sparse formats), root finding (Newton-Raphson, Brent's method), numerical integration (Simpson, Gaussian quadrature, adaptive), FFT/NTT, and...
SKILL.md
Full skill instructions
Numerical Methods
When to use
- Diagnosing floating-point precision issues (cancellation, accumulation error, overflow)
- Choosing between LU, QR, or SVD decomposition for a linear algebra problem
- Implementing root-finding when an analytical solution does not exist
- Selecting a numerical integration method based on smoothness and accuracy requirements
- Using FFT for polynomial multiplication, convolution, or signal analysis
- Reviewing cryptographic primitive usage for correctness and safety
Core principles
- Never compare floats with == — always use absolute or relative epsilon; this is not optional
- Partial pivoting is not optional in Gaussian elimination — skipping it breaks numerical stability
- SVD is the Swiss Army knife — when LU and QR fail or when rank matters, SVD solves it
- Brent's method is the default root finder — combines guaranteed convergence with superlinear speed
- Never implement your own crypto — use libsodium, OpenSSL, or Web Crypto API; rolling your own is a security incident waiting to happen
Reference Files
references/floating-point-and-matrix.md— IEEE 754 pitfalls, Kahan summation, precision type selection, Gaussian elimination with pivoting, LU/QR/SVD decompositions, sparse matrix formats (COO/CSR/CSC)references/root-finding-and-integration.md— bisection, Newton-Raphson, secant, Brent's method, trapezoidal/Simpson's/Gaussian quadrature, adaptive integration, Cooley-Tukey FFT, NTT for exact polynomial multiplicationreferences/cryptographic-foundations.md— secure hash functions (SHA-256, BLAKE3), HMAC, AES-256-GCM, ChaCha20-Poly1305, key derivation (Argon2, bcrypt), RSA/ECDSA/ECDH, timing-safe comparisons, CSPRNG usage