role-algorithms:numerical-methods

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

1. Never compare floats with == — always use absolute or relative epsilon; this is not optional
2. Partial pivoting is not optional in Gaussian elimination — skipping it breaks numerical stability
3. SVD is the Swiss Army knife — when LU and QR fail or when rank matters, SVD solves it
4. Brent's method is the default root finder — combines guaranteed convergence with superlinear speed
5. 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 multiplication
• references/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