Iterative Numerical Methods & Stability

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Abstractions about iterative acceleration, harmonic balance, residuals, recurrence diagnostics, truncation error, and numerical stability.

7 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Aitken's delta-squared process — A nonlinear sequence transformation that accelerates approximately linear convergence by extrapolating from three consecutive terms and canceling the leading error mode.
• Harmonic Balance — Approximate a nonlinear system's periodic steady state by a truncated Fourier series and solve for coefficients whose retained residual harmonics balance to zero.
• Iterative method — A numerical procedure that repeatedly updates an approximation from prior iterates toward a solution under a convergence rule.
• Recurrence plot — A square matrix visualization marking pairs of observation times whose reconstructed system states are equal or sufficiently close under a declared metric and threshold.
• Residual (numerical analysis) — The discrepancy obtained by substituting an approximate solution into the original equation, usually b−f(x₀) or its signed convention.
• Truncation error — The discrepancy introduced when an exact infinite, limiting or continuous mathematical process is replaced by a finite approximation, distinct from finite-precision roundoff and analyzable through omitted terms or local discretization expansions.
• Von Neumann stability analysis — A Fourier-mode method for testing linear finite-difference schemes by requiring their amplification factors not to grow beyond the stability bound.