Iterative Methods Introduction

Numerical Analysis · Axiom Academy

LESSON Introduction to Iterative Methods When direct methods aren't practical Direct methods like Gaussian elimination have a problem: they require O(n³) operations and O(n²) storage. For a sparse 1,000,000 × 1,000,000 matrix (common in PDEs), that's: All iterative methods follow the same pattern: start with a guess x⁽⁰⁾ and improve it step by step: The iteration matrix G and vector c depend on the specific method. The key question: Does x⁽ᵏ⁾ converge to the true solution? Most classical iterative methods come from splitting A = M - N, where M is easy to invert: Here, A = D + L + U where D is diagonal, L is strictly lower triangular, and U is strictly upper triangular. The iteration x⁽ᵏ⁺¹⁾ = Gx⁽ᵏ⁾ + c converges for any starting point if and only if: where ρ(G) is the spectral radius —the largest eigenvalue magnitude of G. You now understand why iterative methods matter and when they converge!

This is the written version of the interactive lesson above. See the full Numerical Analysis course.