Iterative Refinement - Error Analysis

Error Analysis

As a rule of thumb, iterative refinement for Gaussian elimination produces a solution correct to working precision if double the working precision is used in the computation of r, e.g. by using quad or double extended precision IEEE 754 floating point, and if A is not too ill-conditioned (and the iteration and the rate of convergence are determined by the condition number of A).

More formally, assuming that each solve step is reasonably accurate, i.e., in mathematical terms, for every m, we have

A(I + F_m)d_m = r_m

where ‖F_m‖_∞ < 1, the relative error in the th iterate of iterative refinement satisfies

where

• ‖·‖_∞ denotes the ∞-norm of a vector,
• κ_∞(A) is the ∞-condition number of A,
• is the order of A,
• ε₁ and ε₂ are unit round-offs of floating-point arithmetic operations,
• σ, μ₁ and μ₂ are constants depending on A, ε₁ and ε₂

if A is “not too badly conditioned”, which in this context means

0 < σκ_∞(A)ε₁ ≪ 1

and implies that μ₁ and μ₂ are of order unity.

The distinction of ε₁ and ε₂ is intended to allow mixed-precision evaluation of r_m where intermediate results are computed with unit round-off ε₂ before the final result is rounded (or truncated) with unit round-off ε₁. All other computations are assumed to be carried out with unit round-off ε₁.