Iterative refinement

Iterative refinement is an iterative method proposed by James H. Wilkinson to improve the accuracy of numerical solutions to systems of linear equations. Iterative refinement is an iterative method proposed by James H. Wilkinson to improve the accuracy of numerical solutions to systems of linear equations. When solving a linear system Ax = b, due to the presence of rounding errors, the computed solution x̂ may sometimes deviate from the exact solution x*. Starting with x1 = x̂, iterative refinement computes a sequence {x1,x2,x3,...} which converges to x* when certain assumptions are met. For m = 1,2,..., the m {displaystyle m} th iteration of iterative refinement consists of three steps: 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 where ‖Fm‖∞ < 1, the relative error in the m {displaystyle m} th iterate of iterative refinement satisfies