Iterative refinement is an iterative method proposed by James H. Wilkinson to improve the accuracy of numerical solutions to systems of linear equations.

due to the compounded accumulation of rounding errors when certain assumptions are met.

Description

the th iteration of iterative refinement consists of three steps:

The crucial reasoning for the refinement algorithm is that although the solution for in step, the calculation of the residual in step(i), in comparison, is numerically nearly exact: You may not know the right answer very well, but you know quite accurately just how far the solution you have in hand is from producing the correct outcome (). If the residual is small in some sense, then the correction must also be small, and should at the very least steer the current estimate of the answer,

The iterations will stop on their own when the residual is zero, or close enough to zero that the corresponding correction is too small to change the solution which produced it; alternatively, the algorithm stops when is too small to convince the linear algebraist monitoring the progress that it is worth continuing with any further refinements.

for each iteration. If the matrix equation is solved using a direct method, such as Cholesky or LU decomposition is done once and is reused for the relatively inexpensive forward and back substitution to solve for at each iteration.

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 , e.g. by using quad or double extended precision IEEE 754 floating point, and if is not too ill-conditioned (and the iteration and the rate of convergence are determined by the condition number of ).

More formally, assuming that each step(ii) can be solved reasonably accurately, i.e., in mathematical terms, for every

where , the relative error in the

where

• denotes the ∞-norm of a vector,

• is the -condition number of ,

• is the order of ,

• and are unit round-offs of floating-point arithmetic operations,

• , and are constants that depend on , and

if is "not too badly conditioned", which in this context means

and implies that and are of order unity.

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

References