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.^[1][2]

When solving a linear system ${\displaystyle A\mathbf {x} =\mathbf {b} \,,}$ due to the compounded accumulation of rounding errors, the computed solution ${\displaystyle {\hat {\mathbf {x} }}}$ may sometimes deviate from the exact solution ${\displaystyle \mathbf {x} _{\star }\,.}$ Starting with ${\displaystyle \mathbf {x} _{1}={\hat {\mathbf {x} }}\,,}$ iterative refinement computes a sequence ${\displaystyle \{\mathbf {x} _{1},\,\mathbf {x} _{2},\,\mathbf {x} _{3},\dots \}}$ which converges to ${\displaystyle \mathbf {x} _{\star }\,,}$ when certain assumptions are met.