Lesson plan
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. When solving a linear system A \mathbf{x} = \mathbf{b} \,, due to the compounded accumulation of rounding errors, the computed solution \hat{\mathbf{x}} may sometimes deviate from the exact solution \mathbf{x}_\star\,. Starting with \mathbf{x}_1 = \hat{\mathbf{x}}\,, iterative refinement computes a sequence \{\mathbf{x}_1,\, \mathbf{x}_2,\, \mathbf{x}_3,\dots\} which converges to \mathbf{x}_\star\,, when certain assumptions are met.
Article summary
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 A \mathbf{x} = \mathbf{b} \,, due to the compounded accumulation of rounding errors, the computed solution \hat{\mathbf{x}} may sometimes deviate from the exact solution \mathbf{x}_\star\,. Starting with \mathbf{x}_1 = \hat{\mathbf{x}}\,, iterative refinement computes a sequence \{\mathbf{x}_1,\, \mathbf{x}_2,\, \mathbf{x}_3,\dots\} which converges to \mathbf{x}_\star\,, when certain assumptions are met.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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