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'''Iterative refinement''' is an [[iterative method]] proposed by James H. Wilkinson to improve the accuracy of numerical solutions to [[systems of linear equations]].
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When solving a linear system {{math|'''<var>Ax</var>''' {{=}} '''<var>b</var>'''}}, due to the presence of [[Round-off error|rounding error]]s, the computed solution {{math|'''<var>x</var>&#x302;'''}} may sometimes deviate from the exact solution {{math|'''<var>x</var>'''<sup>*</sup>}}. Starting with {{math|'''<var>x</var>'''<sub>1</sub> {{=}} '''<var>x</var>&#x302;'''}}, iterative refinement computes a sequence {{math|{'''<var>x</var>'''<sub>1</sub>,'''<var>x</var>'''<sub>2</sub>,'''<var>x</var>'''<sub>3</sub>,…&#x7d;}} which converges to {{math|'''<var>x</var>'''<sup>*</sup>}} when certain assumptions are met.
 
==Description==
For {{math|<var>m</var> {{=}} 1,2,&hellip;}}, the <math>m</math>th iteration of iterative refinement consists of three steps:
# Compute the residual<br>{{math|'''<var>r</var>'''<sub><var>m</var></sub> {{=}} '''<var>b</var>''' &minus; '''<var>Ax</var>'''<sub><var>m</var></sub>}}
# Solve the system<br>{{math|'''<var>Ad</var>'''<sub><var>m</var></sub> {{=}} '''<var>r</var>'''<sub><var>m</var></sub>}}
# Add the correction<br>{{math|'''<var>x</var>'''<sub><var>m</var>+1</sub> {{=}} '''<var>x</var>'''<sub><var>m</var></sub> + '''<var>d</var>'''<sub><var>m</var></sub>}}
 
==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 {{math|<var>r</var>}}, e.g. by using [[Quadruple-precision floating-point format|quad]] or [[extended precision|double extended]] precision [[IEEE 754]] [[floating point]], and if {{math|<var>A</var>}} is not too ill-conditioned (and the iteration  and the rate of convergence are determined by the condition number of {{math|<var>A</var>}}).<ref>{{cite book|first=Nicholas | last=Higham |title=Accuracy and Stability of Numerical Algorithms (2 ed)| publisher=SIAM|year=2002 | pages=232  }}</ref>
 
More formally, assuming that each solve step is reasonably accurate, i.e., in mathematical terms, for every {{math|<var>m</var>}}, we have
 
:{{math|'''<var>A</var>'''('''<var>I</var>''' + <var>'''F'''<sub>m</sub></var>)<var>'''d'''<sub>m</sub></var> {{=}} <var>'''r'''<sub>m</sub></var>}}
 
where {{math|&#x2016;<var>'''F'''<sub>m</sub></var>&#x2016;<sub>&infin;</sub> < 1}}, the [[Approximation error|relative error]] in the <math>m</math>th iterate of iterative refinement satisfies
 
:<math>\frac{\lVert\boldsymbol{x}_m-\boldsymbol{x}^\ast\rVert_\infty}{\lVert\boldsymbol{x}^\ast\rVert_\infty}\leq\bigl(\sigma\kappa_\infty(\boldsymbol{A})\epsilon_1\bigr)^m+\mu_1\epsilon_1+\mu_2n\kappa_\infty(\boldsymbol{A})\epsilon_2</math>
 
where
* {{math|&#x2016;&middot;&#x2016;<sub>&infin;</sub>}} denotes the [[Uniform norm|{{math|&infin;}}-norm]] of a vector,
* {{math|<var>&kappa;</var><sub>&infin;</sub>('''<var>A</var>''')}} is the {{math|&infin;}}-[[condition number]] of {{math|'''<var>A</var>'''}},
* <math>n</math> is the order of {{math|'''<var>A</var>'''}},
* {{math|<var>&epsilon;</var><sub>1</sub>}} and {{math|<var>&epsilon;</var><sub>2</sub>}} are [[Machine epsilon|unit round-offs]] of [[Floating point|floating-point]] arithmetic operations,
* {{math|<var>&sigma;</var>}}, {{math|<var>&mu;</var><sub>1</sub>}} and {{math|<var>&mu;</var><sub>2</sub>}} are constants depending on {{math|'''<var>A</var>'''}}, {{math|<var>&epsilon;</var><sub>1</sub>}} and {{math|<var>&epsilon;</var><sub>2</sub>}}
if {{math|'''<var>A</var>'''}} is “not too badly conditioned”, which in this context means
 
:{{math|0 &lt; <var>&sigma;&kappa;</var><sub>&infin;</sub>('''<var>A</var>''')<var>&epsilon;</var><sub>1</sub> &#x226a; 1}}
 
and implies that {{math|<var>&mu;</var><sub>1</sub>}} and {{math|<var>&mu;</var><sub>2</sub>}} are of order unity.
 
The distinction of {{math|<var>&epsilon;</var><sub>1</sub>}} and {{math|<var>&epsilon;</var><sub>2</sub>}} is intended to allow mixed-precision evaluation of {{math|'''<var>r</var>'''<sub><var>m</var></sub>}} where intermediate results are computed with unit round-off {{math|<var>&epsilon;</var><sub>2</sub>}} before the final result is rounded (or truncated) with unit round-off {{math|<var>&epsilon;</var><sub>1</sub>}}. All other computations are assumed to be carried out with unit round-off {{math|<var>&epsilon;</var><sub>1</sub>}}.
 
==Notes==
{{Reflist}}
 
==References==
*{{cite book|title=Rounding Errors in Algebraic Processes|first=James H.|last=Wilkinson|publisher=[[Prentice Hall]]|location= Englewood Cliffs, NJ|year=1963}}
*{{cite journal|title=Iterative Refinement in Floating Point|first=Cleve B.|last=Moler|publisher=[[Association for Computing Machinery]]|location=New York, NY|date=April 1967|journal=[[Journal of the ACM]]|volume=14|issue=2|pages=316&ndash;321|url=http://portal.acm.org/citation.cfm?id=321394|doi=10.1145/321386.321394}}
 
{{Numerical linear algebra}}
 
[[Category:Numerical linear algebra]]
 
 
{{comp-sci-stub}}

Latest revision as of 07:44, 23 October 2014

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