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Iterative refinement
Method to improve accuracy of numerical solutions to systems of linear equations
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🏷️ Keywords
RLIE framework (1) · Large Language Models (1) · Probabilistic modeling (1) · Rule learning (1) · Iterative refinement (1) · Logistic regression (1) · Rule interactions (1) · arXiv (1)
📖 Key Information
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
x
=
b
,
{\displaystyle A\mathbf {x} =\mathbf {b} \,,}
due to the compounded accumulation of rounding errors, the computed solution
x
^
{\displaystyle {\hat {\mathbf {x} }}}
may sometimes deviate from the exact solution
x
⋆
.
{\displaystyle \mathbf {x} _{\star }\,.}
Starting with
x
1
=
x
^
,
{\displaystyle \mathbf {x} _{1}={\hat {\mathbf {x} }}\,,}
iterative refinement computes a sequence
{
x
1
,
x
2
,
x
3
,
…
}
{\displaystyle \{\mathbf {x} _{1},\,\mathbf {x} _{2},\,\mathbf {x} _{3},\dots \}}
which converges to
x
⋆
,
{\displaystyle \mathbf {x} _{\star }\,,}
when certain assumptions are met.
📰 Related News (1)
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🇺🇸 RLIE: Rule Generation with Logistic Regression, Iterative Refinement, and Evaluation for Large Language Models
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