ACCELERATING THE SOLUTION OF LINEAR SYSTEMS BY ITERATIVE REFINEMENT IN THREE PRECISIONS
ACCELERATING THE SOLUTION OF LINEAR SYSTEMS BY ITERATIVE REFINEMENT IN THREE PRECISIONS
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DOI：
10.1137/17m1140819
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发表时间：
2018-01-01
影响因子：
3.1
通讯作者：
Higham, Nicholas J.
中科院分区：
文献类型：
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作者：
Carson, Erin;Higham, Nicholas J.
提出了一种求解n×n非奇异线性方程组Ax=b的通用算法，该算法基于三种精度的迭代求精.工作精度与用于求解校正项和计算残差的可能不同的精度相结合。通过对算法的舍入误差分析，我们得到了算法收敛的充分条件和可达到的前向误差、法向和分量向后误差的界。我们的结果推广和统一了许多已有的用于迭代求精的舍入误差分析。以单精度为工作精度，证明了用IEEE半精度的LU分解作为求解器，计算双精度的残差，可以将Ax=b解为无穷范数条件数kappa(无穷)(A)的全单精度。
We propose a general algorithm for solving an n x n nonsingular linear system Ax = b based on iterative refinement with three precisions. The working precision is combined with possibly different precisions for solving for the correction term and for computing the residuals. Via rounding error analysis of the algorithm we derive sufficient conditions for convergence and bounds for the attainable forward error and normwise and componentwise backward errors. Our results generalize and unify many existing rounding error analyses for iterative refinement. With single precision as the working precision, we show that by using LU factorization in IEEE half precision as the solver and calculating the residuals in double precision it is possible to solve Ax = b to full single precision accuracy for infinity-norm condition numbers kappa(infinity)(A)