冗余数表示下迭代法的数字稳定性推断
Digit Stability Inference for Iterative Methods Using Redundant Number Representation
摘要：
在我们最近的工作迭代计算的硬件，我们表明，任意精度求解器可以执行更有利的比传统的算术等价物时，后者的精度是低于或超过预算的解决方案的问题。这些性能改进的很大一部分源于推断迭代之间存在相同的最高有效数字的能力。这种技术利用了对冗余表示的数字进行运算的算法的特性，允许跳过这些数字的生成，从而提高了效率。然而，它不能保证数字将稳定，即，永远不要在未来的迭代中改变。在这篇文章中，我们解决这个缺点，使用区间和向前误差分析证明，高意义的数字将变得稳定时，使用固定的迭代法计算线性方程组的近似。我们形式化的矩阵调理和增长率之间的关系，在最有效数字稳定，使用这些信息，以更快地收敛到我们想要的结果。与我们以前的工作相比，这种新技术的示例性硬件实现在使用Jacobi方法的一组不同条件的系统的解决方案中实现了高达2.2倍的加速。
英文摘要：
In our recent work on iterative computation in hardware, we showed that arbitrary-precision solvers can perform more favorably than their traditional arithmetic equivalents when the latter's precisions are either under- or over-budgeted for the solution of the problem at hand. Significant proportions of these performance improvements stem from the ability to infer the existence of identical most-significant digits between iterations. This technique uses properties of algorithms operating on redundantly represented numbers to allow the generation of those digits to be skipped, increasing efficiency. It is unable, however, to guarantee that digits will stabilize, i.e., never change in any future iteration. In this article, we address this shortcoming, using interval and forward error analyses to prove that digits of high significance will become stable when computing the approximants of systems of linear equations using stationary iterative methods. We formalize the relationship between matrix conditioning and the rate of growth in most-significant digit stability, using this information to converge to our desired results more quickly. Versus our previous work, an exemplary hardware realization of this new technique achieves an up-to 2.2x speedup in the solution of a set of variously conditioned systems using the Jacobi method.
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