SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A258-A277, January 2021.
What is the fastest way to solve a linear system $Ax= b$ in arithmetic of a given precision when $A$ is symmetric positive definite and otherwise unstructured? The usual answer is by Cholesky factorization, assuming that $A$ can be factorized. We develop an algorithm that can be faster, given an arithmetic of precision lower than the working precision as well as (optionally) one of higher precision. The arithmetics might, for example, be of precisions half, single, and double; half and double, possibly with quadruple; or single and double, possibly with quadruple. We compute a Cholesky factorization at the lower precision and use the factors as preconditioners in GMRES-based iterative refinement. To avoid breakdown of the factorization we shift the matrix by a small multiple of its diagonal. We explain why this is preferable to the common approach of shifting by a multiple of the identity matrix. We also incorporate scaling in order to avoid overflow and reduce the chance of underflow when working in IEEE half precision arithmetic. We extend the algorithm to solve a linear least squares problem with a well-conditioned coefficient matrix by forming and solving the normal equations. In both algorithms most of the work is done at low precision provided that iterative refinement and the inner iterative solver converge quickly. We explain why replacing GMRES by the conjugate gradient method causes convergence guarantees to be lost, but we show that this change has little effect on convergence in practice. Our numerical experiments confirm the potential of the new algorithms to provide faster solutions in environments that support multiple precisions of arithmetic.

中文翻译：

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在求解对称正定线性系统和最小二乘问题时采用低精度算法

SIAM科学计算杂志，第43卷，第1期，第A258-A277页，2021年1月。
当$ A $是对称正定且无结构时，以给定精度算出线性系统$ Ax = b $的最快方法是什么？通常的答案是通过Cholesky分解，假设$ A $可以分解。我们开发了一种算法，该算法可以更快，给定的精度算法要低于工作精度，以及（可选）更高的精度。举例来说，算术的精度可以是半精度，单精度和双精度。一半和两倍，可能是四倍；或单人和双人，可能是四人。我们以较低的精度计算Cholesky因子分解，并将这些因子用作基于GMRES的迭代优化中的前置条件。为了避免分解，我们将矩阵对角线的一小部分进行移位。我们解释了为什么这比以单位矩阵倍数移动的通用方法更好。我们还合并了缩放比例，以便在使用IEEE半精度算法时避免溢出并减少下溢的机会。我们扩展算法，通过形成和求解法线方程来解决条件良好的系数矩阵的线性最小二乘问题。在这两种算法中，只要迭代细化和内部迭代求解器快速收敛，大多数工作都是在低精度下完成的。我们解释了为什么用共轭梯度法替换GMRES会导致收敛性保证丢失，但我们证明了这种变化在实践中对收敛性影响很小。