Efficient multiple precision linear numerical computation libraries such as MPLAPACK are critical in dealing with ill-conditioned problems. Specifically, there are optimization methods for matrix multiplication, such as the Strassen algorithm and the Ozaki scheme, which can be used to speed up computation. For complex matrix multiplication, the 3M method can also be used, which requires only three multiplications of real matrices, instead of the 4M method, which requires four multiplications of real matrices. In this study, we extend these optimization methods to arbitrary precision complex matrix multiplication and verify the possible increase in computation speed through benchmark tests. The optimization methods are also applied to complex LU decomposition using matrix multiplication to demonstrate that the Ozaki scheme can be used to achieve higher computation speeds.

翻译：高效的多种精度线性数值计算库（如MPLAPACK）对于处理病态问题至关重要。具体而言，矩阵乘法存在诸如Strassen算法和Ozaki方案等优化方法，可用于加速计算。针对复矩阵乘法，还可采用3M方法，该方法仅需三次实矩阵乘法，而4M方法则需要四次实矩阵乘法。本研究将这些优化方法推广至任意精度复矩阵乘法，并通过基准测试验证了计算速度的可能提升。此外，我们将优化方法应用于基于矩阵乘法的复LU分解，结果表明Ozaki方案能够实现更高的计算速度。