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.

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