Adaptive accuracy-driven splitting in Ozaki matrix multiplication and application to block LU factorization

The Ozaki scheme for matrix multiplication computes an accurate matrix product by splitting the input matrices and evaluating a sequence of low-precision matrix products. Its accuracy and computational cost depend strongly on the number of splits. We develop accuracy-driven criteria for selecting this number in both Ozaki matrix multiplication and in block LU factorization using Ozaki trailing updates. For matrix multiplication, we derive normwise and componentwise criteria that determine how many split products are required to achieve a given accuracy requirement. For block LU, we present a fixed normwise strategy with essentially no selection cost, an LU-aware adaptive normwise strategy that can reduce the number of splits in subsequent trailing updates, and a componentwise strategy that uses information accumulated from preceding updates to guide the later selections. In the componentwise strategy, when the required number of splits exceeds a prescribed maximum, the corresponding update is instead carried out using ordinary matrix multiplication. We also provide rigorous backward error bounds for these approaches. Numerical experiments show that the adaptive normwise strategy can reduce the number of splits while preserving normwise accuracy, whereas the componentwise strategy can provide smaller componentwise backward errors when stronger local accuracy is needed.