High-performance Algorithm For Large-scale Exponential-type Matrix Functions Based On Block Krylov Subspace

The exponential matrix function is mainly used for solving the exponential integration of semi-linear rigidity problems and its calculation speed determines the efficiency of the exponential integration method directly.In this paper,an effective numerical method is designed to solve the problem which shapes likeφ0(tA)b0+tφ1(tA)b1+…+tpφp(tA)bp,the linear combination of the exponential matrix function and the product of vectors for large sparse matrices and rigid matrices.For large sparse matrices,an adaptive block Krylov subspace method is constructed.The forward error analysis theory and reliable posterior error estimation are established.By automatically selecting the value of the subspace dimension m and the time step τ,the algorithm can meets the predetermined accuracy requirements with the smallest possible amount of operation.The operation amount of this method is mainly reflected in the product of each exponential matrix function and vector.The numerical experiments results show that the computational accuracy of this method is high and the computational efficiency is higher than that of phipm,kiops and block Krylov subspace methods.For large rigid matrices,a rational block Krylov subspace method is constructed and a posterior error estimation is established.Rigid matrices mainly occur in infinite-dimensional operator discrete,which are not only pathological but also have a large norm.The standard block Krylov subspace method produces a large number of subspace dimensions and the effect is often not good.Numerical experiments results show that the computational efficiency of the algorithm significantly exceeds that of phipm,kiops and other algorithms under the given accuracy.