I have the system $A^kx=b$ and the $LU$ factorization $A=LU$. How can I solve the system without actually calculating $A^k$?
2026-05-16 05:17:10.1778908630
Solve $A^kx=b$ system using $LU$
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Using $A=LU$, you can calculate the sequence $Ax_1=b,\cdots Ax_i=Ax_{i-1},\cdots,Ax_k=Ax_{k-1}$. The complexity of each step (regarding only the mult-divis) is $2.\dfrac{n^2}{2}=n^2$; thus, the complexity of the calculation of $x_k$ is $kn^2$. That is interesting only if $k\ll n$.