The eigenpair here means the twins of eigenvalue and corresponding eigenvector. The talk introduces the three steps of our study on computing the maximal eigenpair. In the first two steps, we construct efficient initials for a known but dangerous algorithm, first for tridiagonal matrices and then for the irreducible matrices, having nonnegative off-diagonal elements. In the third step, we present two global algorithms which are still efficient and work well for a quite large class of matrices, even complex for instance.The topic was talked several times last year at several universities and conferences, some updates are included.

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