Computing permanents of complex diagonally dominant matrices and tensors

Computing permanents of complex diagonally dominant matrices and tensors

We prove that for any λ > 1, fixed in advance, the permanent of an n × n complex matrix, where the absolute value of each diagonal entry is at least λ times bigger than the sum of the absolute values of all other entries in the same row, can be approximated within any relative error 0 < ǫ < 1 in quasi-polynomial nO(lnn−ln ǫ) time. We extend this result to multidimensional permanents of tensors ...

Doubly Diagonally Dominant Matrices

We consider the class of doubly diagonally dominant matrices (A = [ ajj] E C”, ‘, la,,1 l”jjl > Ck+ i laiklCk+ jlajkl. i #j) and its subclasses. We give necessary and sufficient conditions in terms of the directed graph for an irreducibly doubly diagonally dominant matrix to be a singular matrix or to be an H-matrix. As in the case of diagonal dominance, we show that the Schur complements of do...

Relative Perturbation Theory for Diagonally Dominant Matrices

OF DISSERTATION RELATIVE PERTURBATION THEORY FOR DIAGONALLY DOMINANT MATRICES Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well condi...

Subdirect sums of doubly diagonally dominant matrices

The problem of when the k-subdirect sum of a doubly diagonally dominant matrix (DDD matrix) is also a DDD matrix is studied. Some sufficient conditions are given. The same situation is analyzed for diagonally dominant matrices and strictly diagonally dominant matrices. Additionally, some conditions are also derived when card(S)>card(S1) which was not studied by Bru, Pedroche and Szyld [Electron...

A note on nonnegative diagonally dominant matrices