Inclusion sets for singular values

In this paper, the following inclusion sets are under certain conditions presented for singular values of a matrix A=[aij]∈Cn×nA=[aij]∈Cn×n:D(A)≔⋃γ∈C(A)0:∏i∈γ∣z-ai∣⩽∏i∈γ(ρ(B)-bii)}andKP(A)≔⋃i=1n⋃j∈Pi(A){z⩾0:∣z-ai∣∣z-aj∣⩽(ρ(B)-bii)(ρ(B)-bjj)},where ai=∣aii∣ai=∣aii∣ and Pi(A)≔{j∣aij≠0oraji≠0,j≠i} for any i∈{1,2,⋯,n}i∈{1,2,⋯,n}, ρ(B  ) and C(A)C(A) denote the Perron root of the nonnegative matrix B=[bij]B=[bij] satisfying ∣aij∣,∣aji∣⩽bij∣aij∣,∣aji∣⩽bij for all i,j∈{1,2,⋯,n}i,j∈{1,2,⋯,n} with i≠ji≠j and the set of all nontrivial circuits γ in Γ(A)—the associated directed digraph of A, respectively. Subsequently, relations between these two results are investigated. Theoretic analysis and numerical examples show that these estimates are more precise than recent corresponding results in some cases.