Fine spectral estimates with applications to the optimally fast solution of large FDE linear systems

Abstract In the present article we consider a type of matrices stemming in context numerical approximation distributed order fractional differential equations (FDEs). From one side they could look standard, since are real, symmetric and positive definite. On other hand cause specific difficulties which prevent successful use classical tools. particular associated matrix-sequence, with respect to matrix-size, is ill-conditioned it such that generating function does not exists, but face problem dealing sequence functions an intricate expression. Nevertheless, obtain real interval where smallest eigenvalue belongs to, showing also its asymptotic behavior. We observe new bounds improve those already literature give more accurate pieces spectral information, fact used design fast algorithms for large linear systems, approximating given FDEs. Very satisfactory results presented critically discussed, while section conclusions open problems ends current work.