On the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problems

Abstract

The analysis of the spectral features of a Toeplitz matrix-sequence Tn(f ) n∈N , generated by a symbol f ∈ L 1 ([−π, π]), real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form Tn = c0Tn(f0) + c1h h Tn(f1) + c2h 2h Tn(f2) + • • • + cn−1h (n−1)h Tn(fn−1), c0, c1, . . . , cn−1 ∈ [c * , c * ], c * c * > 0, independent of n, h = 1 n , fj ∼ gj, gj = |θ| 2−jh , j = 0, . . . , n − 1. Since the resulting generating function depends on n, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed together with open questions and future investigations.

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

“…Regarding [21], based on [26], the only relevant observation is that the minimal eigenvalue of T −1 n (|θ| 2 )h α T n (|θ| 2−α ) is well separated from zero, for any choice of α ∈ (0, 2), and this provides a qualitative indication that the minimal eigenvalue of An n converges to zero with a speed of 1 n 2 . Concerning [5], the latter claim is indeed proved formally, but the constants are not computed, while in the present note we improve the findings, by determining quite precise lowerbounds and upperbounds (see Figure 2).…”

“…In the current note we have considered a type of matrix stemming when considering the numerical approximations of distributed order FDEs (see [5,21] for example). The main contribution relies in precise bounds for the minimal eigenvalue of the involved matrices.…”

“…which satisfies k 2 m, and take p as with respect to those available in the literature and which amount so far to only two works [5,21]:…”

“…In this work, starting from [5,21], we improve the bounds present in the literature. We take into account the techniques already developed in [4,6]: the additional difficulty, in the present setting, relies on the fact that the generating function is not fixed, since it depends on the matrix order n, which is somehow a challenging novelty with respect to the classical work on the subject [4,6,27].…”

Eigenvalue superposition expansion for Toeplitz matrix-sequences, generated by linear combinations of matrix-order dependent symbols, and applications to fast eigenvalue computations

“…Beside the technical results, which are mathematically non trivial, the new findings have application to the design of fast eigensolvers (of matrix-less type; see [19,20] and references therein) for the computation in linear time of the eigenvalues of large matrices, stemming e.g. from the numerical approximation via local methods, like Finite Differences, Finite Elements, Isogeometric Analysis etc (see [12,13,34] and references there reported), of coercive differential equations like diffusion-advection, or from distributed order fractional differential equations again approximated by using local methods (see [1,6,28] and references therein). In Subsection 1.2 we also present a brief account on the theory of Generalized Locally Toeplitz (GLT) matrix-sequences [2,3,23,24], which is of support in interpreting our findings from a different perspective.…”