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The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol$f$, satisfying some conditions and tracing out a simple loop over the interval$[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[\lambda_{j}(T_{n}(f))=f(d_{j,n})+c_{1}(d_{j,n})h+c_{2}(d_{j,n})h^{2}+O(h^{3}),\] where $h=\frac{1}{n+1}$, $d_{j,n}=\pi j h$, and $c_k$ are some boundedcoefficients depending only on $f$. The numerical results presented in theliterature suggests that the effective conditions for the expansion to hold areweaker and reduce to an even character of $f$, to a fixed smoothness, and toits monotonicity over $[0,\pi]$. \\ In this note we investigate thesuperposition caused over this expansion, when considering a linear combinationof symbols that is \[ \lambda_{j}\big(T_{n}(f_0)+\beta_{n}^{(1)} T_{n}(f_{1}) +\beta_{n}^{(2)} T_{n}(f_{2}) +\cdots\big), \] where $\beta_{n}^{(t)}=o\big(\beta_{n}^{(s)}\big)$ if $t>s$ and the symbols $f_{j}$are either simple loop or satisfy the weaker conditions mentioned before. Weprove that the asymptotic expansion holds also in this setting under mildassumptions and we show numerically that there is much more to investigate,opening the door to linear in time algorithms for the computation ofeigenvalues of large matrices of this type.  The problem is of concrete interest in particular in the case where thecoefficients of the linear combination are functions of $h$, consideringspectral features of matrices stemming from the numerical approximation ofstandard differential operators and distributed order fractional differentialequations, via local methods such as Finite Differences, Finite Elements,Isogeometric Analysis etc.

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