Spectral properties of flipped Toeplitz matrices and related preconditioning
 139 Downloads
Abstract
In this work, we investigate the spectra of “flipped” Toeplitz sequences, i.e., the asymptotic spectral behaviour of \(\{Y_nT_n(f)\}_n\), where \(T_n(f)\in \mathbb {R}^{n\times n}\) is a real Toeplitz matrix generated by a function \(f\in L^1([\pi ,\pi ])\), and \(Y_n\) is the exchange matrix, with 1s on the main antidiagonal. We show that the eigenvalues of \(Y_nT_n(f)\) are asymptotically described by a \(2\times 2\) matrixvalued function, whose eigenvalue functions are \(\pm \, f\). It turns out that roughly half of the eigenvalues of \(Y_nT_n(f)\) are well approximated by a uniform sampling of f over \([\,\pi ,\pi ]\), while the remaining are well approximated by a uniform sampling of \(\,f\) over the same interval. When f vanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative. Some insights on the spectral distribution of related preconditioned sequences are provided as well. Finally, a wide number of numerical results illustrate our theoretical findings.
Keywords
Toeplitz matrices Spectral symbol GLT theory Hankel matricesMathematics Subject Classification
15A18 15B05 65F081 Introduction
One reason for characterizing the spectra of these flipped matrices relates to the solution of linear systems with Toeplitz coefficient matrices. Since \(Y_nT_n(f)\) is symmetric, the resulting linear system may be solved by the MINRES method [18, 21] or by preconditioned MINRES [16, 17], with its descriptive convergence rate bounds based on eigenvalues (see, e.g., [2, Chapters 2 and 4]). However, whilst there has been significant interest in relating the eigenvalues and singular values of Toeplitz sequences to generating functions, analogous results have not been proved for flipped Toeplitz sequences and corresponding preconditioned ones.
This paper aims to fill this gap. To do so, in Sect. 2 we describe the tools we require, specifically we introduce the class of Generalized locally Toeplitz matrixsequences and related properties [6]. The main results, that describe the spectra of sequences of (preconditioned) flipped Toeplitz matrices can be found in Sect. 3. Examples that illustrate these theoretical results are in Sect. 4.
2 Preliminaries
In this section we formalize the definition of block Toeplitz and Hankel sequences associated to a matrixvalued Lebesgue integrable function. Moreover, we introduce a class of matrixsequences containing both block Toeplitz and Hankel sequences known as the block Generalized Locally Toeplitz (GLT) class [5, 6]. The properties of block GLT sequences will be used to derive the spectral distribution of (preconditioned) flipped Toeplitz sequences (cf. Definition 2.2).
2.1 Block Toeplitz and Hankel matrices, and their spectral distributions
Let us denote by \(L^1([\pi ,\pi ],s)\) the space of \(s\times s\) matrixvalued functions \(f:[\pi ,\pi ]\rightarrow \mathbb {C}^{s\times s}\), \(f=[f_{ij}]_{i,j=1}^s\) with \(f_{ij}\in L^1([\pi ,\pi ])\), \(i,j=1,\dots ,s\). In Definition 2.1 we introduce the notion of Toeplitz and Hankel matrixsequences generated by f.
Definition 2.1
The generating function f provides a description of the spectrum of \(T_{n}(f)\), for n large enough in the sense of the following definition.
Definition 2.2
 We say that \(\{A_n\}_{n}\) is distributed as f over [a, b] in the sense of the eigenvalues, and we write \(\{A_n\}_{n}\sim _\lambda (f,[a,b]),\) iffor every continuous function F with compact support. In this case, we say that f is the symbol of \(\{A_{n}\}_{n}\).$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{d_n}\sum _{j=1}^{d_n}F(\lambda _j(A_n))= \frac{1}{b  a} \int _a^b \frac{\sum _{i=1}^sF(\lambda _i(f(t)))}{s} \mathrm {d}t, \end{aligned}$$(2.1)
 We say that \(\{A_n\}_{n}\) is distributed as f over [a, b] in the sense of the singular values, and we write \(\{A_n\}_{n}\sim _\sigma (f,[a,b]),\) iffor every continuous function F with compact support.$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{d_n}\sum _{j=1}^{d_n}F(\sigma _j(A_n))= \frac{1}{b  a} \int _a^b \frac{\sum _{i=1}^sF(\sigma _i(f(t)))}{s} \mathrm {d}t, \end{aligned}$$(2.2)
Remark 2.1
If f is smooth enough, an informal interpretation of the limit relation (2.1) (resp. (2.2)) is that when n is sufficiently large, then \(d_n/s\) eigenvalues (resp. singular values) of \(A_{n}\) can be approximated by a sampling of \(\lambda _1(f)\) (resp. \(\sigma _1(f)\)) on a uniform equispaced grid of the domain [a, b], and so on until the last \(d_n/s\) eigenvalues (resp. singular values), which can be approximated by an equispaced sampling of \(\lambda _s(f)\) (resp. \(\sigma _s(f)\)) in the domain.
Remark 2.2
Both Definitions 2.1 and 2.2 can be generalized to the case where \(f:[\pi , \pi ]^d\rightarrow \mathbb {C}^{s\times s}\), \(d>1\). In this case a Toeplitz (resp. Hankel) sequence associated to f is referred to as multilevel block Toeplitz (resp. Hankel) sequence. Theorems 2.2–2.3 and Proposition 2.2 hold true for \(d>1\), as well.
For Toeplitz matrixsequences, the following theorem (due to Szegő, Tilli, Zamarashkin, Tyrtyshnikov, ...) holds.
Theorem 2.1
(see [8, 20, 22]) Let \(\{T_n(f)\}_n\) be a Toeplitz sequence generated by \(f\in L^1([\pi ,\pi ])\). Then, \(\left\{ T_n(f)\right\} _{n}\sim _\sigma (f,[\pi ,\pi ]).\) Moreover, if f is realvalued, then \(\left\{ T_n(f)\right\} _{n}\sim _\lambda (f,[\pi ,\pi ]).\)
In the case where f is a Hermitian matrixvalued function, the previous theorem can be extended as follows:
Theorem 2.2
(see [20]) Let \(f\in L^1([\pi ,\pi ],s)\) be a Hermitian matrixvalued function. Then, \(\{T_{n}(f)\}_{{n}}\sim _\lambda (f,[\pi , \pi ]).\)
We end this subsection with a theorem that is a useful tool for computing the spectral distribution of a sequence of Hermitian matrices. For the related proof, see [11, Theorem 4.3]. From now on, the conjugate transpose of the matrix A is denoted by \(A^*\).
Theorem 2.3
Let \(\{X_n\}_n\) be a sequence of matrices, with \(X_n\) Hermitian of size \(d_n\), and let \(\{P_n\}_n\) be a sequence such that \(P_n\in \mathbb C^{d_n\times \delta _n}\), \(P_n^*P_n=I_{\delta _n}\), \(\delta _n\le d_n\) and \(\delta _n/d_n\rightarrow 1\) as \(n\rightarrow \infty \). Then \(\{X_n\}_n\sim _{\lambda }f\) if and only if \(\{P_n^*X_nP_n\}_n\sim _{\lambda }f\).
2.2 Block generalized locally Toeplitz class
In the sequel, we introduce the block GLT class [5], a \(*\)algebra of matrixsequences containing both block Toeplitz and Hankel matrixsequences. The formal definition of block GLT matrixsequences is rather technical and involves somewhat cumbersome notation: therefore we just give and briefly discuss a few properties of the block GLT class, which are sufficient for studying the spectral features of (preconditioned) flipped Toeplitz matrices.
 GLT1

Let \(\{A_{ n}\}_{ n}\sim _\mathrm{GLT}\kappa \) with \(\kappa :G\rightarrow \mathbb {C}^{s\times s}\), \(G=[0,1]\times [\pi ,\pi ]\), then \(\{A_{ n}\}_{ n}\sim _\sigma (\kappa ,G)\). If the matrices \(A_{ n}\) are Hermitian, then it also holds that \(\{A_{ n}\}_{ n}\sim _\lambda (\kappa ,G)\).
 GLT2

The set of block GLT sequences forms a \(*\)algebra, i.e., it is closed under linear combinations, products, inversion, conjugation. In formulae, let \(\{ A_{ n} \}_{ n} \sim _\mathrm{GLT}\kappa _1\) and \(\{ B_{ n} \}_{ n} \sim _\mathrm{GLT}\kappa _2\), then

\(\{\alpha A_{ n} + \beta B_{ n}\}_{ n} \sim _\mathrm{GLT}\alpha \kappa _1+\beta \kappa _2, \quad \alpha , \beta \in \mathbb {C};\)

\(\{A_{ n}B_{ n}\}_{ n} \sim _\mathrm{GLT}\kappa _1 \kappa _2;\)

\(\{A^{1}_{ n}\}_{ n} \sim _\mathrm{GLT}\kappa _1^{1}\) provided that \(\kappa _1\) is invertible a.e.;

\(\{ A_{ n}^{*} \}_{ n} \sim _\mathrm{GLT}{\kappa ^*_1}.\)
 GLT 3

Any sequence of block Toeplitz matrices \(\{ T_{ n}(f) \}_{ n}\) generated by a function \(f \in L^1([\pi , \pi ],s)\) is a \(s\times s\)block GLT sequence with symbol \(\kappa (x, \theta ) = f(\theta )\).
 GLT4

Let \(\{A_{ n}\}_{ n}\sim _\sigma 0\). We say that \(\{A_{ n}\}_{ n}\) is a zerodistributed matrixsequence. Note that for any \(s>1\) \(\{A_{ n}\}_{ n}\sim _\sigma O_s\), with \(O_s\) the \(s\times s\) null matrix, is equivalent to \(\{A_{ n}\}_{ n}\sim _\sigma 0\). Every zerodistributed matrixsequence is a block GLT sequence with symbol \(O_s\) and viceversa, i.e., \(\{A_{ n}\}_{ n}\sim _\sigma 0\) \(\iff \) \(\{A_{ n}\}_{ n}\sim _\mathrm{GLT}O_s\).
According to Definition 2.2, in the presence of a zerodistributed sequence the singular values of the nth matrix (weakly) cluster around 0. This is formalized in the following result [6].
Proposition 2.1
where \(\Vert \cdot \Vert \) is the spectral norm.
We next recall a result on the spectral distribution of Hankel sequences associated to \(f\in L^1([\pi ,\pi ],s)\).
Proposition 2.2
(see [3]) If \(\{ H_n(f) \}_n\) is an Hankel sequence generated by \(f \in L^1([\pi ,\pi ],s)\), then \(\{ H_n(f) \}_n \sim _\sigma 0\).
Proposition 2.2 together with GLT4 tell us that \(\{ H_n(f) \}_n\) is a \(s\times s\)block GLT sequence with symbol \(O_s\).
We end this preliminary section with a theorem that is very useful in the context of GLT preconditioning. It is obtained as a straightforward extension of Theorem 1 in [7] to the block GLT case where the symbol of the preconditioning sequence is a multiple of the identity.
Theorem 2.4
3 Spectral distribution of (preconditioned) flipped Toeplitz sequences
Lemma 3.1
Proof
The following lemma concerns the spectral distribution of the matrixsequence \(\{\widehat{T}_n\}_n\).
Lemma 3.2
Proof
We are now ready to present our main results.
Theorem 3.1
Remark 3.1
Remark 3.2
Remark 3.3
Based on Theorem 3.1 and on Remark 2.1, we expect that asymptotically almost half of the eigenvalues of \(Y_nT_n(f)\) are well approximated by a uniform sampling of f over \([\pi ,\pi ]\), while the remaining are well approximated by a uniform sampling of \(f\) over the same interval. Specifically, a certain number of outliers whose ratio is infinitesimal in the matrixsize is allowed. When f vanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative.
Remark 3.4
Independently, a result equivalent to Theorem 3.1 has been proved in [4]; see Theorem 3.2 and Corollary 3.3 therein. The main differences between the approach used here and the one in [4] can be summarized as follows: in [4] the authors use the notion of approximating class of sequences, a technical tool behind the GLT construction, and they write the symbol as a scalar function (devoting some extra attention to how to define its domain). Here we leverage the block GLT algebra as a blackbox tool and we naturally get a matrixvalued symbol with two eigenvalue functions that, in line with Remark 2.1, immediately fits with the quasihalf positive/negative nature of the spectrum of \(Y_nT_n(f)\).
We end this section by providing the spectral distribution of a preconditioned sequence of flipped Toeplitz matrices.
Theorem 3.2
Proof
The thesis easily follows from the combination of Theorem 2.4 with Theorem 3.1 by noticing that \(\{\mathscr {P}_n\}_n\sim _\mathrm{GLT}h\) is equivalent to \(\{\mathscr {P}_n\}_n\sim _\mathrm{GLT}h\cdot I_2\), and recalling that \(Y_nT_n(f)\) is real symmetric.
In the next section we give a variety of examples that validate the theoretical findings in Theorems 3.1 and 3.2.
4 Numerical experiments
Example 4.1
Our first example is the banded Toeplitz matrix generated by \(f(\theta ) = 2+\mathrm {e}^{\varvec{\mathrm {i}}\theta }\). We see from Fig. 1 that, even for small matrices, the sampling of the eigenvalue functions of g collected in \(\varLambda \) accurately describes the eigenvalues of \(Y_nT_n(f)\). Also Fig. 2a confirms a good matching between the eigenvalues of \(Y_nT_n(f)\) and the spectrum of g when \(n=100\).
Example 4.2
After discretizing these derivatives by a shifted GrünwaldLetnikov finite difference method [12, 13], we obtain an approximation of \(\frac{\mathrm {d}^\alpha u(x)}{\mathrm {d}_{\pm } x^\alpha }\). For instance, when the step size is constant, the approximation of \(\frac{\mathrm {d}^\alpha u(x)}{\mathrm {d}_{+} x^\alpha }\) is a dense lower Hessenberg Toeplitz matrix \(T_n(f)\), with symbol \(f(\theta ) = \mathrm {e}^{\varvec{\mathrm {i}}\theta }(1+\mathrm {e}^{\varvec{\mathrm {i}}(\pi + \theta )})^\alpha \) as described in [1]. Figure 3 shows that when \(n=100\) the eigenvalues of the flipped Toeplitz matrix \(Y_nT_n(f)\) are well described by the sampling of the eigenvalue functions of g collected in \(\varLambda \), and when \(n=300\) the two are visually indistinguishable. Similar results can be inferred from Fig. 2b when comparing the eigenvalues of \(Y_nT_n(f)\) directly with the spectrum of g.
Example 4.3
This example from [9, Example 2] has dense Toeplitz matrices with slowly decaying entries. These entries are defined by the generating function \(f(\theta ) = (22\cos (\theta ))(1+\varvec{\mathrm {i}}x)\). Figure 4 illustrates that even for such matrices the sampling of the eigenvalue functions of g collected in \(\varLambda \) accurately describes the eigenvalues of \(Y_nT_n(f)\). We refer the reader to Fig. 2c for the direct comparison with the spectrum of g.
Example 4.4
The eigenvalues of \(Y_nT_n(f)\) are again characterized by a uniform sampling of the eigenvalues of g (see Fig. 5), even though the spectrum is somewhat more complicated than in previous examples (see Fig. 2d).
Example 4.5
Let \(\tilde{g}\) be the symbol of \(\{Y_{n+p}T_{n+p}(a_p)\}_n\). In Fig. 6a, we set p to 4 and we compare the imaginary part of the eigenvalues of \(Y_{n+p}T_{n+p}(a_p)\) with a sampling of \(\pm a_p\), i.e., with the imaginary part of \(\lambda _i(\tilde{g})\), \(i=1,2\) over \([0,\pi ]\). We refer to the resulting set ordered in an ascending way as \(\tilde{\varLambda }\). As in all previous examples, the matching between the two is quite good when \(n=300\). Similar results can be obtained when comparing the eigenvalues of \(Y_{n+p}T_{n+p}(m_p)\) with a sampling of \(\pm m_p\) over \([0,\pi ]\) (refer to Fig. 6b).
Example 4.6
In all previous examples, the matching between eigenvalues and the sampling of the eigenvalue functions was “exact”, in the sense that there were no outliers. In our final example we show a case where the outliers come into play. More precisely, we show how to build sequences of matrices with a constant number of outliers that can be chosen a priori.
Outliers for \(T_{n+p}(m_p+r_{p,t})\) when: (a) \(p=3\), \(t=2\), \(n=100,200,400,800,1600\); (b) \(n=300\), \(t=2\), \(p=1,\ldots ,5\); (c) \(n=300\), \(p=4\), \(t=5,10,15,20,25 \)
Let us define \(r_{p,t}(\theta )=\mathrm {e}^{(p+t)\varvec{\mathrm {i}}\theta }\), with \(t>1\) and \(p\ge 1\), and let us consider the Toeplitz sequence \(\{T_{n+p}(m_p+r_{p,t})\}_n\), with \(m_p\) as in the previous example. As shown in Table 1a, with the fixed values of \(p=3,t=2\) and varying n, we get a constant number of outliers equal to 5. More generally, the number of outliers of \(T_{n+p}(m_p+r_{p,t})\) equals \(p+t\) and can be decided a priori by changing either p or t as illustrated in Table 1b, c.
Example 4.7
Our final example shows how the spectral results in Sect. 3 can be used to describe the convergence rate of preconditioned MINRES, which depends heavily on the spectral properties of the coefficient matrix (see, e.g., [2, Chapters 2 and 4]).

\(\mathscr {P}_n= T_n(f_R)\), where \(f_R = (f + f^*)/2\), so that \(h = f_R\);

\(\mathscr {P}_n= T_n(f)\), so that \(h = f\), and

the absolute value Strang circulant preconditioner \(\mathscr {P}_n= C_n\) described in [17]. To compute \(C_n\) we first form \(C_n\), the Strang circulant preconditioner for \(T_n(f)\) [19]. Then \(C_n = (C_n^TC_n)^{1/2}\) can be cheaply evaluated using fast Fourier transforms. Since \(C_n\) is normal its singular values must equal the eigenvalues of \(C_n\). The fact that \(\{C_n\}_n\sim _{\sigma } f\) implies \(h = f\).
Given the eigenvalues in Fig. 7a and the fact that f has a zero at 0, it is not surprising that MINRES applied to Example 4.7 does not converge. On the other hand, from Fig. 7b–d and since \(\lambda _i(h^{1}g)\) is either bounded or equal to \(\pm 1\), we expect that preconditioned MINRES with preconditioners \(T_n(f_R)\), \(T_n(f)\) or \(C_n\) will converge rapidly. To test this, we apply (preconditioned) MINRES to Example 4.7, stopping when the residual norm is reduced by eight orders of magnitude, i.e, when \(\Vert r_k\Vert _2/\Vert r_0\Vert _2 < 10^{8}\). We see from Table 2 that these iteration counts are exactly what we might expect from the above spectral results: all three preconditioners are optimal, with \(T_n(f)\) resulting in the lowest iteration counts.
Preconditioned MINRES iterations for Example 4.7
n  \(T_n(f_R)\)  \(T_n(f)\)  \(C_n\) 

100  15  7  12 
300  16  8  14 
500  17  8  15 
1000  17  9  15 
2000  18  9  17 
4000  18  9  17 
5 Conclusions
We have investigated the spectra of flipped Toeplitz sequences, i.e., the asymptotic spectral behaviour of \(\{Y_nT_n(f)\}_n\), where \(T_n(f)\in \mathbb {R}^{n\times n}\) is a real Toeplitz matrix generated by a function \(f\in L^1([\pi ,\pi ])\), and \(Y_n\) is the exchange matrix, with 1s on the main antidiagonal. Using the GLT machinery, we have shown that the eigenvalues of \(Y_nT_n(f)\) are asymptotically described by a \(2\times 2\) matrixvalued function, whose eigenvalue functions are \(\pm f\). When f vanishes only on a set of measure zero, this motivates that roughly half of the eigenvalues of \(Y_nT_n(f)\) are positive, while the remaining are negative. The GLT theory allows us to describe also the asymptotic spectral behaviour of \(\{\mathscr {P}_n^{1}Y_nT_n(f)\}_n\), when \(\mathscr {P}_n\) is Hermitian positive definite, and hence predict the convergence rate of preconditioned MINRES.
Notes
Acknowledgements
Open access funding provided by Max Planck Society.
References
 1.Donatelli, M., Mazza, M., SerraCapizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)MathSciNetCrossRefGoogle Scholar
 2.Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014)CrossRefGoogle Scholar
 3.Fasino, D., Tilli, P.: Spectral clustering properties of block multilevel Hankel matrices. Linear Algebra Appl. 306, 155–163 (2000)MathSciNetCrossRefGoogle Scholar
 4.Ferrari, P., Furci, I., Hon, S., Mursaleen, M.A., SerraCapizzano, S.: The eigenvalue distribution of special \(2 \)by\(2 \) block matrix sequences, with applications to the case of symmetrized Toeplitz structures, arXiv:1810.03326 (2018)
 5.Garoni, C., Mazza, M., SerraCapizzano, S.: Block generalized locally Toeplitz sequences: from the theory to the applications. Axioms 7(3), 49 (2018)CrossRefGoogle Scholar
 6.Garoni, C., SerraCapizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. I. Springer, Cham (2017)CrossRefGoogle Scholar
 7.Garoni, C., SerraCapizzano, S.: Generalized locally Toeplitz sequences: a spectral analysis tool for discretized differential equations. In: Lyche, T., Manni, C., Speleers, H. (eds.) Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, pp. 161–236. Springer (2018)Google Scholar
 8.Grenander, U., Szegő, G.: Toeplitz Forms and Their Applications, vol. 321, 2nd edn. Chelsea, New York (1984)zbMATHGoogle Scholar
 9.Huckle, T., Serra Capizzano, S., TablinoPossio, C.: Preconditioning strategies for nonHermitian Toeplitz linear systems. Numer. Linear Algebra Appl. 12, 211–220 (2005)MathSciNetCrossRefGoogle Scholar
 10.Massei, S., Mazza, M., Robol, L.: Fast solvers for 2D fractional diffusion equations using rank structured matrices, arXiv:1804.05522 (2018)
 11.Mazza, M., Ratnani, A., SerraCapizzano, S.: Spectral analysis and spectral symbol for the 2d curlcurl (stabilized) operator with applications to the related iterative solutions. Comput. Math. (2018). https://doi.org/10.1090/mcom/3366
 12.Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advectiondispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)MathSciNetCrossRefGoogle Scholar
 13.Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for twosided spacefractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)MathSciNetCrossRefGoogle Scholar
 14.Oseledets, I., Tyrtyshnikov, E.: A unifying approach to the construction of circulant preconditioners. Linear Algebra Appl. 418, 435–449 (2006)MathSciNetCrossRefGoogle Scholar
 15.Pan, J., Ng, M., Wang, H.: Fast preconditioned iterative methods for finite volume discretization of steadystate spacefractional diffusion equations. Numer. Alg. 74, 153–173 (2017)MathSciNetCrossRefGoogle Scholar
 16.Pestana, J.: Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices, arXiv:1812.02479 (2018)
 17.Pestana, J., Wathen, A.J.: A preconditioned MINRES method for nonsymmetric Toeplitz matrices. SIAM J. Matrix Anal. Appl. 36, 273–288 (2015)MathSciNetCrossRefGoogle Scholar
 18.Sogabe, T., Zheng, B., Hashimoto, K., Zhang, S.L.: A preconditioner of permutation matrix for solving nonsymmetric Toeplitz linear systems. Trans. Jpn. Soc. Ind. Appl. Math. 15, 159–168 (2004)Google Scholar
 19.Strang, G.: A proposal for Toeplitz matrix calculations. Stud. Appl. Math. 74, 171–176 (1986)CrossRefGoogle Scholar
 20.Tilli, P.: A note on the spectral distribution of Toeplitz matrices. Linear Multilin. Algebra 45(2–3), 147–159 (1998)MathSciNetCrossRefGoogle Scholar
 21.Wang, S.F., Huang, T.Z., Gu, X.M., Luo, W.H.: Fast permutation preconditioning for fractional diffusion equations. SpringerPlus 5, 1109 (2016)CrossRefGoogle Scholar
 22.Zamarashkin, N.L., Tyrtyshnikov, E.E.: Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function. Sbornik Math. 188(8), 1191 (1997)MathSciNetCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.