Garoni, Carlo; Serra-Capizzano, Stefano Generalized locally Toeplitz sequences: theory and applications. Volume I. (English) Zbl 1376.15002 Cham: Springer (ISBN 978-3-319-53678-1/hbk; 978-3-319-53679-8/ebook). xi, 312 p. (2017). Suppose that a differential equation \(Au=g\) and a numerical method to solve it are both linear. In such a case, the actual computation of the numerical solution reduces to solving a family of linear systems \(A_{n}u_{n}= g_{n}\) whose size \(d_{n}\) increases with \(n\) and tends to infinity when \(n\rightarrow\infty\). Thus, we have a sequence of discretization matrices such that in the practice it enjoys an asymptotical spectral distribution which is somehow connected to the spectrum of the differential operator \(A\). It happens that, for a large set of test functions, typically the set \(C_{c}(R)\) of continuous complex-valued functions compactly supported in the real line, there exists a linear functional \(\phi\) in the dual space of \(C_{c}(R)\) such that \(\lim _{n\rightarrow \infty} \frac{1}{d_{n}} \sum_{k=1}^{d_{n}} F[\lambda_{k}(A_{n})] = \phi (F)\), where \(F\in C_{c}(R)\) (eigenvalue distribution of the sequence) or \(\lim _{n\rightarrow \infty} \frac{1}{d_{n}} \sum_{k=1}^{d_{n}} F[\sigma_{k}(A_{n})] = \phi (F)\), where \(\lambda_{k} (A_{n})\) and \(\sigma_{k}(A_{n})\), \(k=1, \dots, d_{n}\), are the eigenvalues and singular values, respectively, of the matrix \(A_{n}\). The above convergence will be denoted by \(A_{n} \sim _{\lambda} \phi\) and \(A_{n}\sim_{\sigma} \phi\), respectively. If the linear functional \(\phi\) has an integral representation \(\phi(F)= \frac{1}{\mu_{m}}\int_{D} F (\kappa(\mathbf{y})) d\mathbf{y}\), where \(D\) is a subset of some Euclidean space \(\mathbb R^{m}\) and \(\mu_{m}\) is the volume of \(D\), the function \(\kappa\) is said to be the spectral symbol of the sequence \((A_{n})_{n}\).In this book, the authors deal with the analysis of the spectral and singular value distribution of sequences of matrices related with Toeplitz matrices, as well as the so-called locally Toeplitz and generalized locally Toeplitz matrices, which appear in the discretization of boundary value problems for linear differential equations when finite difference methods and finite element methods are used.In the sequel, I will describe the contents of this book.Chapter 2 presents the basic background in measure theory, topology and matrix analysis which will be useful in the next chapters. In particular, the discussion about Schatten \(p\)-norms and the results about singular values and perturbation theory of matrices are very illustrative.Chapters 3 and 4 are focused on the notions of singular value and eigenvalue distribution of sequences of matrices \((A_{n})_{n}\) when \(d_{n}=n\). Clustering and spectral attraction for a matrix sequence are defined as well as the spectral distribution of sequences of perturbed matrices is analyzed.Chapter 5 deals with the study of approximating classes of sequences. Let \((A_{n})_{n}\) be a sequence of matrices and let \(((B_{m,n})_{n})_{m}\) be a sequence of matrix sequences. This double indexed sequence is said to be an approximating class of sequences (a.c.s., in short) of \((A_{n})_{n}\) if for every \(m\) there exists \(n(m)\), a positive integer number depending only of \(m\), such that, for \(n\geq n(m)\), \(A_{n}= B_{n,m} + R_{n,m} + N_{n,m}\), with rank \(R_{n,m} \leq c(m) n, \) \(||N_{n,m}|| \leq w(m)\), and \( c(m)\rightarrow 0\), \(w(m)\rightarrow 0\), when \(m\rightarrow\infty\). The corresponding topology is studied. The a.c.s. tools for the computation of singular value and eigenvalue distributions are presented. The case of Hermitian matrices is deeply analyzed. Some criteria to identify a.c.s are stated.Chapters 6, 7, and 8 constitute the theoretical core of the book. Chapter 6 is focused on Toeplitz sequences of matrices. An alternative proof of a classical result concerning the asymptotic distribution of the singular values and eigenvalues of a sequence of Toeplitz matrices defined by \(T_{n} (f)= \sum _{k=-(n-1)}^{n-1} t_{k} J_{n}^{(k)}\), where \(f(\theta)= \sum_{n= -\infty}^{\infty} t_{n} e^{in \theta}\) is the Fourier expansion of a function \(f\in L^1[-\pi, \pi]\) and \(J_{n}^{(k)}\) is a \(n\times n\) matrix with entries \(c_{i,j}=1\), when \(i-j=k\), and \(0\) otherwise. Indeed, \(T_{n}(f)\sim_{\sigma} \phi\), where \(\phi (F)= \frac{1}{2\pi}\int_{-\pi}^{\pi} F(|f(\theta)|) d\theta\). If \(f\) is real, then \(T_{n}(f)\sim_{\lambda} \phi\), where \(\phi (F)= \frac{1}{2\pi}\int_{-\pi}^{\pi} F(f(\theta)) d\theta\) (see [U. Grenander and G. Szegö, Toeplitz forms and their applications. 2nd ed. New York: Chelsea Publishing Company (1984; Zbl 0611.47018)].In Chapter 7, locally Toeplitz (LT) matrices are introduced. Let \(m,n\) be positive integer numbers, \(a\), a Riemann integrable complex-valued function defined in \([0,1]\), and \(f\in L^{1}[-\pi, \pi]\). Let define the LT operator \(\operatorname{LT}_{n}^{m} (a,f)= \operatorname{diag}_{k=1,2, \dots, m} ( a(\frac{k}{m}) T_{\lfloor n/m\rfloor}(f)) \oplus \mathbf{O}_{n, \mathrm{mod} (m)}\). A sequence \((A_{n})_{n}\) is said to be LT, with symbol \(a\otimes f\) and denoted by \((A_{n})_{n}\sim _{\mathrm{LT}} a\otimes f\), if \((\operatorname{LT}_{n}^{m} (a,f))_{n} \rightarrow (A_{n})_{n}\) a.c.s. The functions \(a\) and \(f\) are called the weight function and the generating function of the sequence, respectively, of the sequence \((A_{n})_{n}\). Zero-distributed sequences, sequences of diagonal sampling matrices, and the Toeplitz sequences themselves are fundamental examples of LT sequences. A characterization of LT sequences is given in Theorem 7.10. The theory of LT sequences dates back to P. Tilli [Linear Algebra Appl. 278, No. 1–3, 91–120 (1998; Zbl 0934.15009)]. For an updated summary about this theory, see [C. Garoni and S. Serra-Capizzano, Bol. Soc. Mat. Mex., III. Ser. 22, No. 2, 529–565 (2016; Zbl 1360.65110)].Chapter 8 analyzes the generalized locally Toeplitz (GLT) sequences introduced in [S. Serra Capizzano, Linear Algebra Appl. 366, 371–402 (2003; Zbl 1028.65109)]. A sequence of matrices \((A_{n})_{n}\) is said to be GLT with symbol \(\kappa\) and will be denoted by \(A_{n}\sim_{\mathrm{GLT}} \kappa\) if for every \(m\in N\), there exists a finite number of LT sequences \((A^{(j,m)}_{n})_{n}\sim_{\mathrm{LT}} a_{j,m}\otimes f_{j,m}\), \(j=1,2, \dots, N(m)\), such that (i) \(\sum_{j=1}^{N(m)} A_{j,m}\otimes f_{j,m} \rightarrow \kappa\) in measure and (ii) \((\sum_{j=1}^{N(m)} A^{(j,m)})_{n} \rightarrow (A_{n})_{n} a.c.s\). Singular value and spectral distribution of GLT sequences are deduced as well as some approximation results. Theorem 8.6 provides a characterization result for GLT sequences.Chapter 9 summarizes the main results of the theory and constitutes a useful tool to handle it.Chapter 10 shows several applications of GLT sequences. The numerical analysis of boundary value problems for univariate convection-diffusion-reaction equations by using several standard methods (FD, FE and isogeometric analysis) is implemented. Chapter 11 announces some future developments as the theory of multivariate GLT sequences and their applications in the discretization methods of linear partial differential equations, among others.The presentation of the book is very friendly for any reader interested both in computational methods and perturbations of Toeplitz matrices. Furthermore, a careful selection of exercises, with proofs given in Chapter 12, allows to learn in practice the basic ideas contained in the book. A careful selection of 129 references shows the active research in the fields covered by this book and where the leadership of the authors is emphasized according to the number of their contributions. Reviewer: Francisco Marcellán (Leganes) Cited in 2 ReviewsCited in 117 Documents MSC: 15-02 Research exposition (monographs, survey articles) pertaining to linear algebra 15B05 Toeplitz, Cauchy, and related matrices 34A30 Linear ordinary differential equations and systems 15A18 Eigenvalues, singular values, and eigenvectors 35K20 Initial-boundary value problems for second-order parabolic equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 15B57 Hermitian, skew-Hermitian, and related matrices Keywords:singular values; eigenvalues; symbol; approximating classes of matrix sequences; Toeplitz matrices; locally Toeplitz matrices; generalized locally Toeplitz matrices; convection-diffusion-reaction differential equations; boundary value problems; finite difference discretization methods; finite element discretization methods; isogeometric analysis discretization; Hermitian matrices Citations:Zbl 0611.47018; Zbl 0934.15009; Zbl 1360.65110; Zbl 1028.65109 × Cite Format Result Cite Review PDF Full Text: DOI