For a given nonnegative integer α, a matrix A n of size n is called α-Toeplitz if its entries obey the rule A n = [a r−αs ] n−1 r,s=0 . Analogously, a matrix A n again of size n is called α-circulant if A n = a (r−αs) mod n n−1 r,s=0 . Such kind of matrices arises in wavelet analysis, subdivision algorithms and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of α-circulants and we provide an asymptotic analysis of the distribution results for the singular values of α-Toeplitz sequences in the case where {a k } can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain (−π, π). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered.Theorem 2.1. Let B be an arbitrary (complex) m × n matrix. Then: (a) There exists a unitary m × m matrix U and a unitary n × n matrix V such that U * BV = Σ is an m × n "diagonal matrix" of the following form: Σ = D 0 0 0 , D := diag(σ 1 , . . . , σ r ), σ 1 ≥ σ 2 ≥ · · · ≥ σ r > 0.Here σ 1 , . . . , σ r are the nonvanishing singular values of B, and r is the rank of B.

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Starting from the spectral analysis of g-circulant matrices, we study the convergence of a multigrid method for circulant and Toeplitz matrices with various size reductions. We assume that the size n of the coefficient matrix is divisible by g ≥ 2 such that at the lower level the system is reduced to one of size n/g, by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the two-grid method and of the multigrid method is proved, when the number θ ∈ N of recursive calls is such that 1 < θ < g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located at "mirror points" and the standard two-grid method with g = 2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas, both for circulant and Toeplitz matrices.

This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractIn this paper we introduce a new fast and accurate numerical method for pricing exotic derivatives when discrete monitoring occurs, and the underlying evolves according to a Markov one-dimensional stochastic processes. The approach exploits the structure of the matrix arising from the numerical quadrature of the pricing backward formulas to devise a convenient factorization that helps greatly in the speed-up of the recursion. The algorithm is general and is examined in detail with reference to the CEV (Constant Elasticity of Variance) process for pricing different exotic derivatives, such as Asian, barrier, Bermudan, lookback and step options for which up to date no efficient procedures are available. Extensive numerical experiments confirm the theoretical results. The MATLAB code used to perform the computation is available online at

A linear full elliptic second order Partial Differential Equation (PDE), defined on a d-dimensional domain Ω, is approximated by the isogeometric Galerkin method based on uniform tensor-product Bsplines of degrees (p 1 ,. .. , p d). The considered approximation process leads to a d-level stiffness matrix, banded in a multilevel sense. This matrix is close to a d-level Toeplitz structure when the PDE coefficients are constant and the physical domain Ω is just the hypercube (0, 1) d without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiffness matrices has been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coefficients and an arbitrary domain Ω, parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiffness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical technique used for computing the symbol is based on the theory of Generalized Locally Toeplitz (GLT) sequences.