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The implicit finite difference scheme with the shifted Gr u ¨ nwald formula for discretizing the two-dimensional spatial fractional diffusion equations can result in discrete linear systems whose coefficient matrices are the sum of the identity matrix and a block-Toeplitz with Toeplitz-block matrix. For these symmetric positive definite coefficient matrices, we construct block-circulant with circulant-block preconditioners to further accelerate the convergence rate of the CG method. We analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that the eigenvalues of the preconditioned matrices are weakly clustered around 1 and the convergence of the preconditioned CG method with the block-circulant with circulant-block preconditioner is weakly dependent on the mesh size. Numerical experiments demonstrate that this structured preconditioner can improve the convergence behavior of the CG method. Moreover, this approach is superior to the preconditioned CG methods incorporated with the block diagonal and block tridiagonal preconditioners in both iteration counts and computing times.

For a jump diffusion with upward phase-type jumps with p phases, the maximum before an independent Erlang time e η q with q stages and rate parameter η is again phase-type with q ( p + 1 ) phases. An iterative scheme for computing the phase generator is presented and applied to representing the price of a barrier option with time horizon e η q as a single ordinary integral. Canadization then means to approximate a fixed horizon T with an e η q satisying E e η q = T for a sufficiently large q . Similar results holds for Greeks like the delta and the gamma. A numerical example is given for a down-and-in call option and the Canadization is combined with Richardson extrapolation. Finally, a recursion is developed that only requires the iteration to be performed in p + 1 dimensions.

Abstract We prove that every element of the special linear group can be represented as the product of at most six block unitriangular matrices, and that there exist matrices for which six products are necessary, independent of indexing. We present an analogous result for the general linear group. These results serve as general statements regarding the representational power of alternating linear updates. The factorizations and lower bounds of this work immediately imply tight estimates on the expressive power of linear affine coupling blocks in machine learning.

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite.

A novel method of message steganography is introduced to solve the disadvantages of traditional least significant bit (LSB) based methods by dividing the covering-stego image into a secret number of blocks. A chaotic logistic map model was performed using the chaotic parameters and the number of image blocks for generating a chaotic key. This key was then sorted, and the locations of blocks 1 to 8 were used to select the required blocks to be used as covering-stego blocks. The introduced method simplifies the process of message bits hiding and extracting by adopting a batch method of bits hiding and extracting. A comparative analysis was conducted between the outcomes of proposed method and those of prevalent approaches to outline the enhancements in both speed and quality of message steganography.

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

The expressions for the Drazin inverse of two kinds of anti-triangular block matrices are developed under new and weaker assumptions relative to those already used recently in this subject. Applying our results concerning the Drazin inverse and anti-triangular block matrices, we propose some characterizations and representations of the Drazin inverse of a 2 × 2 block matrix. In this way, we expand some notable achievements in characterizing and representing generalized inverses of partitioned matrices.

This paper is devoted to studying the Drazin inverse of certain structured matrices under newly introduced restrictive conditions. Specifically, we focus on the Drazin inverse of two types of anti-triangular block matrices. Moreover, several special cases of these results are also discussed. New representations for the Drazin inverse of an arbitrary block matrix are provided under certain conditions, extending recent results in the literature. Furthermore, a numerical example is presented to illustrate the theoretical findings.