Compared to Krylov space methods based on orthogonal or oblique projection, the Chebyshev iteration does not require inner products and is therefore particularly suited for massively parallel computers with high communication cost. Here, six different algorithms that implement this method are presented and compared with respect to roundoff effects, in particular, the ultimately achievable accur...

This paper presents the XAMG library for solving large sparse systems of linear algebraic equations with multiple right-hand side vectors. The specializes but is not limited to solution obtained from discretization elliptic differential equations. A corresponding set numerical methods includes Krylov subspace, multigrid, Jacobi, Gauss-Seidel, and Chebyshev iterative methods. parallelization imp...

The Chebyshev or L∞ estimator minimizes the maximum absolute residual and is useful in situations where the error distribution has bounded support. In this paper, we derive the asymptotic distribution of this estimator in cases where the error distribution has bounded and unbounded support. We also consider the asymptotics of set-membership estimators such as the Chebyshev centre and maximum in...

In this paper Chebyshev wavelet and their properties are employed for deriving Chebyshev wavelet operational matrix of fractional derivatives and a general procedure for forming this matrix is introduced. Then Chebyshev wavelet expansion along with this operational matrix are used for numerical solution of Bagley-Torvik boundary value problems. The error analysis and convergence properties of t...

A weighted orthogonal system on the half-line based on the Chebyshev rational functions is introduced. Basic results on Chebyshev rational approximations of several orthogonal projections and interpolations are established. To illustrate the potential of the Chebyshev rational spectral method, a model problem is considered both theoretically and numerically: error estimates for the Chebyshev ra...

Odd degree Chebyshev polynomials over a ring of modulo 2 have two kinds of period. One is an “orbital period”. Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate affecting a factor of the ring, we can observe an orbit over the ring. The “ orbital period ” is a period of the orbit. The other is a “degree period”. It is observe...

In this paper, we propose a method to approximate the solution of a linear Fredholm integro-differential equation by using the Chebyshev wavelet of the first kind as basis. For this purpose, we introduce the first Chebyshev operational matrix of integration. Chebyshev wavelet approximating method is then utilized to reduce the integro-differential equation to a system of algebraic equations. Il...

Random projections are able to perform dimension reduction efficiently for datasets with nonlinear low-dimensional structures. One well-known example is that random matrices embed sparse vectors into a subspace nearly isometrically, known as the restricted isometric property in compressed sensing. In this paper, we explore some applications of deep neural networks. We provide expressive power f...

In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate va...

We study bicolored configurations of points in the Euclidean $n$-space that are constrained to remain either inside or outside a fixed $m$-subspace, with $n - m \ge 2$. define higher-codimensional variant Swiss-Cheese operad, called complementarily disks operad $\mathsf{CD}_{mn}$, associated such configurations. The $\mathsf{CD}_{mn}$ is weakly equivalent locally constant factorization algebras...