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In this paper, we present a new family of two-step iterative methods for solving nonlinear equations. The order of convergence of the new family without memory is four requiring three functional evaluations, which implies that this family is optimal according to Kung and Traubs conjecture Kung and Traub (J Appl Comput Math 21:643–651, 1974). Further accelerations of convergence speed are obtained...
In this paper, we consider applying the preconditioned conjugate gradient (PCG) method to solve system of linear equations $$T x = \mathbf b $$ where $$T$$ is a block Toeplitz matrix with Toeplitz blocks (BTTB). We first consider Level-2 circulant preconditioners based on generalized Jackson kernels. Then, BTTB preconditioners based on a splitting of BTTB matrices are proposed. We show that...
We present the iterative methods of fourth and sixth order convergence for solving systems of nonlinear equations. Fourth order method is composed of two Jarratt-like steps and requires the evaluations of one function, two first derivatives and one matrix inversion in each iteration. Sixth order method is the composition of three Jarratt-like steps of which the first two steps are that of the proposed...
We study the time discretization of the Cauchy problem $$\begin{aligned} u_{t}+\int _{0}^{t}\,\beta (t-s)\,L\,u\,(s)\;ds = 0,\quad t>0, \quad u(0)=u_{0}, \end{aligned}$$ where $$L$$ is a self-adjoint densely defined linear operator on a Hilbert space H with a complete eigen system $$\{\lambda _{m},\; \varphi _{m}\}_{m=1}^{\infty }$$ , and the subscript denotes differentiation with...
In this paper, an asymptotic expansion is constructed to solve second-order differential equation systems with highly oscillatory forcing terms involving multiple frequencies. An asymptotic expansion is derived in inverse of powers of the oscillatory parameter and its truncation results in a very effective method of dicretizing the differential equation system in question. Numerical experiments illustrate...
The main purpose of this paper is to propose the Legendre spectral-collocation method to solve the Volterra integral equations of the second kind with non-vanishing delay. We divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay. In each subinterval, where the solution is smooth enough, we can apply Legendre spectral-collocation...
We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax–Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $$\ell ^2$$ -decreasing. These results improve on a series of partial results on strong stability...
In this paper, we present a family of optimal, in the sense of Kung–Traub’s conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun’s fourth-order method. We use the Ostrowski’s efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the...
This paper aims to develop a fully discrete local discontinuous Galerkin finite element method for numerical simulation of the time-fractional telegraph equation, where the fractional derivative is in the sense of Caputo. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. The stability and convergence of this discontinuous approach are discussed...
Different convergence and comparison theorems for proper regular splittings and proper weak regular splittings are discussed. The notion of double splitting is also extended to rectangular matrices. Finally, convergence and comparison theorems using this notion are presented.
Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape...
For a given $$\theta \in (a,b)$$ , we investigate the question whether there exists a positive quadrature formula with maximal degree of precision which has the prescribed abscissa $$\theta $$ plus possibly $$a$$ and/or $$b$$ , the endpoints of the interval of integration. This study relies on recent results on the location of roots of quasi-orthogonal polynomials. The above positive...
In this paper, we derive a new similarity transformation from the discrete Lotka–Volterra system by taking an infinite limit of the discretization parameter. The proposed similarity transformation is shown to be applicable for computing singular values of a bidiagonal matrix. The forward stability of it is also verified in floating point arithmetic.
Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper...
Using S.L. Sobolev’s method, we construct the interpolation splines minimizing the semi-norm in $$K_2(P_2)$$ , where $$K_2(P_2)$$ is the space of functions $$\phi $$ such that $$\phi ^{\prime } $$ is absolutely continuous, $$\phi ^{\prime \prime } $$ belongs to $$L_2(0,1)$$ and $$\int _0^1(\varphi ^{\prime \prime }(x)+\varphi (x))^2dx<\infty $$ . Explicit formulas...
This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $$W_2^{(m,m-1)}(0,1)$$ space. Using the Sobolev’s method we obtain new optimal quadrature formulas of such type for $$N+1\ge m$$ , where $$N+1$$ is the number of the nodes. Moreover, explicit formulas of the optimal coefficients are obtained. We investigate the order of convergence...
Given a multigrid procedure for linear systems with coefficient matrices $$A_n,$$ we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $$B_n$$ : we assume that both $$A_n$$ and $$B_n$$ are Hermitian positive definite with $$A_n\le \vartheta B_n,$$...
The $$\varepsilon $$ ε -weighted energy norm is the natural norm for singularly perturbed convection-diffusion problems with exponential layers. But, this norm is too weak to recognise features of characteristic layers. We present an error analysis in a differently weighted energy norm—a balanced norm—that overcomes this drawback.
In this paper, we investigate the superconvergence of a variational discretization approximation for parabolic optimal control problems with control constraints. The state and the adjoint state are approximated by piecewise linear functions and the control is not directly discretized. The time discretization is based on difference methods. We derive the superconvergence between the numerical solution...
In this paper, we give some results for the Drazin inverse of a modified matrix $$M=A-CD^dB$$ M = A - C D d B with the generalized Schur complement $$Z=D-BA^dC$$ Z = D - B A d C under some conditions. Further, we present some new results for the Drazin inverse of the modified matrix $$M=A-CD^dB$$ M = A - C D d B , when the generalized Schur complement $$Z=0$$...
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