Numerical Algorithms最新文献

The present study aims to introduce a numerical approach based on the hybrid of block-pulse functions (BPFs), Bernoulli polynomials (BPs), and hypergeometric function for analyzing a class of fractional variational problems (FVPs). The FVPs are made by the Caputo derivative sense. To analyze this problem, first, we create an approximate for the Riemann-Liouville fractional integral operator for BPFs and BPs of the fractional order. In this framework and using the Gauss-Legendre points, the main problem is converted into a system of algebraic equations. In the follow-up, an accurate upper bound is obtained and some theorems are established on the convergence analysis. Moreover, the computational order of convergence and solvability of the proposed approach are displayed and approximated theoretically and numerically. Meanwhile, the thrust of the proposed scheme is compared with other sophisticated examples in the literature, demonstrating that the process is accurate and efficient.

For solving horizontal linear complementarity problem (HLCP), we propose a general double-relaxation two-sweep modulus-based matrix splitting iteration method and a double-relaxation two-sweep modulus-based matrix splitting iteration method which contain a series of methods, by using different splittings. When the system matrices are (H_+)-matrices, we analyze convergence theory of these methods. Numerical examples in this paper illustrate that these methods are more efficient than modulus-based matrix splitting iteration method and general modulus-based matrix splitting iteration method.

In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices (textbf{B}_n). First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence ({textbf{B}_n}_n), which is described via two eigenvalue functions of a (2times 2) matrix-valued symbol. In connection with the above findings, we provide a characterization of the eigenvalues and eigenvectors of real tridiagonal 2-Toeplitz matrices (textbf{B}_n) of even order, that can be turned into a numerical effective scheme for the computation of all the entries of (textbf{B}_n^l), n even and l positive and small compared to n. We recall that a corresponding eigenvalue decomposition for odd order tridiagonal 2-Toeplitz matrices was found previously, while, for even orders, an implicit formula for all the eigenvalues is obtained.

In this manuscript, we introduce a novel hybrid iteration process called the Picard–SP iteration process. We apply this new iteration process to approximate fixed points of generalized (alpha )–nonexpansive mappings. Convergence analysis of our newly proposed iteration process is discussed in the setting of uniformly convex Banach spaces and results are correlated with some other existing iteration processes. The dominance of the newly proposed iteration process is exhibited with the help of a new numerical example. In the end, the comparison of polynomiographs generated by other well-known iteration processes with our proposed iteration process has been presented to make a strong impression of our proposed iteration process.

We present an (L^2) norm convergence of the implicit-explicit BDF2 scheme with variable-step for the unsteady incompressible Navier-Stokes equations with an inf-sup stable FEM for the space discretization. Under a weak step-ratio constraint (0<r_k:=tau _k/tau _{k-1}<4.864), our error estimate is mesh-robust in the sense that it completely removes the possibly unbounded quantities, such as (Gamma _N=sum _{k=1}^{N-2}max {0,r_{k}-r_{k+2}}) and (Lambda _N=sum _{k=1}^{N-1}(|r_{k}-1|+|r_{k+1}-1|)) included in previous studies. In this analysis, we integrate our recent theoretical framework that employs discrete orthogonal convolution (DOC) kernels with an auxiliary Stokes problem to split the convergence analysis into two distinct parts. In the first part, we address intricate consistency error estimates for the velocity, pressure and nonlinear convection term. The resulting estimates allow us to utilize the conventional methodologies within the DOC framework to preserve spatial accuracy. In the second part, through the use of the DOC technique, we prove that the proposed variable-step BDF2 scheme is of second-order accuracy in time with respect to the (L^2) norm. Extensive numerical simulations coupled with an adaptive time-stepping algorithm are performed to show the accuracy and efficiency of the proposed variable-step method for the incompressible flows.

Let (A_alpha ) be the semi-infinite tridiagonal matrix having subdiagonal and superdiagonal unit entries, ((A_alpha )_{11}=alpha ), where (alpha in mathbb C), and zero elsewhere. A basis ({P_0,P_1,P_2,ldots }) of the linear space (mathcal {P}_alpha ) spanned by the powers of (A_alpha ) is determined, where (P_0=I), (P_n=T_n+H_n), (T_n) is the symmetric Toeplitz matrix having ones in the nth super- and sub-diagonal, zeros elsewhere, and (H_n) is the Hankel matrix with first row ([theta alpha ^{n-2}, theta alpha ^{n-3}, ldots , theta , alpha , 0, ldots ]), where (theta =alpha ^2-1). The set (mathcal {P}_alpha ) is an algebra, and for (alpha in {-1,0,1}), (H_n) has only one nonzero anti-diagonal. This fact is exploited to provide a better representation of symmetric quasi-Toeplitz matrices (mathcal{Q}mathcal{T}_S), where, instead of representing a generic matrix (Ain mathcal{Q}mathcal{T}_S) as (A=T+K), where T is Toeplitz and K is compact, it is represented as (A=P+H), where (Pin mathcal {P}_alpha ) and H is compact. It is shown experimentally that the matrix arithmetic obtained this way is much more effective than that implemented in the toolbox CQT-Toolbox of Numer. Algo. 81(2):741–769, 2019.

A fourth-order improvised cubic B-spline collocation method (ICSCM) is proposed to numerically solve the equal width (EW) equation and the modified equal width (MEW) equation. The discretization of the spatial domain is done using the ICSCM and the Crank-Nicolson scheme is used for the discretization of the temporal domain. The nonlinear terms are processed using quasi-linearization techniques and the stability analysis of this method is performed using Fourier series analysis. The validity and accuracy of this method are verified through several numerical experiments using a single solitary wave, two solitary waves, Maxwellian initial condition, and an undular bore. Since there is an exact solution for the single wave, the error norms (varvec{L_2}) and (varvec{L_{infty }}) are first calculated and compared with some previous studies published in journal articles. In addition, the three conserved quantities (varvec{Q}), (varvec{M}), and (varvec{E}) of the problems raised during the simulation are also calculated and recorded in the table. Lastly, the comparisons of these error norms and conserved quantities show that the numerical results obtained with the proposed method are more accurate and agree well with the values of the conserved quantities obtained in some literatures using the same parameters. The main advantage of ICSCM is its ability to effectively capture solitary wave propagation and describe solitary wave collisions. It can perform solution calculations at any point in the domain, easily use larger time steps to calculate solutions at higher time levels, and produce more accurate calculation results.

In this paper, we introduce a modified projection method and give a strong convergence theorem for solving variational inequality problems in real Hilbert spaces. Under mild assumptions, there exists a novel line-search rule that makes the proposed algorithm suitable for non-Lipschitz continuous and pseudo-monotone operators. Compared with other known algorithms in numerical experiments, it is shown that our algorithm has better numerical performance.

In this paper, we first propose a version of FISTA, called C-FISTA type gradient projection algorithm, for quasi-variational inequalities in Hilbert spaces and obtain linear convergence rate. Our results extend the results of Nesterov for C-FISTA algorithm for strongly convex optimization problem and other recent results in the literature where linear convergence results of C-FISTA are obtained for strongly convex composite optimization problems. For a comprehensive study, we also introduce a new version of gradient projection algorithm with momentum terms and give linear rate of convergence. We show the adaptability and effectiveness of our proposed algorithms through numerical comparisons with other related gradient projection algorithms that are in the literature for quasi-variational inequalities.