Truncation Error

The boundary integral method for the two dimensional Helmholtz equation requires the approximate evaluation of the integral 1 -1 g(x)H (1) 0 λ (xa) 2 + b 2 dx, where g is a polynomial. In particular, Gauss-Legendre quadrature is considered when the source point is close to the interval of integration; that is -1 ≤ a ≤ 1 and 0 < b ≪ 1 so that the integral is nearly weakly singular. It is shown that the real and imaginary parts of the integral must be considered separately. The sinh transformation can be used to improve the truncation error of the imaginary part, but must not be used for the real part. An asymptotic error analysis is given.

We give an error estimate for the Energy and Helicity Preserving Scheme (EHPS) in second order finite difference setting on axisymmetric incompressible flows with swirling velocity. With careful and detailed truncation error analysis near the geometric singularity and far field decay estimate for the stream function, we have achieved optimal error bound using weighted energy estimate. A key ingredient in our a priori estimate is the permutation identity associated with the Jacobians, which is also a unique feature that distinguishes EHPS from standard finite difference schemes.

Determining the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally challenging. The often-used harmonic mean approximation (HMA) makes no prior assumptions about the character of the distribution but tends to be inconsistent. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample using Lebesgue integration theory in one of two ways: 1) when the HMA integral exists, compute the measure function numerically and analyze the resulting quadrature to control error; 2) compute the measure function numerically for the marginal likelihood integral itself using a space-partitioning tree, followed by quadrature. The first algorithm automatically eliminates the part of the sample that contributes large truncation error in the HMA. Moreover, it provides a simple graphical test for the existence of the HMA integral. The second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly. It uses the posterior sample to discover and tessellate the subset of the sample space that was explored and uses quantiles to compute a representative field value. When integrating directly, this space may be trimmed to remove regions with low probability density and thereby improve accuracy. This second algorithm is consistent for all proper distributions. Error analysis provides some diagnostics on the numerical condition of the results in both cases.

This paper presents a general technique for an effective near-field to far-field transformation based on the complex source beam (CSB) method. In this technique, a set of CSBs launched from the center of antenna under test is used to represent the antenna's radiation. The weightings of CSBs are found by matching the near-fields measured on a pre-described surface in the conventional measurement systems. All three conventional measurement techniques including planar, cylindrical and spherical measurement systems are examined to validate the CSB method with accuracy comparisons in the nearand far-field patterns.

In modern VLSI technology, the occurrence of all kinds of errors has become inevitable. By adopting an emerging concept in VLSI design and test, error tolerance (ET), a novel error-tolerant adder (ETA) is proposed. The ETA is able to ease the strict restriction on accuracy, and at the same time achieve tremendous improvements in both the power consumption and speed performance. When compared to its conventional counterparts, the proposed ETA is able to attain more than 65% improvement in the Power-Delay Product (PDP). One important potential application of the proposed ETA is in digital signal processing systems that can tolerate certain amount of errors.

This type of problem was only investigated in the case of the Navier-Stokes equations without considering rotation of body. This is rst result in case of motion of viscous uids around rotating and translation body with articial boundary condition. The paper is organized as follows: In the rest of this section we introduce notation and give some auxiliary results. The next section 2 deals with pointwise estimates of the pressure of the linear system (1.2). In Section 3 we consider the linear system (1.2) with articial boundary conditions. The error estimate of the velocity is derived comparing to the solution to the system given in the exterior domain. First let us introduce notation: 1

This paper shows the displacements and velocity-stress formulations for the wave propagation problem with the aim of comparing its effectivity when they are implemented with the Generalized Finite Difference Method (GFDM). Schemes in GFD with approximations of the second and fourth order have been used and absorbing boundaries have been simulated with perfectly matched layers. The results obtained using both formulations are compared by using several examples in 2-D. The influence of the type of discretization of the clouds of nodes (regular and irregular) is studied.

The Generalized finite difference method (GFDM) is a meshfree method that can be applied for solving problems defined over irregular clouds of points. The GFDM uses the Taylor series development and the moving least squares approximation to obtain explicit formulae for the partial derivatives. In this paper, this meshfree method is used for solving elliptic and parabolic partial differential equations in 3-D. The influence of the main parameters involved in the approximation and the treatment of the Neumann boundary condition are shown. Parabolic equations have been solved using an explicit method and the criterion for stability has been improved taking into account the irregularity of the cloud of points. The numerical results show the high accuracy obtained.

Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points. In this paper the extension of the GFD to the explicit solution of parabolic and hyperbolic equations has been developed for partial differential equations with constant coefficients in the cases of considering one, two or three space dimensions. The convergence of the method has been studied and the truncation errors over irregular grids are given. Different examples have been solved using the explicit finite difference formulae and the criterion of stability. This has been expressed in function of the coefficients of the star equation for irregular clouds of nodes in one, two or three space dimensions. The numerical results show the accuracy obtained over irregular grids. This paper also includes the study of the maximum local error and the global error for different examples of parabolic and hyperbolic time-dependent equations.

In this paper, experimental results of the application of a new method for the reduction of the truncation error in nearfield measurement are presented. It is shown that the use of the proposed algorithm allows the test of larger antennas obtaining a significant upgrading of near-field system possibilities.

This paper is concerned with response sensitivity analysis of elastoplastic structures regarding geometric shape and external load parameters. Additionally to classic shape optimization, sensitivity information has to be computed concerning external load parameters. Due to the path dependent material behavior, variations of history variables have to be provided likewise. The analytical derivation of sensitivities is based on an enhanced kinematic concept that allows a strict separation of physical and geometrical quantities. With this approach it is possible to compute the structural response and response sensitivity simultaneously within a finite element framework. The main base equations are briefly described and the accuracy of the algorithm is demonstrated in an academic example.

In this paper, a continuous projection method is designed and analyzed. The continuous projection method consists of a set of partial differential equations which can be regarded as an approximation of the Navier-Stokes (N-S) equations in each time interval of a given time discretization. The local truncation error (LTE) analysis is applied to the continuous projection methods, which yields a sufficient condition for the continuous projection methods to be temporally second order accurate. Based on this sufficient condition, a fully second order accurate discrete projection method is proposed. A heuristic stability analysis is performed to this projection method showing that the present projection method can be stable. The stability of the present scheme is further verified through numerical experiments. The second order accuracy of the present projection method is confirmed by several numerical test cases.

We study the dimensions or degrees of freedom of farfield multipath that is observed in a limited, source-free region of space. The multipath fields are studied as solutions to the wave equation in an infinite-dimensional vector space. We prove two universal upper bounds on the truncation error of fixed and random multipath fields. A direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension. For circular and spherical spatial regions, we show that this finite dimension is proportional to the radius and area of the region, respectively. We use the Karhunen-Loève (KL) expansion of random multipath fields to quantify the notion of multipath richness. The multipath richness is defined as the number of significant eigenvalues in the KL expansion that achieve 99% of the total multipath energy. We establish a lower bound on the largest eigenvalue. This lower bound quantifies, to some extent, the well-known reduction of multipath richness with reducing the angular power spread of multipath angular power spectrum.

A new approach in the finite difference framework is developed, which consists of three steps: choosing the dimension of the local approximation subspace, constructing a vector basis for this subspace, and determining the coefficients of the linear combination. New schemes were developed to form the basis of the local approximation subspace, which were derived by approximating only the k^2u term of the Helmholtz equation. The construction of a basis of the local approximation subspace allows the new approach to be able to represent any finite difference scheme that belongs to this subspace. The new method is both consistent and capable of minimizing the dispersion relation for all stencils in all dimensions. In the one-dimensional case and 3-point stencil, pollution error is eliminated. In the two-dimensional (2D) case and 5-point stencil, the Complete Centered Finite Difference Method presents a dispersion relation equivalent to Galerkin/Least-Squares Finite Element Method. In the 2D case and 9-point stencil, two versions were developed using two different bases for the local approximation space. Both versions are equivalent and exhibit a dispersion relation similar to Quasi Stabilized Finite Element Method. Additionally, the dispersion analysis revealed a connection between the coefficients of the linear system and the stencil symmetry.

Using a complete orthonormal system of functions in L2(-8 ,8) a Fourier-Galerkin spectral technique is developed for computing of the localized solutions of equations with cubic nonlinearity. A formula expressing the triple product into series in the system is derived. Iterative algorithm implementing the spectral method is developed and tested on the soliton problem for the cubic Boussinesq equation. Solution is obtained and shown to compare quantitatively very well to the known analytical one. The issues of convergence rate and truncation error are discussed.

In our earlier work , we proposed an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD) of a set of simulation data for a partial differential equation (PDE) without storing the data. In this work, we perform an error analysis of the incremental SVD algorithm. We also modify the algorithm to incrementally update both the SVD and an error bound when a new column of data is added. We show the algorithm produces the exact SVD of an approximate data matrix, and the operator norm error between the approximate and exact data matrices is bounded above by the computed error bound. This error bound also allows us to bound the error in the incrementally computed singular values and singular vectors. We illustrate our analysis with numerical results for three simulation data sets from a 1D FitzHugh-Nagumo PDE system with various choices of the algorithm truncation tolerances.

This work is devoted to the application of the super compact finite difference (SCFDM) and the combined compact finite difference (CCFDM) methods for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence and height. The fourth-order compact, the sixth-order and eighth-order SCFDM and the sixth-order and eighth-order CCFDM schemes are used for the spatial differencing. To advance the solution in time, a semi-implicit Runge-Kutta method is used. In addition, to control the non-linear instability an eighth-order compact spatial filter is employed. For the numerical solution of the elliptic equations in the problem, a direct hybrid method which consists of a high-order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate is utilized. The accuracy and convergence rate for all methods are verified against exact analytical solutions. Qualitative and quantitative assessment of the results for an unstable barotropic mid-latitude zonal jet employed as an initial condition, is addressed. It is revealed that the sixth-order and eighth-order CCFDM and SCFDM methods lead to a remarkable improvement of the solution over the fourth-order compact method.

This article proposes a new approach for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. The approach fits into the long tradition of seeking for slaving relationships between the small scales and the large ones (to be controlled) but differ by the introduction of a new type of manifolds to do so, namely the finite-horizon parameterizing manifolds (PMs). Given a finite horizon [0, T ] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the high modes by the controlled low ones so that the unexplained high-mode energy is reduced -in a mean-square sense over [0, T ] -when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes introduced in [26]. These formulas allow for an effective derivation of reduced systems of ordinary differential equations (ODEs), aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. The design of low-dimensional suboptimal controllers is then obtained by (indirect) techniques from finite-dimensional optimal control theory, applied to the PM-based reduced ODEs. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u * R and the optimal controller u * are derived under a second-order sufficient optimality condition. These estimates demonstrate that the closeness of u * R to u * is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with the suboptimal controller u * R and the optimal controller u * ; and (ii) the energy kept in the high modes of the PDE solution either driven by u * R or u * itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.

A modified solution of the nonlinear singular oscillator has been obtained based on the extended iteration procedure. We have used an appropriate truncation of the obtained Fourier series in each step of iterations to determine the approximate analytic solution of the oscillator. The third approximate frequency of the nonlinear singular oscillator shows a good agreement with its exact values. Earlier different authors presented the analytic solution of the oscillator by using various types of methods. We have compared the results obtained by the modified technique with some of the existing results. We see that some of their techniques deviate from higher-order approximations and the present technique performs comparatively better.  The rate of change of percentage of error of the presented modified solution shows the validity of convergence.

In this study, we develop a concise and efficient formula for determining the order and error constants of fourth-order linear multistep methods used in the numerical solution of ordinary differential equations. Traditional approaches to computing these parameters often involve tedious algebraic manipulations, which can be time-consuming and error prone. This new approach provides a streamlined and systematic method that simplifies the analysis while ensuring accuracy and reliability. By leveraging this new approach, researchers can more efficiently assess the accuracy and stability of fourth-order linear multistep method schemes.

This paper is addressed to the numerical solving of the rendering equation in realistic image creation. The rendering equation is integral equation describing the light propagation in a scene accordingly to a given illumination model. The used illumination model determines the kernel of the equation under consideration. Nowadays, widely used are the Monte Carlo methods for solving the rendering equation in order to create photorealistic images. In this work we consider the Monte Carlo solving of the rendering equation in the context of the parallel sampling scheme for hemisphere. Our aim is to apply this sampling scheme to stratified Monte Carlo integration method for parallel solving of the rendering equation. The domain for integration of the rendering equation is a hemisphere. We divide the hemispherical domain into a number of equal sub-domains of orthogonal spherical triangles. This domain partitioning allows to solve the rendering equation in parallel. It is known that the Neumann series represent the solution of the integral equation as a infinity sum of integrals. We approximate this sum with a desired truncation error (systematic error) receiving the fixed number of iteration. Then the rendering equation is solved iteratively using Monte Carlo approach. At each iteration we solve multi-dimensional integrals using uniform hemisphere partitioning scheme. An estimate of the rate of convergence is obtained using the stratified Monte Carlo method. This domain partitioning allows easy parallel realization and leads to convergence improvement of the Monte Carlo method. The high performance and Grid computing of the corresponding Monte Carlo scheme are discussed.

We propose a second renormalization group method to handle the tensor-network states or models. This method reduces dramatically the truncation error of the tensor renormalization group. It allows physical quantities of classical tensor-network models or tensor-network ground states of quantum systems to be accurately and efficiently determined.

Determining the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally challenging. The often-used harmonic mean approximation (HMA) makes no prior assumptions about the character of the distribution but tends to be inconsistent. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample using Lebesgue integration theory in one of two ways: 1) when the HMA integral exists, compute the measure function numerically and analyze the resulting quadrature to control error; 2) compute the measure function numerically for the marginal likelihood integral itself using a space-partitioning tree, followed by quadrature. The first algorithm automatically eliminates the part of the sample that contributes large truncation error in the HMA. Moreover, it provides a simple graphical test for the existence of the HMA integral. The second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly. It uses the posterior sample to discover and tessellate the subset of the sample space that was explored and uses quantiles to compute a representative field value. When integrating directly, this space may be trimmed to remove regions with low probability density and thereby improve accuracy. This second algorithm is consistent for all proper distributions. Error analysis provides some diagnostics on the numerical condition of the results in both cases.

Weinberg (2012) described a constructive algorithm for computing the marginal likelihood, Z, from a Markov chain simulation of the posterior distribution. Its key point is: the choice of an integration subdomain that eliminates subvolumes with poor sampling owing to low tail-values of posterior probability. Conversely, this same idea may be used to choose the subdomain that optimizes the accuracy of Z. Here, we explore using the simulated distribution to define a small region of high posterior probability, followed by a numerical integration of the sample in the selected region using the volume tessellation algorithm described in Weinberg (2012). Even more promising is the resampling of this small region followed by a naive Monte Carlo integration. The new enhanced algorithm is computationally trivial and leads to a dramatic improvement in accuracy. For example, this application of the new algorithm to a four-component mixture with random locations in 16 dimensions yields accurate evaluation of Z with 5% errors. This enables Bayes-factor model selection for real-world problems that have been infeasible with previous methods.

Computation of the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally difficult. The often-used harmonic mean approximation uses the posterior directly but is unstably sensitive to samples with anomalously small values of the likelihood. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. It is useful, but not general. We need algorithms that apply to general distributions, like the harmonic mean approximation, but do not suffer from convergence and instability issues. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample by careful attention to the numerics of the probability integral. Posing the expression for the marginal likelihood as a Lebesgue integral, we may convert the harmonic mean approximation from a sample statistic to a quadrature rule. As a quadrature, the harmonic mean approximation suffers from enormous truncation error as consequence . This error is a direct consequence of poor coverage of the sample space; the posterior sample required for accurate computation of the marginal likelihood is much larger than that required to characterize the posterior distribution when using the harmonic mean approximation. In addition, I demonstrate that the integral expression for the harmonic-mean approximation converges slowly at best for high-dimensional problems with uninformative prior distributions. These observations lead to two computationally-modest families of quadrature algorithms that use the full generality sample posterior but without the instability. The first algorithm automatically eliminates the part of the sample that contributes large truncation error. The second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly. This eliminates convergence issues. The first algorithm is analogous to standard quadrature but can only be applied for convergent problems. The second is a hybrid of cubature: it uses the posterior to discover and tessellate the subset of that sample space was explored and uses quantiles to compute a representative field value. Qualitatively, the first algorithm improves the harmonic mean approximation using numerical analysis, and the second algorithm is an adaptive version of the Laplace approximation. Neither algorithm makes strong assumptions about the shape of the posterior distribution and neither is sensitive to outliers. Based on numerical tests, we recommend a combined application of both algorithms as consistency check to achieve a reliable estimate of the marginal likelihood from a simulated posterior distribution.

This paper presents a spectrally-weighted balanced truncation technique for RLC interconnects, when the interconnect circuit parameters change as a result of variations in the manufacturing process. The salient features of this algorithm are the inclusion of the parameter variation in the RLC interconnect, the guaranteed stability of the reduced transfer function, and the availability of provable frequency-weighted error bounds for the reduced-order system. This paper shows that the balanced truncation technique is an effective model-order reduction technique when variations in the circuit parameters are taken into consideration. Experimental results show that the new variational spectrally-weighted balanced truncation attains, on average, 20% more accuracy than the variational Krylovsubspace-based model-order reduction techniques while the run-time is also, on average, 5% faster.

We derive rigorous truncation-error bounds for the spin-boson model and its generalizations to arbitrary quantum systems interacting with bosonic baths. For the numerical simulation of such baths the truncation of both, the number of modes and the local Hilbert-space dimensions is necessary. We derive super-exponential Lieb-Robinson-type bounds on the error when restricting the bath to finitely-many modes and show how the error introduced by truncating the local Hilbert spaces may be efficiently monitored numerically. In this way we give error bounds for approximating the infinite system by a finite-dimensional one. As a consequence, numerical simulations such as the time-evolving density with orthogonal polynomials algorithm (TEDOPA) now allow for the fully certified treatment of the system-environment interaction.

An accurate metric for the time step control in the power device transient simulation is proposed. This metric contains an exponential term of the dominant time constant of the whole device structure derived from the matrix exponential term of the linearized device state equation. The proposed metric allows larger time step widths than the conventional metric of 2 nd order approximation of the local truncation error. It focuses on the dominant part of the transient response and its truncation error approximation is more accurate. In the transient device simulation, box integration method and Backward Euler method are used for spatial and temporal discretization, respectively. The discretized nonlinear device equations are solved by using Newton iteration whose initial guess is given by the approximated solution of the linearized device state equation by using the dominant time constant. Total calculation time of the transient simulation of a silicon power DMOSFET by using the proposed method decreases down to 27% of that by the conventional method with keeping the current accuracy of the dominant transient response.

We introduce solution dependent finite difference stencils whose coefficients adapt to the current numerical solution by minimizing the truncation error in the least squares sense. The resulting scheme has the resolution capacity of dispersion relation preserving difference stencils in under-resolved domains, together with the high order convergence rate of conventional central difference methods in well resolved regions. Numerical experiments reveal that the new stencils outperform their conventional counterparts on all grid resolutions from very coarse to very fine.

If F', Fdenote the masses of F( A) at f w , then we have the result Tibor P o g h y Abstract-A very tight truncation error upper bound is established for band-limited weakly stationary stochastic processes if the sampling interval is closed. In particular, the magnitude of the upper bound is 0 (N-'* In2 N ) for a symmetric sampling reconstruction from 2 N + 1 sampled values, where q is an arbitrary positive integer. The results are derived with the help of the Bernstein bound on the remainder of a symmetric complex Fourier series of the function exp ( i k t ) . Finally, convergence rates are given for mean square and almost sure sampling reconstructions.

If F', Fdenote the masses of F( A) at f w , then we have the result Tibor P o g h y Abstract-A very tight truncation error upper bound is established for band-limited weakly stationary stochastic processes if the sampling interval is closed. In particular, the magnitude of the upper bound is 0 (N-'* In2 N ) for a symmetric sampling reconstruction from 2 N + 1 sampled values, where q is an arbitrary positive integer. The results are derived with the help of the Bernstein bound on the remainder of a symmetric complex Fourier series of the function exp ( i k t ) . Finally, convergence rates are given for mean square and almost sure sampling reconstructions.

The numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximatebut fast-methods such as the fast multipole method; however, since these methods are only approximate, it is important to have an estimate for the error introduced in such calculations. An analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtained. The error caused by polynomial interpolation in a multilevel fast multipole algorithm is also analyzed. The total error introduced in a multilevel implementation is also investigated numerically.

The numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximatebut fast-methods such as the fast multipole method; however, since these methods are only approximate, it is important to have an estimate for the error introduced in such calculations. An analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtained. The error caused by polynomial interpolation in a multilevel fast multipole algorithm is also analyzed. The total error introduced in a multilevel implementation is also investigated numerically.

In this paper, we analyze detection of multilevel phase-shift keying (MPSK) signals transmitted over a Gamma shadowed Nakagami-m fading channel. We derive novel analytical expression, in terms of Meijer's G function, for Fourier coefficients of the probability density function of the received signal composite phase. Under the assumption of the imperfect reference signal extraction in the receiver, which is performed from the pilot signal, the analytical expression is derived for the symbol error probability (SEP) in the form of convergent series. The existence of the error floor is identified, and expression for its computation is provided. Mathematical proofs for convergence of Fourier series are provided for both SEP and SEP floor, and novel expressions of upper bounds for truncation errors are derived in terms of elementary mathematical functions. The convergence rate of the derived expressions is examined. Numerical results are confirmed independently by Monte-Carlo simulations.

The boundary integral method for the two dimensional Helmholtz equation requires the approximate evaluation of the integral 1 -1 g(x)H (1) 0 λ (xa) 2 + b 2 dx, where g is a polynomial. In particular, Gauss-Legendre quadrature is considered when the source point is close to the interval of integration; that is -1 ≤ a ≤ 1 and 0 < b ≪ 1 so that the integral is nearly weakly singular. It is shown that the real and imaginary parts of the integral must be considered separately. The sinh transformation can be used to improve the truncation error of the imaginary part, but must not be used for the real part. An asymptotic error analysis is given.

We present a sequence of n-tuple-ζ augmented polarized (nZaP) basis sets designed for extrapolations of both self-consistent field (SCF) and correlation energies to the complete basis set (CBS) limit. These nZaP basis sets (n=2–6) are formulated to give consistent errors throughout the Periodic Table (e.g., a consistent of ∼1 mhartree/electron error for the 2ZaP SCF energy and a consistent of ∼1.4 μhartree/electron error for the 6ZaP SCF energy). The SCF energy exhibits systematic convergence to the CBS limit: ESCF(nZaP)≈ESCF(CBS)+Ae−an. A single parameter, a=6.30, describes the 2ZaP through 6ZaP errors of H through Xe within 10%. The SCF rms basis set truncation errors of H through Xe are 33.5mEh, 4.58mEh, 0.82mEh, 0.18mEh, and 0.047mEh for 2ZaP, 3ZaP, 4ZaP, 5ZaP, and 6ZaP, respectively. Linear extrapolations of the (2,3)ZaP, (3,4)ZaP, (4,5)ZaP, and (5,6)ZaP calculations (all with a=6.30) reduce these errors by an order of magnitude to 0.24mEh, 0.056mEh, 0.020mEh, and 0.005mEh, res...

In 1975 one of the coauthors, Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices. Approximation by truncation is justified but no error estimates are given there. The class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by , where an extremely accurate asymptotic error estimate is shown. In this paper, we apply this error formula to the former case to obtain error formulas in a closed, explicit form.

This paper provides a simple method to estimate both univariate and multivariate MA processes. Similar to Durbin's method, it rests on the recursive relation between the parameters of the MA process and those of its AR representation. This recursive relation is shown to be valid both for invertible / stable and non invertible / unstable processes under the assumption that the process has no constant and started from zero. This makes the method suitable for unit root processes, too.

This paper provides a simple method to estimate both univariate and multivariate MA processes. Similar to Durbin's method, it rests on the recursive relation between the parameters of the MA process and those of its AR representation. This recursive relation is shown to be valid both for invertible / stable and non invertible / unstable processes under the assumption that the process has no constant and started from zero. This makes the method suitable for unit root processes, too.

-In this paper we introduce a new circuit simulation technique based on a stepwise equivalent conductance model of a nonlinear resistive device. The major advantage of this technique is to eliminate the need for employing Newton-Raphson iterations for the implicit integration. The technique, when applicable, is consistent, absolutely stable, and convergent. Furthermore, we demonstrate that a second order of accuracy (the local truncation error for integration is of the cubic order of the time step used) is achieved by solving linear equations for each integration step. When applied to digital MOS circuits, the technique takes advantage of the fact that voltage waveforms can be modeled to a good approximation as piecewise-linear functions and thus achieve further speed-up in the simulation. The program, called SWEC, has been implemented, and has proven to be accurate and efficient on a large number of circuit examples. The comparisons of results with Relax2.3 [7], iSPLICE3.0 [5], [17], XPsim [2], and SPECS2 [3] are given. G. SPICE and ASTAP have been the bread and butter of industry and university researchers in circuit simulation for over two decades. However, as VLSI circuits get larger, new approaches are sorely needed. Timing simulators based on relaxation and other techniques have proven useful, but they lack predictability and are thus not generally trusted or accepted. Another approach is to employ parallel processing. Although some research has been done, its value has been restrictive because parallel machines are not generally available today. In this paper we present a simple method which replaces nonlinear elements with a stepwise equivalent time-varying model. We will demonstrate its generality as well as its effectiveness in handling digital CMOS circuits. The implicit multistep integration algorithms used in SPICE2 and ASTAP are computationally much more expensive than explicit integration schemes, because at each step an implicit solution of a large nonlinear algebraic Manuscript

In this research paper, we present a class of polynomials referred to as Apostol-type Hermite-Bernoulli/Euler polynomials U ν (x, y; ρ; µ), which can be given by the following generating function 2-µ + µ 2 ξ ρe ξ + (1-µ) e xξ+yξ 2 = ∞ ν=0 U ν (x, y; ρ; µ) ξ ν ν! , for some particular values of ρ and µ. Further, the summation formulae and determinant forms of these polynomials are derived. This novel family encompasses both the classical Appell-type polynomials and their noteworthy extensions. Our investigations heavily rely on generating function techniques, supported by illustrative examples to demonstrate the validity of our results. Furthermore, we introduce derivative and multiplicative operators, facilitating the definition of the Apostol-type Hermite-Bernoulli/Euler polynomials as a quasi-monomial set.

In this paper, we analyze detection of multilevel phase-shift keying (MPSK) signals transmitted over a Gamma shadowed Nakagami-m fading channel. We derive novel analytical expression, in terms of Meijer's G function, for Fourier coefficients of the probability density function of the received signal composite phase. Under the assumption of the imperfect reference signal extraction in the receiver, which is performed from the pilot signal, the analytical expression is derived for the symbol error probability (SEP) in the form of convergent series. The existence of the error floor is identified, and expression for its computation is provided. Mathematical proofs for convergence of Fourier series are provided for both SEP and SEP floor, and novel expressions of upper bounds for truncation errors are derived in terms of elementary mathematical functions. The convergence rate of the derived expressions is examined. Numerical results are confirmed independently by Monte-Carlo simulations.

In this paper, we analyze detection of multilevel phase-shift keying (MPSK) signals transmitted over a Gamma shadowed Nakagami-m fading channel. We derive novel analytical expression, in terms of Meijer's G function, for Fourier coefficients of the probability density function of the received signal composite phase. Under the assumption of the imperfect reference signal extraction in the receiver, which is performed from the pilot signal, the analytical expression is derived for the symbol error probability (SEP) in the form of convergent series. The existence of the error floor is identified, and expression for its computation is provided. Mathematical proofs for convergence of Fourier series are provided for both SEP and SEP floor, and novel expressions of upper bounds for truncation errors are derived in terms of elementary mathematical functions. The convergence rate of the derived expressions is examined. Numerical results are confirmed independently by Monte-Carlo simulations.

Many problems in finance, mechanics, biology, medical, social sciences and other disciplines can be modeled by stochastic integral equations (SIEs). Given the wide range of applications of SIEs, solving these type of equations is a great importance. Clearly, obtaining the analytic solution of SIEs is often either complicated or impossible. Therefore, the development of numerical methods for solving these types of equations is inevitable. Hence, many authors have proposed several numerical approaches for solving these equations. In [11], authors applied triangular functions for solving SIEs. Asgari et al. suggested stochastic operational matrix based on Bernstein polynomials for obtaining numerical solution of nonlinear SIEs [12]. Cheraghi et al. [2], used new basis functions for solving linear stochastic Volterra integral equations. Authors in [1] used stochastic operational matrix based on Haar wavelets for obtaining numerical solution of nonlinear SIEs. To see another methods for ...

The reference atmosphere is applied in CPTEC spectral model in order to reduce spectral truncation errors and improve medium-range forecasts. The variables temperature and geopotential height are replaced by their deviations from the reference atmosphere. The pressure-gradient terms are rewritten to remove much of the cancellation between them, and so are the temperature advection terms. A series of experiments has been performed in order to assess the impacts of the reference atmosphere on forecasts. The results reveal that the reference atmosphere introduced in CPTEC model has generally beneficial impacts on forecast quality, especially on that in Southern Hemisphere. 1 INTRODUCTION The sigma coordinate, pressure normalized by its surface value, has become widely used in numerical weather forecast models as a way of conveniently incorporating variations in the height of Earth&#39;s surface. The upper and lower boundary conditions become more simple, but the use of this vertical co...