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The aim of this paper is to recognize the attitude of the conjugate -Gradient Algorithms for solving linear systems Ax=b under the existence of rounding errors. The effect of matrix condition number of A on the relative error of the calculated series of approximations { xk} is analyzed. An especially appealing feature of the algorithm qualified is that error rating can be obtained very easily. Some examples are presented to support the theoretical results and to demonstrate the applicability and efficiency of the methods. The paper ends with some conclusions that sum up the finding of the study. The executed program for calculation is carried using "Matlb7". [ABSTRACT FROM AUTHOR]

Several different techniques intend to improve the accuracy of results computed in floating-point precision. Here, we focus on a method to improve the accuracy of the evaluation of rational functions. We present a compensated algorithm to evaluate rational functions. This algorithm is accurate and fast. The accuracy of the computed result is similar to the one given by the classical algorithm computed in twice the working precision and then rounded to the current working precision. This algorithm runs much more faster than existing implementation producing the same output accuracy. [ABSTRACT FROM AUTHOR]

Compensated algorithms consist in computing the rounding errors of individual operations and then adding them later on to the computed result. This makes it possible to increase the accuracy of the computed result efficiently. Computing the rounding error of an individual operation is possible through the use of a so-called error-free transformation . In this article, we show that it is possible to validate the result of compensated algorithms using stochastic arithmetic. We study compensated algorithms for summation, dot product and polynomial evaluation. We prove that the use of the random rounding mode inherent to stochastic arithmetic does not change much the accuracy of compensated methods. This is due to the fact that error-free transformations are no more exact but still sufficiently accurate to improve the numerical quality of results. [ABSTRACT FROM AUTHOR]

In this paper, accurate argument calculation for sine-fitting algorithms is investigated, assuming floating-point (FP) arithmetic. An easy-to-implement incremental calculation technique is suggested. In order to decrease error propagation, the algorithm is complemented with an advanced summation technique. Theoretical and numerical analyses on computational demand are performed to highlight that incremental argument calculation outperforms the method proposed in former research. Furthermore, an algorithm is implemented to mitigate the effect of imprecise representation of frequency on FP arithmetic. Monte Carlo analyses are carried out to demonstrate the accuracy of the suggested algorithms. Results show that phase information can be evaluated precisely even with single-precision FP arithmetic, applying incremental argument calculation. By this means, the cost of equipment that is needed to perform sine fitting can be reduced significantly. Finally, possible application areas are shown to demonstrate the applicability of the suggested solutions in the state-of-art measurement procedures. [ABSTRACT FROM PUBLISHER]

The roundoff-error-free (REF) LU and Cholesky factorizations, combined with the REF substitution algorithms, allow rational systems of linear equations to be solved exactly and efficiently by working entirely in integer arithmetic. These REF computational tools share two key properties: their constituent divisions are exact, and their matrix entries' bit-lengths (i.e., required number of storage bits) are bounded polynomially. However, as this paper explains, updating the REF factorizations via the common delete-insert-reduce approach turns out to be inefficient in terms of operand bit-length growth and increased computational effort. In fact, we prove that this inefficiency can eliminate the computational savings expected of the factorization update process. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. To achieve this, it introduces the frame matrix methodology for representing and applying the REF factorizations, and then it devises new REF operations and shortcuts for transitioning between "nearby" frame matrices. The presented push-and-swap update approach utilizes these transitions to preserve the special structure of the REF factorizations and to avoid additional growth in their matrix entries' bit-lengths; this update process requires the expected O(n²) operations. The featured updates are column addition, deletion, and replacement with respect to the REF LU factorization. We also prove that analogous row updates can be performed via the column updates and discuss special considerations for updating the REF Cholesky factorization. An accompanying set of computational tests attests that for fully dense random instances, the push-and-swap column replacement update for the REF LU factorization is on average an order of magnitude faster than the corresponding delete-insert-reduce update for the exact rational arithmetic LU factorization. Altogether, the REF update algorithms represent a promising preliminary step toward enhancing the basic solution validation subroutines employed by state-of-the-art mixed-precision methodologies for exact linear and mixed-integer programming. [ABSTRACT FROM AUTHOR]

The numerical errors was used to verify the correctness of key results. The truncation errors, which are larger than the round-off errors by orders of magnitude, have a superlinear relationship with both the simulation time-step and the interparticle collision speed. This remains the case regardless of the simulation details including the chosen contact model, particle size distribution, particle density or stiffness. Hence, the total errors can usually be reduced by choosing a smaller time-step. Increasing the polydispersity in a simulation by including smaller particles necessitates choosing a smaller time-step to maintain simulation stability and reduces the truncation errors in most cases. The truncation errors are increased by the dissipation of energy by frictional sliding or by the inclusion of damping in the system. The number of contacts affects the accuracy, and one can deduce that because 2D simulations contain fewer interparticle contacts than the equivalent 3D simulations, they therefore have lower accrued simulation errors. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper it is shown that the SVD of a matrix can be constructed efficiently in a hierarchical approach. The proposed algorithm is proven to recover the singular values and left singular vectors of the input matrix A if its rank is known. Further, the hierarchical algorithm can be used to recover the d largest singular values and left singular vectors with bounded error. It is also shown that the proposed method is stable with respect to round-off errors or corruption of the original matrix entries. Numerical experiments validate the proposed algorithms and parallel cost analysis. [ABSTRACT FROM AUTHOR]

Floating-point arithmetic is a very efficient solution to perform computations in the real field. However, it induces rounding errors making results computed in floating-point differ from what would be computed with reals. Although numerical analysis gives tools to bound such differences, the proofs involved can be painful, hence error prone. We thus investigate the ability of a proof assistant like Coq to mechanically check such proofs. We demonstrate two different results involving matrices, which are pervasive among numerical algorithms, and show that a large part of the development effort can be shared between them. [ABSTRACT FROM AUTHOR]

Roundoff error is an error. It can be dramatically reduced by the use of additional low-order digits, i.e. guard digits. Although the significant-figures idea in its standard form is incompatible with guard digits, this problem can be neatly solved by underlining the last "significant" digit, and then appending guard digits to the right of there. This clarifies the distinction between roundoff error and overall uncertainty while preserving compatibility with sig figs. The focus here is on reducing roundoff error; we defer discussion of most other contributions to the uncertainty. [ABSTRACT FROM AUTHOR]

This article considers the numerical inversion of the power series p(x)=1 + b1x + b²x² +· · · to compute the inverse series q(x) satisfying p(x)q(x)= 1. Numerical inversion is a special case of triangular back-substitution, which has been known for its beguiling numerical stability since the classic work of Wilkinson (1961, Error analysis of direct methods of matrix inversion. J. Assoc. Comput. Mach., 8, 281-330). We prove the numerical stability of inversion of power series and obtain bounds on numerical error. A range of examples show that these bounds overestimate the error by only a few digits. When p(x) is a polynomial and x = a is a root with p(a) = 0, we show that root deflation via the simple division p(x)/(x - a) can trigger instabilities relevant to polynomial root finding and computation of finite-difference weights. When p(x) is a polynomial, the accuracy of the computed inverse q(x) is connected to the pseudozeros of p(x). [ABSTRACT FROM AUTHOR]

We study the numerical behavior of stationary one-step or two-step matrix splitting iteration methods for solving large sparse systems of linear equations. We show that inexact solutions of inner linear systems associated with the matrix splittings may considerably influence the accuracy of the approximate solutions computed in finite precision arithmetic. For a general stationary matrix splitting iteration method, we analyze two mathematically equivalent implementations and discuss the conditions when they are componentwise or normwise forward or backward stable. We distinguish two different forms of matrix splitting iteration methods and prove that one of them is significantly more accurate than the other when employing inexact inner solves. The theoretical results are illustrated by numerical experiments with an inexact one-step and an inexact two-step splitting iteration method. [ABSTRACT FROM AUTHOR]

Runtime analysis provides an effective method for measuring the sensitivity of programs to rounding errors. To date, implementations have required significant changes to source code, detracting from their widespread application. In this work, we present an open source system that automates the quantitative analysis of floating point rounding errors through the use of C-based source-to-source compilation and a Monte Carlo arithmetic library. We demonstrate its application to the comparison of algorithms, detection of catastrophic cancellation, and determination of whether single precision floating point provides sufficient accuracy for a given application. Methods for obtaining quantifiable measurements of sensitivity to rounding error are also detailed. [ABSTRACT FROM AUTHOR]

We apply round-off to planar rotations, obtaining a one-parameter family of invertible maps of a two-dimensional lattice. As the angle of rotation approaches π / 2 , the fourth iterate of the map produces piecewise-rectilinear motion, which develops along the sides of convex polygons. We characterise the dynamics–which resembles outer billiards of polygons–as the concatenation of so-called strip maps, each providing an elementary perturbation of an underlying integrable system. Significantly, there are orbits which are subject to an arbitrarily large number of these perturbations during a single revolution, resulting in the appearance of a novel discrete-space version of near-integrable Hamiltonian dynamics. We study the asymptotic regime of the limiting integrable system analytically, and numerically some features of its very rich near-integrable dynamics. We unveil a dichotomy: there is one regime in which the nonlinearity tends to zero, and a second where it does not. In the latter case, numerical experiments suggest that the distribution of the periods of orbits is consistent with that of random dynamics; in the former case the fluctuations result in an intricate structure of resonances. [ABSTRACT FROM AUTHOR]

Abstract: Algorithms for enclosing solutions of least squares problems and underdetermined systems are proposed. The results obtained by these algorithms are “verified” in the sense that all the possible rounding errors have been taken into account. In order to develop these algorithms, theories for obtaining componentwise error bounds of numerical solutions are established. The error bounds by the proposed algorithms are vectors, as opposed that those by the fast algorithms proposed by Rump are scalars. The proposed algorithms require computational costs similar to those of Rumpʼs algorithms. It is moreover proved that each component of the error bounds by the proposed algorithms is equal or smaller than the error bounds by Rumpʼs algorithms. Numerical results show the properties of the algorithms. [Copyright &y& Elsevier]

Abstract: We present a roundoff error analysis of the method for solving the linear least squares problem with full column rank matrix A, using only factors Σ and V from the SVD decomposition of . This method (called here) is an analogue of the method of seminormal equations ( ), where the solution is computed from using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A and multiple right-hand sides. However, in general both and are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the method. We show that, under the condition , this method (called ) produces a forward stable solution, where denotes the condition number of the matrix A and u is the unit roundoff. However, for problems with only it is generally not forward stable, and has similar numerical properties to the corresponding method. Our forward error bounds for the are slightly better than for the since the terms are not present. We illustrate our analysis by numerical experiments. [Copyright &y& Elsevier]

In this paper, an efficient algorithm is presented for minimizing $\|A_1X_1B_1 + A_2X_2B_2+\cdots +A_lX_lB_l-C\|$ where $\|\cdot \|$ is the Frobenius norm, $X_i\in R^{n_i \times n_i}(i=1,2,\cdots ,l)$ is a reflexive matrix with a specified central principal submatrix $[x_{ij}]_{r\leq i,j\leq n_i-r}$. The algorithm produces suitable $[X_1,X_2,\cdots ,X_l]$ such that $\|A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l-C\|=\min $ within finite iteration steps in the absence of roundoff errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Given numerical examples show that the algorithm is efficient. [ABSTRACT FROM AUTHOR]

The article presents mathematical equations addressing the two issues of an arbitrary polynomial with complex coefficients. It states that the computation for the larger parameter and factorization of the polynomial are resolved by reduction to matrix eigenvalue problems. In addition, classical Schur algorithm is used to compute for Cholesky decomposition. Moreover, classical Schur algorithm is proved to be stable relative to roundoff errors.

To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained. [ABSTRACT FROM AUTHOR]

Slope and aspect derived from digital elevation models are important variables used regularly in geographical information systems for modelling purposes. Digital elevation models (DEMs) have become a very common and widely used source for creating slope and aspect map surfaces. However, quite often these map surfaces are created and used without due consideration being given to the accuracy of the DEMs and the algorithms used to derive slope and aspect surfaces. The purpose of this paper is to show the flaws in some of the algorithms used to generate aspect and slope information from gridded digital elevation models and to make users aware of problems or errors that may be encountered. The research shows that the precision to which elevation values are stored in a DEM strongly influences the derived products of aspect and, to a lesser extent, slope. [ABSTRACT FROM PUBLISHER]

A general algorithm for finding finite-difference approximations to derivatives is presented. The algorithm uses a linear combination of suitable Taylor expansions, and is sufficiently general to cater for non-equispaced nodes. It also provides analytical expressions for truncation and round-off errors. [ABSTRACT FROM AUTHOR]

An efficient iterative algorithm is presented to solve a system of linear matrix equations A1X1B1 + A2X2B2 = E, C1X1D1 + C2X2D2 = F with real matrices X1 and X2. By this iterative algorithm, the solvability of the system can be determined automatically. When the systemis consistent, for any initial matrices X01 and X02, a solution can be obtained in the absence of roundoff errors, and the least normsolution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions X1 and X2 to the given matrices X1 and X2 in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations A1X1B1 + A2X2B2 = Ē, C1X1D1 + C2X2D2 = F, where Ē = E - A1X1B1 - A2X2B2, F = F - C1X1D1 - C2X2D2. The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices A1, A2, B1, B2, C1, C2, D1, D2 are large, our algorithm is efficient as well. [ABSTRACT FROM AUTHOR]

The existing methods to compute the definite integral of associated Legendre function (ALF) with respect to the argument suffer from a loss of significant figures independently of the latitude. This is caused by the subtraction of similar quantities in the additional term of their recurrence formulas, especially the finite difference of their values between two endpoints of the integration interval. In order to resolve the problem, we develop a recursive algorithm to compute their finite difference. Also, we modify the algorithm to evaluate their definite integrals assuming that their values at one endpoint are known. We numerically confirm a significant increase in computing precision of the integral by the new method. When the interval is one arc minute, for example, the gain amounts to 2-4 digits for the degree of harmonics in the range 2 ≤ n ≤ 2,048. This improvement in precision is achieved at a negligible increase in CPU time, say less than 5%. [ABSTRACT FROM AUTHOR]

Abstract: The integration of problems derived from dynamical systems is an important topic, both in mathematics and physics. In many publications, specific algorithms for each problem are proposed to obtain high accuracy in the integration. In this paper, we study the performance of the Taylor Series Method and the ways to obtain optimal accuracy in the integration of differential equations. We present different sources of rounding errors and how to reduce them. All the different strategies are compared to show their efficiency. [Copyright &y& Elsevier]

Abstract: An efficient iterative method for updating the mass, gyroscopic and stiffness matrices simultaneously using a few of complex measured modal data is developed. By using the proposed iterative method, the unique symmetric solution can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrices. Numerical results show that the presented method can be used to update finite element models to get better agreement between analytical and experimental modal parameters. [Copyright &y& Elsevier]

Abstract: In this paper, two iterative algorithms are proposed to solve the linear matrix equations . When the matrix equations are consistent, by the first algorithm, a solution can be obtained within finite iterative steps in the absence of roundoff-error for any initial value, furthermore, the minimum-norm solution can be got by choosing a special kind of initial matrix. Additionally, the unique optimal approximation solution to a given matrix can be derived by finding the minimum-norm solution of a new matrix equations . When the matrix equations are inconsistent, we present the second algorithm to find the least-squares solution with the minimum-norm. Finally, two numerical examples are tested by MATLAB, the results show that these iterative algorithms are efficient. [Copyright &y& Elsevier]

In this paper we present a novel improvement of Wilkinson's iterative refinement for the solution of linear system by using stability results of numerical solution for a dynamic system associated with the linear system. The convergence analysis is shown and roundoff error analysis is considered for this new refinement. Numerical experiments are accompanied to support our analysis. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, two efficient iterative methods are presented to solve the symmetric and skew symmetric solutions of a linear matrix equation , respectively, with real pair matrices and . By these two iterative methods, the solvability of the symmetric and skew symmetric solutions for the matrix equation can be determined automatically. When the matrix equation has symmetric and skew symmetric solutions, then, for any initial pair matrices and , symmetric and skew symmetric solutions can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum norm of the symmetric and skew symmetric solutions can be obtained by choosing a special kind of initial pair matrices. In addition, the unique optimal approximation pair solution and to the given matrices and in Frobenius norm can be obtained by finding the minimum norm solution of a new matrix equation , where . The given numerical examples demonstrate that the iterative methods are quite efficient. [Copyright &y& Elsevier]

In this paper, an efficient iterative method is presented to solve the linear matrix equation [image omitted] (X) = E with real matrix X. By this iterative method, the solvability of the linear matrix equation can be determined automatically. When the matrix equation is consistent, then, for any initial matrix X0, a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. We also propose an iterative algorithm to obtain the solution or the least norm solution of the consistent matrix system. The given numerical examples demonstrate the efficiency of these two algorithms. [ABSTRACT FROM AUTHOR]

Abstract: A novel structure is derived for digital filter implementation. This structure is actually an improved version of an existing one in terms of implementation efficiency and reducing finite word length (FWL) effects. Expression of roundoff noise gain is obtained for the proposed structure. Design examples are given to demonstrate the performance of this structure and to compare it with the existing one and the classical minimum roundoff state-space realizations. Numerical examples show that the proposed structure outperforms the others in terms of minimizing roundoff noise as well as implementation efficiency. [Copyright &y& Elsevier]

In this paper the problem of selection of the smoothing parameter for the density estimator using the Poisson distribution [see Gowronski and Stadmüller (1980) and Chaubey and Sen (1996)] is considered. Two cross validation methods, namely likelihood based cross validation and integrated squared error cross validation, are compared through a numerical study. It is found that the choice proposed in Chaubey and Sen (1996) may not be appropriate for large samples. Instead, data adaptive choice works well for large as well as small samples. Based on this study we also claim that the smoothing parameters selected using any of the two cross validation methods are asymptotically equivalent and seem to provide the smallest Hellinger distance between the estimator and the true density. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method is constructed to solve the following constrained linear matrix equations system: [A1(X),A2(X), ⋯ ,Ar (X)] = [E1, E2, ⋯ , Er ], X ϵ L = {X∣X = U (X)}, where Ai is a linear operator from Cm×n onto C pi×qi , Ei ϵ C pi×qi , i = 1, 2, ⋯ , r, and U is a linear self-conjugate involution operator. When the above constrained matrix equations system is consistent, for any initial matrix X0ϵ L, a solution can be obtained by the proposed iterative method in finite iteration steps in the absence of roundoff errors, and the least Frobenius norm solution can be derived when a special kind of initial matrix is chosen. Furthermore, the optimal approximation solution to a given matrix can be derived. Several numerical examples are given to show the efficiency of the presented iterative method. [ABSTRACT FROM AUTHOR]

Given generalized reflection matrices R, S, i.e., R* = R, R2 = I, S* = S, S2 = I, a complex matrix A is said to be generalized reflexive (or anti-reflexive), if RAS = A (or RAS = -A). In this paper, Smarandache iterative algorithm is proposed to solve matrix equation AXB + CY D = F with generalized reflexive matrix dual (X,Y ). For any initial iterative matrix pair (X1; Y1), we show that a solution of this equation can be obtained within finite iteration steps in the absence of roundoff errors. Some numerical examples illustrate the feasibility and efficiency of this algorithm. [ABSTRACT FROM AUTHOR]

In this paper we analyze the numerical behavior of several minimum residual methods which are mathematically equivalent to the GMRES method. Two main approaches are compared: one that computes the approximate solution in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of the resulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and they explain the experimentally observed fact that the GCR method delivers very accurate approximate solutions when it converges fast enough without stagnation. [ABSTRACT FROM AUTHOR]

The article discusses how Newton's method illustrates the type of nonuniformity of significant digits referred to Knuth's floating-point arithmetic algorithms especially those that follow Benford's Law. In most calculations that use digital computers and floating-point arithmetic, roundoff errors cannot be avoided, even with the simplest of functions. It is therefore important to study the distribution of significant digits for algorithms used in scientific computations. The Newton's method is used for computing successive approximations of zeros of functions.

For the fan structure, 8-node isoparametric quadrilateral elements are used to discretize the structure based on the fan blade shape while 20-node isoparametric hexahedron elements are used to discretize the surrounding air of the fan blade. The intensity of pressure at an arbitrary point in the surrounding air can be solved by the Galerkin method when the boundary condition is considered. The Wilson-θ method with good stability is used to solve the air-structure coupling equations. The calculated natural modes and the tested natural modes are compared. The mode method is used to calculate and analyze the fan blade dynamic characteristics in order to study the air-structure coupling. [ABSTRACT FROM AUTHOR]

The problem of acoustic wave scattering by closed objects via second kind integral equations, is considered. Based on, combined Helmholtz integral equation formulation (CHIEF) method, an efficient method for choosing and utilizing interior field relations is suggested for solving the non- uniqueness problem at the characteristic frequencies. The implementation of the algorithm fully utilizes previous computation and thus significantly reduces the CPU time compared to the usual least-squares treatment. The method is tested for acoustic wave scattering by both acoustically hard and soft spheres. Accurate results compared to the known exact solutions are obtained. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Three common refinement methods of achieving more accurate finite element solutions are to increase the number of elements, to employ higher-degree interpolation functions and to implement adaptive mesh by moving the nodes but maintaining the same number of elements as well as the degree of interpolation functions. In this paper, these refinement methods are applied to a planar high-speed four-bar mechanism. Since the selection of refinement methods depends on the demands and requirement of application problems, and the refinement mesh are highly dependent on error indicators, some guidelines are presented, which are based on several error indicators, such as natural frequencies, total energy, strains and global variables. Furthermore, their efficiencies are demonstrated through implementing cubic and quintic shape functions on the mechanism. [Copyright &y& Elsevier]

Running error analysis for the bivariate de Casteljau algorithm and the VS algorithm is performed. Theoretical results joint with numerical experiments show the better stability properties of the de Casteljau algorithm for the evaluation of bivariate polynomials defined on a triangle in spite of the lower complexity of the VS algorithm. The sharpness of our running error bounds is shown. [ABSTRACT FROM AUTHOR]

An efficient scheme is presented for implementing the LMS-based transversal adaptive filter in block floating-point (BFP) format, which permits processing of data over a wide dynamic range, at temporal and hardware complexities significantly less than that of a floating-point processor. Appropriate BFP formats for both the data and the filter coefficients are adopted, taking care so that they remain invariant to interblock transition and weight updating operation, respectively. Care is also taken to prevent overflow during filtering, as well as weight updating processes jointly, by using a dynamic scaling of the data and a slightly reduced range for the step size, with the latter having only marginal effect on convergence speed. Extensions of the proposed scheme to the sign-sign LMS and the signed regressor LMS algorithms are taken up next, in order to reduce the processing time further. Finally, a roundoff error analysis of the proposed scheme under finite precision is carried out. It is shown that in the steady state, the quantization noise component in the output mean-square error depends on the step size both linearly and inversely. An optimum step size that minimizes this error is also found out. [ABSTRACT FROM AUTHOR]

Methods based on iterative integration as well as generalized averaging and special derivatives were proposed to smooth and identify the experimental trends. Examples of constructing the mathematical trend models were presented. [ABSTRACT FROM AUTHOR]

Round-off of spherocylindrical powers, to multiples of 0.25 D (for example) in the case of sphere and cylinder, and 1 or 5° in the case of axis, represents a type of distortion of the data. The result can be artefacts in graphical representations, which can mislead the researcher. Lines and clusters can appear, some caused by moiré effects, which have no deeper significance. Furthermore artefacts can obscure meaningful information in the data including bimodality and other forms of departure from normality. A process called unrounding is described which largely eliminates these artefacts; each rounded power is replaced by a power chosen randomly from the powers that make up what is called the error cell of the rounded power. [ABSTRACT FROM AUTHOR]

We describe an efficient and robust algorithm for computing a common refinement of two meshes modeling the same surface of arbitrary shape by overlaying them on lop of each other. A common refinement is an important data structure for transferring data between meshes that have different combinatorial structures. Our algorithm is optimal in time and space, with linear complexity, and is robust even with inexact computations, through the techniques of error analysis, detection of topological inconsistencies, and automatic resolution of such inconsistencies. We present the verification and some further enhancement of robustness in Part II. [ABSTRACT FROM AUTHOR]

We present some remarks on the numerical evaluation of recurrence relations. Rounding error bounds are presented of the numerical scheme and some numerical examples are given, in particular, we analyse conversion recurrences from different families of orthogonal polynomials, the limit case of Jacobi–Sobolev polynomials, random recurrences and perturbed Gegenbauer polynomials. In all these examples the theoretical bounds give sharp relative rounding error estimations. The parallel evaluation of recurrences are also considered and numerical tests on a Cray T3D are presented. [Copyright &y& Elsevier]

Presents information on a study which showed the results of a numerical analysis of roundoff noise effects in cascade realizations of finite-impulse response (FIR) digital filters. Governing equations for roundoff noise gain in cascaded FIR structures; Description of the numerical methods used; Simulation results; Conclusions.

We consider a 4n2-algorithm for computing the Gohberg­Semencul representation of the inverse of a nonsymmetric Toeplitz matrix and for obtaining the solution of a Toeplitz system. The original algorithm is unstable when one or more of the leading principal submatrices are ill-conditioned. In our stabilized version we slightly perturb the matrix when we encounter a division by a small number. As a consequence, the solution is also perturbed, and its accuracy can be improved by taking a small number of iterative refinement steps. We present a roundoff error analysis of the new algorithm as well as numerical results that support our analysis. [ABSTRACT FROM AUTHOR]

It is well known that the solution of the equality constrained least squares (LSE) problem minBx =d b - Ax 2 is the limit of the solution of the unconstrained weighted least squares problem min x µd b - µB A x 2 as the weight µ tends to infinity, assuming that [ BT AT ]T has full rank. We derive a method for the LSE problem by applying Householder QR factorization with column pivoting to this weighted problem and taking the limit analytically, with an appropriate rescaling of rows. The method ob- tained is a type of direct elimination method. We adapt existing error analysis for the unconstrained problem to obtain a row-wise backward error bound for the method. The bound shows that, provided row pivoting or row sorting is used, the method is well-suited to problems in which the rows of A and B vary widely in norm. As a by-product of our analysis, we derive a row-wise backward error bound of precisely the same form for the standard elimination method for solving the LSE problem. We illustrate our results with numerical tests. [ABSTRACT FROM AUTHOR]