Roundoff Error Analysis Research Articles

Round-Off Error Analysis in mMIMO Detector Based on Cholesky Decomposition

Round-Off Error Analysis in mMIMO Detector Based on Cholesky Decomposition

Formally-Verified Round-Off Error Analysis of Runge–Kutta Methods

Formally-Verified Round-Off Error Analysis of Runge–Kutta Methods

Floating-point numbers round-off error analysis by constraint programming

Floating-point numbers round-off error analysis by constraint programming

An effective approach based on Smooth Composite Chebyshev Finite Difference Method and its applications to Bratu-type and higher order Lane–Emden problems

The Smooth Composite Chebyshev Finite Difference method is generalized for higher order initial and boundary value problems. Round-off and truncation error analyses and convergence analysis of the method are also extended to higher order. The proposed method is applied to obtain the highly precise numerical solutions of boundary or initial value problems of the Bratu and higher order Lane Emden types. To visualize the competency of the presented method, the obtained results are compared with nine different methods, namely, Bezier curve method, Adomian decomposition method, Operational matrix collocation method, Direct collocation method, Haar Wavelet Collocation, Bernstein Collocation Method, Improved decomposition method, Quartic B-Spline method and New Cubic B-spline method. The comparisons show that the presented method is highly accurate than the other numerical methods and also gets rid of the singularity of the given problems.

Accurate Goertzel Algorithm: Error Analysis, Validations and Applications

The Horner and Goertzel algorithms are frequently used in polynomial evaluation. Each of them can be less expensive than the other in special cases. In this paper, we present a new compensated algorithm to improve the accuracy of the Goertzel algorithm by using error-free transformations. We derive the forward round-off error bound for our algorithm, which implies that our algorithm yields a full precision accuracy for polynomials that are not too ill-conditioned. A dynamic error estimate in our algorithm is also presented by running round-off error analysis. Moreover, we show the cases in which our algorithms are less expensive than the compensated Horner algorithm for evaluating polynomials. Numerical experiments indicate that our algorithms run faster than the compensated Horner algorithm in those cases while producing the same accurate results, and our algorithm is absolutely stable when the condition number is smaller than 1016. An application is given to illustrate that our algorithm is more accurate than MATLAB’s fft function. The results show that the relative error of our algorithm is from 1015 to 1017, and that of the fft was from 1012 to 1015.

An efficient method for oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

In this paper, a novel numerical technique, the first-order Smooth Composite Chebyshev Finite Difference method, is presented. Imposing a first-order smoothness of the approximation polynomial at the ends of each subinterval is originality of the method. Both round-off and truncation error analyses of the method are performed beside the convergence analysis. Diffusion of oxygen in a spherical cell including nonlinear uptake kinetics is solved by using the method. The obtained results are compared with the existing methods in the literature and it is observed that the proposed method gives more reliable results.

Roundoff Error Analysis of an Algorithm Based on Householder Bidiagonalization for Total Least Squares Problems

For large-scale problems, how to establish an algorithm with high accuracy and stability is particularly important. In this paper, the Householder bidiagonalization total least squares (HBITLS) algorithm and nonlinear iterative partial least squares for total least squares (NIPALS-TLS) algorithm were established, by which the same approximate TLS solutions was obtained. In addition, the propagation of the roundoff error for the process of the HBITLS algorithm was analyzed, and the mixed forward-backward stability of these two algorithms was proved. Furthermore, an upper bound of roundoff error was derived, which presents a more detailed and clearer approximation of the computed solution.

Finite differences for higher order derivatives of low resolution data

The viability of finite difference stencils for higher-order derivatives of low-resolution data, encountered in various applications, is examined. This study’s illustrative example involves the evaluation of the fourth derivative of the digitized free surface radius obtained from pixelated images. Procedures to obtain an optimal approximation to the derivative are made from estimates of truncation and roundoff error, with emphasis on the analysis of roundoff error. A method of successive approximations to evaluate the optimal grid spacing is presented. Higher-order stencils allow for larger optimal grid spacing, thereby reducing the roundoff error that would otherwise dominate. Hence, higher-order stencils are effective for low precision data.

Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications

In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the sense that it may force the finite precision error accumulation to grow arbitrarily large, under specific conditions that are fully described here. First, an ensemble of definitions of general applicability is given for the rigorous determination of the number of erroneous digits accumulated in any quantity of an arbitrary algorithm. Next, the exact number of erroneous digits produced in a single multiplication is given as a function of the involved operands, together with formulae offering the corresponding probabilities. In case the statistical properties of these operands are known, exact evaluation of the aforementioned probabilities takes place. Subsequently, the statistical properties of the accumulated finite precision error during any number of successive multiplications are explicitly analyzed. A method for exact tracking of this accumulated error is presented, together with associated theorems. Moreover, numerous dedicated experiments are developed and the corresponding results that fully support the theoretical analysis are given. Eventually, a number of important, probable and possible applications is proposed, where all of them are based on the methodology and the results introduced in the present work. The proposed methodology is expandable, so as to tackle the round-off error analysis in all arithmetic operations.

First order smooth composite Chebyshev finite difference method for solving coupled Lane-Emden problem in catalytic diffusion reactions

A new effective technique based on Chebyshev Finite Difference Method is introduced. First order smoothness of the approximation polynomial at the end points of each sub-interval is imposed in addition to the continuity condition. Both round-off and truncation error analyses are given besides the convergence analysis. Coupled Lane Emden boundary value problem in Catalytic Diffusion Reactions is investigated by using presented method. The obtained results are compared with the existing methods in the literature and it is observed that the proposed method gives more reliable results than the others.

Fast Bilinear Algorithms for Symmetric Tensor Contractions

Abstract In matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order s + v {s+v} and v + t {v+t} by a factor of ( s + t + v ) ! s ! ⁢ t ! ⁢ v ! {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58 × \times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

Round-Off Error and Exceptional Behavior Analysis of Explicit Runge-Kutta Methods

Numerical integration schemes are mandatory to understand complex behaviors of dynamical systems described by ordinary differential equations. Implementation of these numerical methods involve floating-point computations and propagation of round-off errors. This paper presents a new fine-grained analysis of round-off errors in explicit Runge-Kutta integration methods, taking into account exceptional behaviors, such as underflow and overflow. Linear stability properties play a central role in the proposed approach. For a large class of Runge-Kutta methods applied on linear problems, a tight bound of the round-off errors is provided. A simple test is defined and ensures the absence of underflow and a tighter round-off error bound. The absence of overflow is guaranteed as linear stability properties imply that (computed) solutions are non-increasing.

A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.

Roundoff error analysis of the CholeskyQR2 algorithm in an oblique inner product

Roundoff error analysis of the CholeskyQR2 algorithm in an oblique inner product

Generation of linear frequency modulation signal with reduced round-off error using pulse-output direct digital synthesis technique

The pulse-output Direct Digital Synthesis (DDS), in which the overflow signal of the phase accumulator is used for the pulse output, can be easily implemented due to its simple hardware architecture and low algorithm complexity. This paper introduces the fundamentals for generating Linear Frequency Modulation (LFM) pulse using pulse-output DDS technique. Error introducing mechanisms that affect the accuracy of signal's duration, initial phase, and frequency are studied. Extensive analysis of round-off error is given. A modified hardware architecture for LFM pulse generation with reduced round-off error is proposed. Experiment results are given, which shows that the proposed generator is promising in applications such as sonar transmitters.

A note on iterative refinement for seminormal equations

We present a roundoff error analysis of the method for solving the linear least squares problem minx‖b−Ax‖2 with full column rank matrix A, using only factors Σ and V from the SVD decomposition of A=UΣVT. This method (called SNESVD here) is an analogue of the method of seminormal equations (SNEQR), where the solution is computed from RTRx=ATb 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 SNEQR and SNESVD are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the SNESVD method. We show that, under the condition O(u)κ2(A)<1, this method (called CSNESVD) produces a forward stable solution, where κ(A) denotes the condition number of the matrix A and u is the unit roundoff. However, for problems with only O(u)κ(A)<1 it is generally not forward stable, and has similar numerical properties to the corresponding CSNEQR method. Our forward error bounds for the CSNESVD are slightly better than for the CSNEQR since the terms O(u2)κ3(A) are not present. We illustrate our analysis by numerical experiments.

The functions erf and erfc computed with arbitrary precision and explicit error bounds

The error function erf is a special function. It is widely used in statistical computations for instance, where it is also known as the standard normal cumulative probability. The complementary error function is defined as erfc(x)=1−erf(x).In this paper, the computation of erf(x) and erfc(x) in arbitrary precision is detailed: our algorithms take as input a target precision t′ and deliver approximate values of erf(x) or erfc(x) with a relative error guaranteed to be bounded by 2−t′.We study three different algorithms for evaluating erf and erfc. These algorithms are completely detailed. In particular, the determination of the order of truncation, the analysis of roundoff errors and the way of choosing the working precision are presented. The scheme used for implementing erf and erfc and the proofs are expressed in a general setting, so they can directly be reused for the implementation of other functions.We have implemented the three algorithms and studied experimentally what is the best algorithm to use in function of the point x and the target precision t′.

On Round-off Error for Adaptive Finite Element Methods

Round-off error analysis has been historically studied by analyzing the condition number of the associated matrix. By controlling the size of the condition number, it is possible to guarantee a prescribed round-off error tolerance. However, the opposite is not true, since it is possible to have a system of linear equations with an arbitrarily large condition number that still delivers a small round-off error. In this paper, we perform a round-off error analysis in context of 1D and 2D hp-adaptive Finite Element simulations for the case of Poisson equation. We conclude that boundary conditions play a fundamental role on the round-off error analysis, specially for the so-called ‘radical meshes’. Moreover, we illustrate the importance of the right-hand side when analyzing the round-off error, which is independent of the condition number of the matrix.

The root and Bell’s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial, Part II: Round-off error analysis by use of interval arithmetic

In Part I (Ikhile, 2008) [4], it was established that the root and Bell’s disk/point iteration methods with or without correction term are of the same asymptotic error propagation characteristics in the simultaneous determination of the zeros of a polynomial. This concluding part of the investigation is a study in round-offs, its propagation and its effects on convergence employing interval arithmetic means. The purpose is to consequently draw attention on the effects of round-off errors introduced from the point arithmetic part, on the rate of convergence of the generalized root and Bell’s simultaneous interval iteration algorithms and its enhanced modifications introduced in Part I for the numerical inclusion of all the zeros of a polynomial simultaneously. The motivation for studying the effects of round-off error propagation comes from the fact that the readily available computing devices at the moment are limited in precision, more so that accuracy expected from some programming or computing environments or from these numerical methods are or can be machine dependent. In fact, a part of the finding is that round-off propagation effects beyond a certain controllable order induces overwhelmingly delayed or even a severely retarded convergence speed which manifest glaringly as poor accuracy of these interval iteration methods in the computation of the zeros of a polynomial simultaneously. However, in this present consideration and even in the presence of overwhelming influence of round-offs, we give conditions under which convergence is still possible and derive the error/round-off relations along with the order/ R -order of convergence of these methods with the results extended to similar interval iteration methods for computing the zeros of a polynomial simultaneously, especially to Bell’s interval methods for refinement of zeros that form a cluster. Our findings are instructive and quite revealing and supported by evidence from numerical experiments. The analysis is preferred in circular interval arithmetic.

A new iterative refinement with roundoff error analysis

AbstractIn 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 &amp; Sons, Ltd.