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In this paper, we propose a new approach to justify a roundoff error impact on the accuracy of the linear least squares (LS) solution using QR decomposition. This decomposition is widely employed to invert a matrix in the linear detector of the multi-user multi-antenna receiver. Compared to existing numerical results, the suggested stochastic bound is far closer to actual errors. It was put through a half-precision format test and verified in real-world scenarios generated by the Quadriga 2.0 tool and in random randsvd ensemble. The results of experiments show that our method predicts errors that are extremely similar to those observed in simulations. The suggested method can be used to examine the roundoff error in many other applications that require matrix inverse operations.

The paper considers the numerical stability of the backward recurrence algorithm (BRalgorithm) for computing approximants of the continued fraction with complex elements. The new method establishes sufficient conditions for the numerical stability of this algorithm and the error bounds of the calculation of the nth approximant of the continued fraction with complex elements. It follows from the obtained conditions that the numerical stability of the algorithm depends not only on the rounding errors of the elements and errors of machine operations but also on the value sets and the element sets of the continued fraction. The obtained results were used to study the numerical stability of the BR-algorithm for computing the approximants of the continued fraction expansion of the ratio of Horn's confluent functions H7. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm. The obtained result is applied to the study of the numerical stability of computing approximants of the continued fraction expansion of the ratio of Horn's confluent function H7 with complex parameters. In addition, the analysis of the relative errors arising from the computation of approximants using the backward recurrence algorithm, the forward recurrence algorithm, and Lenz's algorithm is given. The method for studying the numerical stability of the BR-algorithm proposed in the paper can be used to study the numerical stability of the branched continued fraction expansions and numerical branched continued fractions with elements in angular and parabolic domains. [ABSTRACT FROM AUTHOR]

Roundoff errors often disrupt interpolation methods for time-fractional equations, potentially causing suboptimal convergence or even failure. These issues primarily result from catastrophic cancellations. To address this, we introduce a novel framework for computing coefficients in standard and fast interpolation methods on nonuniform meshes. We propose δ -cancellation and associated threshold conditions to prevent such cancellations. If the thresholds aren't met, a Taylor expansion technique can be applied. Numerical experiments demonstrate our method's accuracy, on par with the Gauss–Kronrod quadrature, but significantly more efficient, allowing for extensive simulations with hundreds of thousands of time steps. [ABSTRACT FROM AUTHOR]

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn's hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm. [ABSTRACT FROM AUTHOR]

Geometric predicates are at the core of many algorithms, such as the construction of Delaunay triangulations, mesh processing and spatial relation tests. These algorithms have applications in scientific computing, geographic information systems and computer-aided design. With floating-point arithmetic, these geometric predicates can incur round-off errors that may lead to incorrect results and inconsistencies, causing computations to fail. This issue has been addressed using a combination of exact arithmetic for robustness and floating-point filters to mitigate the computational cost of exact computations. The implementation of exact computations and floating-point filters can be a difficult task, and code generation tools have been proposed to address this. We present a new C++ meta-programming framework for the generation of fast, robust predicates for arbitrary geometric predicates based on polynomial expressions. We combine and extend different approaches to filtering, branch reduction, and overflow avoidance that have previously been proposed. We show examples of how this approach produces correct results for data sets that could lead to incorrect predicate results with naive implementations. Our benchmark results demonstrate that our implementation surpasses state-of-the-art implementations. [ABSTRACT FROM AUTHOR]

In this work, a robust and rigorous procedure for roots calculation and selection for a Cubic Equation of State is presented. Roundoff errors are detected and corrected with a simple test in addition to an iterative procedure, adapted from the fixed-point method. The new approach applies some properties of Cardano's formulation to find the appropriate phase root of the cubic equation. Furthermore, to find out the actual physical state of the system, when the polynomial function presents only one root, special properties of the second derivative of the cubic polynomial in compressibility factor form and its partial derivative with respect to pressure are analyzed on a rigorous thermodynamic basis. These results are applied to outline an efficient algorithm to calculate the saturation pressure of pure components, which is independent of an accurate initial estimate and integrates bisection, successive substitution, and Newton–Raphson methods to find the correct value in few iterations. [ABSTRACT FROM AUTHOR]