High-precision Arithmetic Research Articles

Optimized Analytical Computation of Partial Elements Using a Retarded Taylor Series Expansion

The aim of this article is to efficiently and accurately calculate the integrals of the full-wave (FW) partial element equivalent circuit (PEEC) method. The accuracy of the analytical formulas calculated by the standard precision can be compromised when using nonuniform mesh to properly model the high-frequency effects. The numerical errors can be avoided by using a high-precision arithmetic, i.e., higher number of digits, however, at the expense of significantly higher computation time. This article presents an analytical approach for calculating the FW-PEEC interaction integrals of two elementary volumes/surfaces based on the Taylor expansion, which allows a high computational speed preserving the accuracy with a relative error of less than 0.1%. The proposed solution is verified compared to the high-precision arithmetic and the standard Gaussian integration for two examples of strip lines. Moreover, it is shown that the accuracy of FW-PEEC integrals can affect the convergence of an iterative PEEC matrix solver.

CLARINET: A quire-enabled RISC-V-based framework for posit arithmetic empiricism

Many applications require high-precision arithmetic. IEEE 754-2019 compliant (floating-point) arithmetic is the de facto standard for performing these computations. Recently, posit-arithmetic has been proposed as a drop-in replacement for floating-point arithmetic. The posit data representation and arithmetic claim several advantages over the floating-point format and arithmetic, including higher dynamic range, better accuracy, and superior performance-area trade-offs. However, very few accessible and holistic frameworks facilitate the validation of these claims of posit-arithmetic, especially with long accumulations (quire).We present a consolidated general-purpose processor-based framework to support posit-arithmetic empiricism. Users can seamlessly experiment using posit and floating-point arithmetic in their applications since the framework is designed for the two number systems to coexist. Melodica is a posit-arithmetic core that implements parametric fused operations on the quire data type. Clarinet is a Melodica-enabled processor based on the RISC-V ISA. To the best of our knowledge, this is the first-ever integration of the quire in a RISC-V core. We report results from application studies on Clarinet and benchmark common linear algebra and computer vision kernels. We synthesize Clarinet on a Xilinx FPGA and present utilization and timing data. Clarinet and Melodica remain actively under development and are available as open-source.

On the application of the analytical discrete ordinates method to the solution of nonclassical transport problems in slab geometry

In this work we investigate the use of the Analytical Discrete Ordinates (ADO) method when solving the spectral approximation of the nonclassical transport equation. The spectral approximation is a recently developed method based on the representation of the nonclassical angular flux as a series of Laguerre polynomials. This representation generates, as outcome, a system of equations that have the form of classical transport equations and can therefore be solved by current deterministic algorithms. Thus, the investigation of efficient approaches to solve the nonclassical transport equation is of interest and shall be pursued. This is the case of the ADO method which has been successfully used to solve a wide class of problems in the general area of particle transport. Numerical results are presented for two nonclassical test problems in slab geometry. These nonclassical transport problems are chosen in such way that their solution exactly reproduces the solution of the classical diffusion problem. Very accurate results are obtained for both test problems. However, the use of high precision arithmetic is sometimes required as illustrated in the second test problem. Limitations of the spectral approximation are also analyzed and discussed.

Using Ginkgo's memory accessor for improving the accuracy of memory‐bound low precision BLAS

AbstractThe roofline model not only provides a powerful tool to relate an application's performance with the specific constraints imposed by the target hardware but also offers a graphic representation of the balance between memory access cost and compute throughput. In this work, we present a strategy to break up the tight coupling between the precision format used for arithmetic operations and the storage format employed for memory operations. (At a high level, this idea is equivalent to compressing/decompressing the data in registers before/after invoking store/load memory operations.) In practice, we demonstrate that a “memory accessor” that hides the data compression behind the memory access, can virtually push the bandwidth‐induced roofline, yielding higher performance for memory‐bound applications using high precision arithmetic that can handle the numerical effects associated with lossy compression. We also demonstrate that memory‐bound applications operating on low precision data can increase the accuracy by relying on the memory accessor to perform all arithmetic operations in high precision. In particular, we demonstrate that memory‐bound BLAS operations (including the sparse matrix‐vector product) can be re‐engineered with the memory accessor and that the resulting accessor‐enabled BLAS routines achieve lower rounding errors while delivering the same performance as the fast low precision BLAS.

Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians

A family of non-Hermitian real and tridiagonal-matrix candidates H(N)(λ)=H0(N)+λW(N)(λ) for a hiddenly Hermitian (a.k.a. quasi-Hermitian) quantum Hamiltonian is proposed and studied. Fairly weak assumptions are imposed upon the unperturbed matrix [the square-well-simulating spectrum of H0(N) is not assumed equidistant)] and upon its maximally non-Hermitian N-parametric antisymmetric-matrix perturbations [matrix W(N)(λ) is not even required to be PT-symmetric]. Despite that, the “physical” parametric domain D[N] is (constructively) shown to exist, guaranteeing that in its interior, the spectrum remains real and non-degenerate, rendering the quantum evolution unitary. Among the non-Hermitian degeneracies occurring at the boundary ∂D[N] of the domain of stability, our main attention is paid to their extreme version corresponding to Kato’s exceptional point of order N (EPN). The localization of the EPNs and, in their vicinity, of the quantum-phase-transition boundaries ∂D[N] is found feasible, at the not too large N, using computer-assisted symbolic manipulations, including, in particular, the Gröbner-basis elimination and the high-precision arithmetics.

Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation

We numerically realize a breather gas for the focusing nonlinear Schrödinger equation. This is done by building a random ensemble of N∼50 breathers via the Darboux transform recursive scheme in high-precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma, and Peregrine breathers as elementary quasiparticles of the respective gases. The interaction properties of the constructed breather gases are investigated by propagating through them a "trial" generic (Tajiri-Watanabe) breather and comparing the mean propagation velocity with the predictions of the recently developed spectral kinetic theory [El and Tovbis, Phys. Rev. E 101, 052207 (2020)2470-004510.1103/PhysRevE.101.052207].

Single-precision arithmetic in ECHAM radiation reduces runtime and energy consumption

Abstract. We converted the radiation part of the atmospheric model ECHAM to a single-precision arithmetic. We analyzed different conversion strategies and finally used a step-by-step change in all modules, subroutines and functions. We found out that a small code portion still requires higher-precision arithmetic. We generated code that can be easily changed from double to single precision and vice versa, basically using a simple switch in one module. We compared the output of the single-precision version in the coarse resolution with observational data and with the original double-precision code. The results of both versions are comparable. We extensively tested different parallelization options with respect to the possible runtime reduction, at both coarse and low resolution. The single-precision radiation itself was accelerated by about 40 %, whereas the runtime reduction for the whole ECHAM model using the converted radiation achieved 18 % in the best configuration. We further measured the energy consumption, which could also be reduced.

ESTIMATION OF COMPUTING ALGORITHMS QUICKLY DEVELOPMENT BY USING HIGH-PRECISION ARITHMETIC SOFTWARE

One of the ways to reduce the computational error of computer simulation results is the use of software tools implemented in specialized libraries of high-precision arithmetic of modern universal high-level programming languages, which is associated with a significant increase in the duration of the computational experiment. This paper is devoted to the study of the degree of influence of a given computational accuracy on the speed of the software implementation of a method for solving a SLAE of a different order and with different digit capacity of coefficients using modern means of high-precision computation of the Java language.

Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned Hankel matrices. It is based on the LDLT decomposition and involves finding a k × k sub-matrix of the inverse of the original N × N Hankel matrix HN−1. The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitutes a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight w(x)=e−xβ, supported on [0, ∞) and β > 0. Such weight generates a Hankel determinant, a fundamental object in random matrix theory. In the situation where β > 1/2, the smallest eigenvalue tends to 0 exponentially fast. If β < 1/2, which is the situation where the classical moment problem is indeterminate, then the smallest eigenvalue is bounded from below by a positive number. If β=1/2, it is conjectured that the smallest eigenvalue tends to 0 algebraically, with a precise exponent. The algorithm run on the HPCC producing a fantastic match between the theoretical value of 2/π and the numerical result.

Extracting analytical one-loop amplitudes from numerical evaluations

In this article we present a method to generate analytic expressions for the integral coefficients of loop amplitudes using numerical evaluations only. We use highprecision arithmetics to explore the singularity structure of the coefficients and decompose them into parts of manageable complexity. To illustrate the usability of our method we provide analytical expressions for all helicity configurations of the colour-ordered six-point gluon amplitudes at one loop with a gluon in the loop.

Numerical aspects of integration in semi-closed option pricing formulas for stochastic volatility jump diffusion models

ABSTRACTIn mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time. In this paper, we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases.

Implementing a chaotic cryptosystem in a 64-bit embedded system by using multiple-precision arithmetic

This paper proposes a new chaotic cryptosystem for the encryption of very high-resolution digital images based on the design of a digital chaos generator by using arbitrary precision arithmetic. This can be taken as an alternative to reduce the dynamic degradation that chaotic models present when they are implemented in digital devices and to increase the security of the cryptosystems. The obtained results show that when using high-precision arithmetic, the generated sequences provide good randomness and security during a greater number of iterations of the implemented chaotic maps in comparison with the generated sequences by using the standard of simple precision or double precision according to the IEEE 754 standard for floating-point arithmetic. The proposed method does not require high-cost hardware for increasing the numerical accuracy and security. As an advantage versus other recent works, using high precision, in relation to the methods that use simple precision or double precision, it awards an exponential increase in the key space. In this manner, it is demonstrated that using multiple-precision arithmetic, a key space of $$2^{33,268}$$ or higher can be obtained, depending on the level of high precision configured. The security analysis confirms that the proposed chaotic cryptosystem is secure and robust against several known attacks, as well as statistical tests of NIST and TestU01, proving that high-precision arithmetic helps to enhance the security of the cryptosystems.

Practical rules for summing the series of the Tweedie probability density function with high-precision arithmetic.

For some ranges of its parameters and arguments, the series for Tweedie probability density functions are sometimes exceedingly difficult to sum numerically. Existing numerical implementations utilizing inversion techniques and properties of stable distributions can cope with these problems, but no single one is successful in all cases. In this work we investigate heuristically the nature of the problem, and show that it is not related to the order of summation of the terms. Using a variable involved in the analytical proof of convergence of the series, the critical parameter for numerical non-convergence ("alpha") is identified, and an heuristic criterion is developed to avoid numerical non-convergence for a reasonably large sub-interval of the latter. With these practical rules, simple summation algorithms provide sufficiently robust results for the calculation of the density function and its definite integrals. These implementations need to utilize high-precision arithmetic, and are programmed in the Python programming language. A thorough comparison with existing R functions allows the identification of cases when the latter fail, and provide further guidance to their use.

Eighth order family of iterative methods for nonlinear equations and their basins of attraction

We present a new family of eighth order methods for solving nonlinear equations. The order of convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of four parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, demonstrate convergence and efficacy of our family of methods. For some combinations of parameter values, the new eighth order methods produce very good results on tested examples compared to the results produced by some of the eighth order methods existing in the related literature. An exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials.

A Family of Combined Iterative Methods for Solving Nonlinear Equations

In this article we construct some higher-order modifications of Newton’s method for solving nonlinear equations, which is based on the undetermined coefficients. This construction can be applied to any iteration formula. It can be found that per iteration the resulting methods add only one additional function evaluation, their order of convergence can be increased by two or three units. Higher order convergence of our methods is proved and corresponding asymptotic error constants are expressed. Numerical examples, obtained using Matlab with high precision arithmetic, are shown to demonstrate the convergence and efficiency of the combined iterative methods. It is found that the combined iterative methods produce very good results on tested examples, compared to the results produced by the existing higher order schemes in the related literature.

Efficient Complex High-Precision Computations on GPUs without Precision Loss

General-purpose computing on graphics processing units is the utilization of a graphics processing unit (GPU) to perform computation in applications traditionally handled by the central processing unit. Many attempts have been made to implement well-known algorithms on embedded and mobile GPUs. Unfortunately, these applications are computationally complex and often require high precision arithmetic, whereas embedded and mobile GPUs are designed specifically for graphics, and thus are very restrictive in terms of input/output, precision, programming style and primitives available. This paper studies how to implement efficient and accurate high-precision algorithms on embedded GPUs adopting the OpenGL ES language. We discuss the problems arising during the design phase, and we detail our implementation choices, focusing on the SIFT and ALP key-point detectors. We transform standard, i.e., single (or double) precision floating-point computations, to reduced-precision GPU arithmetic without precision loss. We develop a desktop framework to simulate Gaussian Scale Space transforms on all possible target embedded GPU platforms, and with all possible range and precision arithmetic. We illustrate how to re-engineer standard Gaussian Scale Space computations to mobile multi-core parallel GPUs using the OpenGL ES language. We present experiments on a large set of standard images, proving how efficiency and accuracy can be maintained on different target platforms. To sum up, we present a complete framework to minimize future programming effort, i.e., to easily check, on different embedded platforms, the accuracy and performance of complex algorithms requiring high-precision computations.

Design and implementation of the modified signed digit multiplication routine on a ternary optical computer.

Multiplication with traditional electronic computers is faced with a low calculating accuracy and a long computation time delay. To overcome these problems, the modified signed digit (MSD) multiplication routine is established based on the MSD system and the carry-free adder. Also, its parallel algorithm and optimization techniques are studied in detail. With the help of a ternary optical computer's characteristics, the structured data processor is designed especially for the multiplication routine. Several ternary optical operators are constructed to perform M transformations and summations in parallel, which has accelerated the iterative process of multiplication. In particular, the routine allocates data bits of the ternary optical processor based on digits of multiplication input, so the accuracy of the calculation results can always satisfy the users. Finally, the routine is verified by simulation experiments, and the results are in full compliance with the expectations. Compared with an electronic computer, the MSD multiplication routine is not only good at dealing with large-value data and high-precision arithmetic, but also maintains lower power consumption and fewer calculating delays.

An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned Conjugate Gradient Method using high-precision arithmetic

In this work, four heuristics for bandwidth and profile reductions are evaluated. Specifically, the results of a recent proposed heuristic for bandwidth and profile reductions of symmetric and asymmetric matrices using a one-dimensional self-organising map is evaluated against the results obtained from the variable neighbourhood search for bandwidth reduction heuristic, the original reverse Cuthill-McKee method, and the reverse Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. These four heuristics were applied to three datasets of linear systems composed of sparse symmetric positive-definite matrices arising from discretisations of the heat conduction and Laplace equations by finite volumes. The linear systems are solved by the Jacobi-preconditioned conjugate gradient method when using high-precision numerical computations. The best heuristic in the simulations performed with one of the datasets used was the Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. On the other hand, no gain was obtained in relation to the computational cost of the linear system solver when a heuristic for bandwidth and profile reduction is applied to instances contained in two of the datasets used.

An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned conjugate gradient method using high-precision arithmetic

An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned conjugate gradient method using high-precision arithmetic

The Role of High Precision Arithmetic in Calculating Numerical Laplace and Inverse Laplace Transforms

In order to find stable, accurate, and computationally efficient methods for performing the inverse Laplace transform, a new double transformation approach is proposed. To validate and improve the inversion solution obtained using the Gaver-Stehfest algorithm, direct Laplace transforms are taken of the numerically inverted transforms to compare with the original function. The numerical direct Laplace transform is implemented with a composite Simpson’s rule. Challenging numerical examples involving periodic and oscillatory functions, are investigated. The numerical examples illustrate the computational accuracy and efficiency of the direct Laplace transform and its inverse due to increasing the precision level and the number of terms included in the expansion. It is found that the number of expansion terms and the precision level selected must be in a harmonious balance in order for correct and stable results to be obtained.