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Transcendental functions are an important part of algorithms in many fields. However, the hardware accelerators available today for transcendental functions typically only support one such function. Hardware accelerators that can support miscellaneous transcendent functions are a waste of hardware resources. In order to solve these problems, this paper proposes a reconfigurable hardware architecture for miscellaneous floating-point transcendental functions. The hardware architecture supports a variety of transcendental functions, including floating-point sine, cosine, arctangent, exponential and logarithmic functions. It adopts the method of a lookup table combined with a polynomial computation and reconfigurable technology to achieve the accuracy of two units of least precision (ulp) with 3.75 KB lookup tables and one core computing module. In addition, the hardware architecture uses retiming technology to realize the different operation times of each function. Experiments show that the hardware accelerators proposed can operate at a maximum frequency of 220 MHz. The full-load power consumption and areas are only 0.923 mW and 1.40×104μm2, which are reduced by 47.99% and 38.91%, respectively, compared with five separate superfunction hardware accelerators.

In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article.

The research aims to estimate the productivity function of the potato crop according to the cultivar grown in the fall season 2021 in Baghdad Governorate - Abu Ghraib district, measure its elasticity and find out the impact of the independent factors on the dependent factor, (181 questionnaires) were obtained through a random personal interview conducted by the researcher with farmers affiliated with the Nasr and Peace Agriculture Division in Abu Ghraib district, It represents (30.2%) of the sample farmers who numbered (599 farmers) according to the statistics of the mentioned division, Use the Eviews 10 program to estimate the transcendental function and to represent the relationship between the productivity of a dunum of crop and the independent factors (Amount of seeds, urea fertilizer, Dab fertilizer, human labor and mechanical work), The modified coefficient of determination showed that 91% of the change in the level of productivity is caused by the independent variables, and that 9% of the change in productivity is caused by variables not included in the model, The total production elasticity of 0.641 indicates that production takes place according to diminishing returns to capacity, The results also showed the superiority of the Purina variety and similar varieties over the rest of the varieties in terms of productivity, so we recommend planting this variety and similar varieties.

The performance of transcendental functions is important to high-performance applications. Various customized implementations of transcendental functions in practice to obtain better performance for some particular applications, but specializing the transcendental functions for each individual application cannot scale with the increase and diversity of the latter. It is thus of vital importance to design and implement an automatic generator for transcendental functions. This article implements an automatic generator TGen (transcendental function generator) for variable precision transcendental functions. Given a specific computation interval, TGen can automatically generate independent implementations with different precisions for a transcendental function, and select the most suitable version for the target by means of a code filtering strategy. Our generator can now support the automatic generation of 18 representative transcendental functions. The experimental results show that TGen can generate code with the unfixed accuracy, and the performance of the generated code is superior to that of the code generated by metalibm. More specifically, when compiled with the GCC compiler, the code generated by TGen can provide a mean speedup of 1.84 × over that of a metalibm’s version, while the speedup of TGen’s code over that of the metalibm is 1.27 × when compiled with the SW5CC compiler.

Transcendental functions are important functions in various high performance computing applications. Because these functions are time-consuming and the vector units on modern processors become wider and more scalable, there is an increasing demand for developing and using vector transcendental functions in such performance-hungry applications. However, the performance of vector transcendental functions as well as their accuracy remain largely unexplored. To address this issue, we perform a comprehensive evaluation of two Single Instruction Multiple Data (SIMD) intrinsics based vector math libraries on two ARMv8 compatible processors. We first design dedicated microbenchmarks that help us understand the performance behavior of vector transcendental functions. Then, we propose a piecewise, quantitative evaluation method with a set of meaningful metrics to quantify their performance and accuracy. By analyzing the experimental results, we find that vector transcendental functions achieve good performance speedups thanks to the vectorization and algorithm optimization. Moreover, vector math libraries can replace scalar math libraries in many cases because of improved performance and satisfactory accuracy. Despite this, the implementations of vector math libraries are still immature, which means further optimization is needed, and our evaluation reveals feasible optimization solutions for future vector math libraries.

Let $\\unicode[STIX]{x1D70C}\\in (0,\\infty ]$ be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc. 94 (2016), 15–19] by proving that any subset of $\\overline{\\mathbb{Q}}\\cap B(0,\\unicode[STIX]{x1D70C})$ , which is closed under complex conjugation and contains $0$ , is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence $\\unicode[STIX]{x1D70C}$ . This solves the question posed by K. Mahler completely.

Let f be an entire transcendental function and let (λn) be a sequence of positive numbers increasing to +∞. Suppose that the series AZ=∑n=1∞anfλnz is regularly convergent in ℂ, i.e., ð\"(r, A) := ∑n=1∞anMfrλn< + ∞ for all r ∈ [0,+ ∞). For a positive function l continuous on [0, + ∞), the function A is called a function of bounded l-ð\"-index if there exists N ∈ ℤ+ such that Mr,Ann!lnr≤maxMr,Akk!lkr:0≤k≤N for all n ∈ ℤ+ and all r ∈ [0,+ ∞). We study the properties of growth of the functions of bounded l- ð\"-index and formulate some unsolved problems.

In recent years, the recreational and commercial use of flight and driving simulators has become more popular. All these applications require the calculation of orientation in either two or three dimensions. Besides the Euler angles notation, other alternatives to represent rigid body rotations include axis-angle notation, homogeneous transformation matrices, and quaternions. All these methods involve transcendental functions in their calculations, which represents a disadvantage when these algorithms are implemented in hardware. The use of transcendental functions in software-based algorithms may not represent a significant disadvantage, but in hardware-based algorithms, the potential of rational models stands out. Generally, to calculate transcendental functions in hardware, it is necessary to utilize algorithms based on the CORDIC algorithm, which requires a significant amount of hardware resources (parallel) or the design of a more complex control unit (pipelined). This research presents a new procedure for model orientation using rational trigonometry and quaternion notation, avoiding trigonometric functions for calculations. We describe the orientation of a gimbal mechanism presented in many applications, from autonomous vehicles such as cars or drones to industrial manipulators. This research aims to compare the efficiency of a rational implementation to classical modeling using the techniques mentioned above. Furthermore, we simulate the models with software tools and propose a hardware architecture to implement our algorithms.

BackgroundA semi-analytical approach is used for low- and high-frequency vibration for the thin plate. This analysis is essential for a wide range of engineering applications. However, very few numerical approaches are available which can be used with certainty to a variety of structures for higher frequency vibration. The present work proposes an innovative attempt to solve this, by combining the transcendental and polynomial functions to construct the complete sets of admissible functions in the Rayleigh–Ritz method.PurposeThe main purpose of this work is to construct a set of complete admissible functions to be used in the Rayleigh–Ritz method for the free vibration of plates. They comprise three types of trial functions, i.e. (1) transcendental, (2) simplest lower order polynomial, and (3) cosine functions.MethodThe Rayleigh–Ritz method is used to construct the Eigen value problem to verify the accuracy of the admissible functions.ResultsVarious combinations of hinged, clamped, guided, and free plates are studied by enforcing the geometric boundary conditions by applying the spring constants in the semi-analytical method to study the lower and higher modes. To verify the accuracy of the admissible functions, the frequency parameters are compared with the available literature. We illustrate the efficacy of present admissible functions for high-frequency analysis for plates pulsating in the first 4000th modes.ConclusionsThe present study proposes a novel approach for constructing the transcendental function from asymptotic an approximate form of rigid body beam mode shapes. However, the combination of transcendental and polynomial functions forms complete sets of admissible functions to be used in the Rayleigh–Ritz method. The convergence rate of such a function in the Rayleigh–Ritz method is faster as compared to other methods for lower and higher modes.

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain hypergeometric varieties.