Showing papers on "Inverse trigonometric functions published in 2020"
TL;DR: A quantum algorithm for solving Poisson equation, as well as a complete and modular circuit design based on the classical non-restoring method, which reduces the algorithm’s complexity and makes the circuit more complete and practical.
Abstract: The Poisson equation has applications across many areas of physics and engineering, such as the dynamic process simulation of ocean current. Here we present a quantum algorithm for solving Poisson equation, as well as a complete and modular circuit design. The algorithm takes the HHL algorithm as the framework (where HHL is for solving linear equations). A more efficient way of implementing the controlled rotation, one of the crucial steps in HHL, is developed based on the arc cotangent function. The key point is that the inverse trigonometric function can be evaluated in a very simple recursive way by a binary expansion method. Quantum algorithms for solving square root and reciprocal functions are proposed based on the classical non-restoring method. These advances not only reduce the algorithm’s complexity, but more importantly make the circuit more complete and practical. We demonstrate our circuits on a quantum virtual computing system installed on the Sunway TaihuLight supercomputer. This is an important step toward practical applications of the present circuits as a fast Poisson solver in the near-term hybrid classical/quantum devices.
48 citations
TL;DR: In this article, the best possible upper and lower bounds for the Sandor-Yang means in terms of the families of one-parameter geometric and quadratic means are presented.
Abstract: In the article, we present the best possible upper and lower bounds for the Sandor–Yang means in terms of the families of one-parameter geometric and quadratic means, and discover new bounds for the inverse tangent and inverse hyperbolic sine functions.
45 citations
TL;DR: Comparison results between minimax concave, arctan and L1 norm also prove the superior high-amplitude preserving ability of the proposed method.
40 citations
TL;DR: The qFBE method provides a unified and programmed solution for the evaluation of transcendental functions, and it will be an important building block for many quantum algorithms.
Abstract: Quantum arithmetic in the computational basis constitutes the fundamental component of many circuit-based quantum algorithms. There exist a lot of studies about reversible implementations of algebraic functions, while research on the higher-level transcendental functions is scant. We propose to evaluate the transcendental functions using a novel methodology, which is called quantum function-value binary expansion (qFBE) method. This method transforms the evaluation of transcendental functions to the computation of algebraic functions, and output the binary solution digit-by-digit in a simple recursive way. The quantum circuits for solving the logarithmic, exponential, trigonometric and inverse trigonometric functions are presented based on the qFBE method. The efficiency of the circuits is demonstrated on a quantum virtual computing system installed on the Sunway TaihuLight supercomputer. The qFBE method provides a unified and programmed solution for the evaluation of transcendental functions, and it can be the essential building block for many quantum algorithms.
18 citations
TL;DR: In this paper, a modified generalized Prandtl-Ishlinskii (GPI) model is proposed for the length-pressure hysteresis modeling of pneumatic artificial muscles.
Abstract: This paper presents a method for the length-pressure hysteresis modeling of pneumatic artificial muscles (PAMs) by using a modified generalized Prandtl-Ishlinskii (GPI) model. Different from the approaches for establishing the GPI models by replacing the linear envelope functions of operators with hyperbolic tangent and exponential envelop functions, the proposed model is derived by modifying the envelope functions of operators into arc tangent functions, which shows an improvement in the modeling accuracy. The effectiveness of the proposed model is verified by the experimental data of a PAM. Furthermore, its capacity in capturing the hysteresis relationship between length and pressure is testified by giving different input pressure signals. With regard to the computational efficiency, the influence of the number of operators on the modeling accuracy is discussed. Furthermore, the inversion of the GPI model is derived. Its capability of compensating the hysteresis nonlinearities is confirmed via the simulation and experimental study.
16 citations
TL;DR: In this paper, the authors present an innovative model that they have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy, and illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers' instruction on inverse trigonometric functions and solving trigonometrical equations.
16 citations
TL;DR: In this paper, a sharp double inequality between the inverse tangent and inverse hyperbolic sine functions was obtained, where the inverse sine function is the inverse of the tangent function.
Abstract: In this paper, we obtain a sharp double inequality between the inverse tangent and inverse hyperbolic sine functions. At the same time, we give a sharp double inequality between the inverse hyperbolic tangent and inverse sine functions.
10 citations
18 Mar 2020
TL;DR: It is proved that the modified robust PID controller by inverse tangent function meets the requirement of energy-saving capacity and is energy efficient and concise for its convenient use, which is feasible to expand its utility to other applications.
Abstract: This paper explores the significance and feasibility of addressing a notion that the system error of a nonlinear feedback control can be decorated by an inverse tangent function in order to attain a sound energy-efficient performance. The related mathematical model and relevant evaluation of this concept are further illustrated by demonstrating a case study about the control performance of water tank level. The rationale of robust control and theoretical algorithm of Lyapunov stability theorem are outlined to evaluate the effectiveness of nonlinear feedback with inverse tangent function in terms of improving robustness of PID (Proportional–Integral–Derivative) controller and energy-saving capability. By demonstrating five simulations of different scenarios, it ultimately proves that the modified robust PID controller by inverse tangent function meets the requirement of energy-saving capacity. Comparing with the routine PID control, the mean control input of controlling water tank level can be reduced up to 39.2% by using modified nonlinear feedback controller. This nonlinear feedback PID controller is energy efficient and concise for its convenient use, which is feasible to expand its utility to other applications.
9 citations
TL;DR: A new non-linear variant of a quantitative extension of the uniform boundedness principle is used to show sharpness of error bounds for univariate approximation by sums of sigmoid and ReLU functions.
Abstract: A new non-linear variant of a quantitative extension of the uniform boundedness principle is used to show sharpness of error bounds for univariate approximation by sums of sigmoid and ReLU functions. Single hidden layer feedforward neural networks with one input node perform such operations. Errors of best approximation can be expressed using moduli of smoothness of the function to be approximated (i.e., to be learned). In this context, the quantitative extension of the uniform boundedness principle indeed allows to construct counterexamples that show approximation rates to be best possible. Approximation errors do not belong to the little-o class of given bounds. By choosing piecewise linear activation functions, the discussed problem becomes free knot spline approximation. Results of the present paper also hold for non-polynomial (and not piecewise defined) activation functions like inverse tangent. Based on Vapnik–Chervonenkis dimension, first results are shown for the logistic function.
8 citations
TL;DR: In this paper, the authors proved the Wilker and Huygens type inequalities for inverse trigonometric functions, which solves two conjectures proposed by Chao-Ping Chen.
Abstract: In this paper, we prove Wilker and Huygens type inequalities for inverse trigonometric functions. This solves two conjectures proposed by Chao-Ping Chen. Also, we present new sharp Wilker and Huygens type inequalities for trigonometric functions.
7 citations
TL;DR: In this paper, the authors investigated the efficient expansion of the Taylor series that should be included in applications, such that numerical bias is not present, and showed that five functions require more than five terms, whereas the arcsine, arccosine, and arctangent did not.
Abstract: The parameters of nonlinear forest models are commonly estimated with heuristic techniques, which can supply erroneous values. The use of heuristic algorithms is partially rooted in the avoidance of transformation of the dependent variable, which introduces bias when back-transformed to original units. Efforts were placed in computing the unbiased estimates for some of the power, trigonometric, and hyperbolic functions since only few transformations of the predicted variable have the corrections for bias estimated. The approach that supplies unbiased results when the dependent variable is transformed without heuristic algorithms, but based on a Taylor series expansion requires implementation details. Therefore, the objective of our study is to investigate the efficient expansion of the Taylor series that should be included in applications, such that numerical bias is not present. We found that five functions require more than five terms, whereas the arcsine, arccosine, and arctangent did not. Furthermore, the Taylor series expansion depends on the variance. We illustrated the results on two forest modeling problems, one at the stand level, namely site productivity, and one at individual tree level, namely taper. The models that are presented in the paper are unbiased, more parsimonious, and they have a RMSE comparable with existing less parsimonious models.
TL;DR: In this paper, the inverse hyperbolic tangent and the inverse tangent functions are shown to have the same relationship as the corresponding functions in this paper, and the analogue for the inverse transitive tangent function is given.
Abstract: In this paper, we present new inequalities which reveal further relationship for both the inverse tangent function $\arctan (x)$ and the inverse hyperbolic function $\operatorname{arctanh}(x)$. At the same time, we give the analogue for inverse hyperbolic tangent and other corresponding functions.
01 Apr 2020
TL;DR: This work proposes method for modulus approximation that uses addition, scaling and comparison, and provides also argument information, and proposes algorithm that reduces error at each additional level.
Abstract: Determination of complex number modulus and argument is commonly encountered task in digital signal processing. According to definition, this task requires evaluation of square root and inverse tangent functions. When computing hardware resources are limited, e.g. in real-time applications, an approximation by basic arithmetic and logical operations are of interest. We propose method for modulus approximation that uses addition, scaling and comparison, and provides also argument information. At each additional level of our algorithm, modulus approximation error decreases by factor of 4 and argument error by factor of 8.
Posted Content•
TL;DR: In this article, the authors present a combinatorial study of sums of integer powers of the cotangent which is a popular theme in classical calculus, and they use the trace method to draw conclusions about integer values of the sums and expand generating functions to obtain explicit evaluations.
Abstract: This paper presents a combinatorial study of sums of integer powers of the cotangent which is a popular theme in classical calculus. Our main tool the realization of cotangent values as eigenvalues of a simple self-adjoint matrix with integer matrix. We use the trace method to draw conclusions about integer values of the sums and expand generating functions to obtain explicit evaluations. It is remarkable that throughout the calculations the combinatorics are governed by the higher tangent and arctangent numbers exclusively. Finally we indicate a new approximation of the values of the Riemann zeta function at even integer arguments.
Posted Content•
TL;DR: The qFBE method as discussed by the authors transforms the evaluation of transcendental functions to the computation of algebraic functions in a simple recursive way, and the quantum circuits for solving the logarithmic, exponential, trigonometric and inverse trigonometrical functions are presented.
Abstract: Quantum arithmetic in the computational basis constitutes the fundamental component of many circuit-based quantum algorithms. There exist a lot of studies about reversible implementations of algebraic functions, while research on the higher-level transcendental functions is scant. We propose to evaluate the transcendental functions based on a novel methodology, which is called qFBE (quantum Function-value Binary Expansion) method. This method transforms the evaluation of transcendental functions to the computation of algebraic functions in a simple recursive way. We present the quantum circuits for solving the logarithmic, exponential, trigonometric and inverse trigonometric functions based on the qFBE method. The efficiency of the circuits is demonstrated on a quantum virtual computing system installed on the Sunway TaihuLight supercomputer. The qFBE method provides a unified and programmed solution for the evaluation of transcendental functions, and it will be an important building block for many quantum algorithms.
Posted Content•
TL;DR: In this paper, several convolution identities of poly-Cauchy numbers with level $2$ were shown, in particular that of three poly-cauchy number with level$2$.
Abstract: Poly-Cauchy numbers with level $2$ are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we show several convolution identities of poly-Cauchy numbers with level $2$. In particular, that of three poly-Cauchy numbers with level $2$ can be expressed as a simple form. In the sequel, we introduce the Stirling numbers of the first kind with level $2$
TL;DR: The burden of the software application of the proposed algorithm is greatly reduced by not using the sector selection algorithm, the region selection algorithms, and the on-state time allocation process as the proposed modulation can directly control the switch on- state time.
Abstract: The present paper proposes a direct space vector modulation and novel balance algorithm for easy software application of three-level converters which operate in three-phase. In the case of the conventional space vector modulation, to get the on-state times of the switches, the dwell times of the three nearest stationary vectors, which are obtained after sector and region selection algorithms, should be rearranged. These processes, therefore, contain diverse conditional statements and complicated calculations such as inverse trigonometric functions and square roots. However, the burden of the software application of the proposed algorithm is greatly reduced by not using the sector selection algorithm, the region selection algorithm, and the on-state time allocation process as the proposed modulation can directly control the switch on-state time. In a three-level topology, it is required to balance top and bottom capacitor voltages because the DC-link voltage is composed of two capacitor voltages; the unbalanced voltage of each DC-link capacitor causes the overvoltage of the switching devices. Thus, the DC-link voltage balancing algorithm is proposed, and it is also very simple and effective without additional circuits because it controls the switch on-state time directly as well. The 5-kW prototype proved the validity of the proposed algorithm with its feasibility.
TL;DR: In this article, the phase shift angle of the output signals of the induction sensor is determined by comparing the obtained arrangements of signal values with a circular discrete convolution in order to achieve the most precise approximation of the obtained signal values to cosine and sine.
Abstract: Goal. Representation of a special mathematical software for determining the angular displacements of the rotor of the induction angle sensor – resolver (rotating transformer) for applications in which the speed of the sensor's rotor is close to zero. As well as performing its experimental verification. Methodology. The presented method is based on the determination of the phase shift angle of the output signals of the induction sensor, which is determined by comparing the obtained arrangements of signal values with a circular discrete convolution in order to achieve the most precise approximation of the obtained signal values to cosine and sine. The conversion of orthogonal components to an angle is based on the use of a digital phase detector which is use of a software comparator and inverse trigonometric functions. Results. Based on the obtained results of mathematical modeling and experimental research, the characteristic dependencies of the angle of rotation of the rotor of the induction sensor relative to its stator, the nature of which is linear, were obtained. In addition, the estimation of measurement errors of angular displacements is carried out that occur when defining such angles by the method offered. The obtained results of the computer simulation taking into account the high signal noise, as well as the results of experimental investigations, confirm the high precision of this method and the fact that it can be used in systems where high positioning accuracy is required and the speed of the sensor shaft is close to zero. Originality. This article introduces, for the first time, special mathematical software for a new method of determining the angular displacements of the rotor of an induction sensor, which is based on the determination of the orthogonal components of the signal in combination with the use of a circular discrete convolution in the determination of the phase shift angle of the induction sensor signals. Practical meaning. The proposed method does not require the use of demodulators, counters and quadrant tables associated with conventional methods for determining the phase shift of signals. The presented method can be used to measure the full range of 0-2p angular displacements in real time, is simple and can be easily implemented using digital electronic circuitry.
Patent•
28 Apr 2020
TL;DR: In this paper, an arbitrary power square root solving method for a single-precision floating-point number and a solver of the arbitrary Power Square Root solving method are presented.
Abstract: The invention provides an arbitrary power square root solving method for a single-precision floating-point number and a solver of the arbitrary power square root solving method. The solver comprises:a division calculation module for carrying out division operation on an input power root value N; an arc tangent value calculation module for carrying out arc tangent value calculation operation on amantissa part M of the input single-precision floating-point number and obtaining a log2M; a calculation module for carrying out multiplication and addition operation on an exponent part E of a single-precision floating-point number, a reciprocal 1/N of the root value N of the power and the log2M of the numerical value; a sine and cosine calculation module for solving hyperbolic sine and cosine values with 2 as the bottom for the calculation result obtained by the calculation module; and a calculation result integration module for summing the solved hyperbolic sine and hyperbolic cosine valuesand integrating the sum with an intermediate calculation result of the index part E to obtain a final calculation result in a single-precision floating-point number format. The solver provided by theinvention can calculate any power root value of any single-precision floating-point number, and has certain universality.
TL;DR: In this paper, the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number, such that a function exists with respect to the leaf function through the imaginary number $i.
Abstract: Addition formulas exist in trigonometric functions. Double-angle and half-angle formulas can be derived from these formulas. Moreover, the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number. The inverse hyperbolic function $\mathrm{arsinh}(r)=\int_{0}^{r} \frac{1}{\sqrt{1+t^2} }\mathrm{d}t$ is similar to the inverse trigonometric function $\mathrm{arcsin}(r)=\int_{0}^{r} \frac{1}{\sqrt{1-t^2} }\mathrm{d}t$, such as the second degree of a polynomial and the constant term 1, except for the sign $-$ and $+$. Such an analogy holds not only when the degree of the polynomial is 2, but also for higher degrees. As such, a function exists with respect to the leaf function through the imaginary number $i$, such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number. In this study, we refer to this function as the hyperbolic leaf function. By making such a definition, the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas, such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions. Using the addition formulas, we can also derive the double angle and half-angle formulas. We then verify the consistency of these formulas by constructing graphs and numerical data.
Patent•
30 Jul 2020
TL;DR: In this article, an acoustic direction finder determining direction of acoustic harmonic signal source consists of three microphones forming three orthogonal measuring bases, three preprocessing units including preliminary amplifiers, band-pass filters, limit amplifiers and inverters and shapers of short pulses, rectifiers, comparison circuit, divider by two, logic unit, counters, triggers, parallel registers, clock oscillator, microcontroller.
Abstract: FIELD: robotics.SUBSTANCE: invention relates to robotics, namely to equipment for orientation of transport robots based on acoustic signal of stationary beacon or beacon of a leading transport device. Acoustic direction finder determining direction of acoustic harmonic signal source consists of three microphones forming three orthogonal measuring bases, three pre-processing units including preliminary amplifiers, band-pass filters, limit amplifiers, inverters and shapers of short pulses, rectifiers, comparison circuit, divider by two, logic unit, counters, triggers, parallel registers, clock oscillator, microcontroller.EFFECT: direction finder technical result is elimination of ambiguity of bearing due to determination of sign of phase shift along two orthogonal bases due to application of logic unit and sign triggers, as well as reducing the direction-finding error by selecting a table of inverse trigonometric functions from the microcontroller memory in comparison to absolute values of phase shifts.1 cl, 3 dwg, 2 tbl
01 Oct 2020
TL;DR: In this article, three weight functions: Inverse trigonometric function, Hyperbolic tangent function and Log function are proposed for the purpose of improving the robustness and time efficiency of IR3C algorithm.
Abstract: The need for occlusion face recognition in daily life, professional fields like criminal investigation and the recent epidemic is increasing, so it is still a hot issue in the current research and many researchers are pursuing the technical excellence for the face recognition of random occlusion. In the previous research, RRC model is proposed for the excellent performance in the occlusion face recognition and the IR3C algorithm is adopted to solve this model. Among them, the weight function in this algorithm plays a big role, which will directly affect the recognition rate and running time of the experiment. In this paper, three weight functions: Inverse trigonometric function, Hyperbolic tangent function and Log function are proposed for the purpose of improving the robustness and time efficiency of IR3C algorithm. Applying these functions on the comparable types and levels of random occlusion and then comparing them with the previous research, the experimental results indicate that the inverse trigonometric function is superior to other functions with better recognition rates and shorter running time.
Posted Content•
TL;DR: This paper developed nested representations for cosine and inverse cosine functions, which is a generalization of Viete's formula for $\pi$ and showed that these nested formulas are valid for complex arguments and a $k$th branch.
Abstract: In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Viete's formula for $\pi$. We explore a natural inverse relationship between these representations and develop numerical algorithms to compute them. Throughout this article, we perform numerical computation for various test cases, and demonstrate that these nested formulas are valid for complex arguments and a $k$th branch. We further extend the presented results to hyperbolic cosine and logarithm functions, and using additional trigonometric identities, we explore the sine and tangent functions and their inverses.
01 Jan 2020
TL;DR: The design implementation of CORIDC algorithm thus reduces the hardware implementation by using shift registers and adders, thereby increases its speed and the proposed system can generate sine and cosine signals in signal processing applications.
Abstract: There are several ways to define trigonometric signals; CORDIC is the more efficient algorithm to perform trigonometric operations for generating sine and cosine waveforms. By using rotation and vectoring modes in the coordinate system implementation in hardware becomes easy, and also different scaling factors and its compensated techniques can also be calculated further. The proposed system can generate sine and cosine signals in signal processing applications. Also, we can perform hyperbolic and exponential calculations. It can be further enhanced by extending for extended hyperbolic and linear coordinates for the trigonometric functions and inverse trigonometric functions. The proposed project is implemented in Xilinx ISE 13.2, and the output waveforms are simulated in Questasim 10.0b. The design implementation of CORIDC algorithm thus reduces the hardware implementation by using shift registers and adders, thereby increases its speed.
Posted Content•
TL;DR: In this paper, the higher derivatives of the tangent and hyperbolic tangent functions are determined using Chebyshev polynomials of the first and second kind.
Abstract: The higher derivatives of the tangent and hyperbolic tangent functions are determined. Formulas for the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions as polynomials are stated and proved. Using another formula for the higher derivatives of the inverse tangent function from literature, two known formulas for the Chebyshev polynomials of the first and second kind are proved. From these formulas the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions in terms of the Chebyshev polynomial of the second kind are provided.
Patent•
30 Jul 2020
TL;DR: In this article, a calibration method for an extrinsic camera parameter of an on-board camera system, applicable to electronic apparatuses, is presented. But this method requires the use of a front camera to acquire an image of a road ahead, and extracts information of lanes and a target vehicle position from the image of the road, and uses lane information in combination with a static calibration result and an inverse trigonometric function to solve an angle of a camera parameter dynamically, such that a more accurate angle of the camera parameter is obtained.
Abstract: A calibration method for an extrinsic camera parameter of an on-board camera system, applicable to electronic apparatuses. The method comprises: acquiring image information of one frame, and performing, by means of a lane detection method, global identification on the image information to obtain lane information (101); selecting, from each lane, at least two feature points at random, calculating coordinates of the feature points in a global coordinate system, and incorporating a camera height to find an angle correction amount by means of an inverse trigonometric function (102); and performing, by means of an iterative method, iterative compensation on an angle of an extrinsic camera parameter, and obtaining an accurate angle of the extrinsic camera parameter (103). The invention uses a front camera to acquire an image of a road ahead, extracts information of lanes and a target vehicle position from the image of the road, and uses lane information in combination with a static calibration result and an inverse trigonometric function to solve an angle of an extrinsic camera parameter dynamically, such that a more accurate angle of the extrinsic camera parameter is obtained, thereby solving a distance of a target with respect to a vehicle body according to the target vehicle position in the image.