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Kuiper's $V_n$ statistic, a measure for comparing the difference of ideal distribution and empirical distribution, is of great significance in the goodness-of-fit test. However, Kuiper's formulae for computing the cumulative distribution function, false positive probability and the upper tail quantile of $V_n$can not be applied to the case of small sample capacity $n$ since the approximation error is $\mathcal{O}(n^{-1})$. In this work, our contributions lie in three perspectives: firstly the approximation error is reduced to $\mathcal{O}(n^{-(k+1)/2})$ where $k$ is the expansion order with the \textit{high order expansion} (HOE) for the exponent of differential operator; secondly, a novel high order formula with approximation error $\mathcal{O}(n^{-3})$ is obtained by massive calculations; thirdly, the fixed-point algorithms are designed for solving the Kuiper pair of critical values and upper tail quantiles based on the novel formula. The high order expansion method for Kuiper's $V_n$-statistic is applicable for various applications where there are more than $5$ samples of data. The principles, algorithms and code for the high order expansion method are attractive for the goodness-of-fit test., Comment: arXiv admin note: text overlap with arXiv:2308.09463

The Hurst exponent is a significant indicator for characterizing the self-similarity and long-term memory properties of time sequences. It has wide applications in physics, technologies, engineering, mathematics, statistics, economics, psychology and so on. Currently, available methods for estimating the Hurst exponent of time sequences can be divided into different categories: time-domain methods and spectrum-domain methods based on the representation of time sequence, linear regression methods and Bayesian methods based on parameter estimation methods. Although various methods are discussed in literature, there are still some deficiencies: the descriptions of the estimation algorithms are just mathematics-oriented and the pseudo-codes are missing; the effectiveness and accuracy of the estimation algorithms are not clear; the classification of estimation methods is not considered and there is a lack of guidance for selecting the estimation methods. In this work, the emphasis is put on thirteen dominant methods for estimating the Hurst exponent. For the purpose of decreasing the difficulty of implementing the estimation methods with computer programs, the mathematical principles are discussed briefly and the pseudo-codes of algorithms are presented with necessary details. It is expected that the survey could help the researchers to select, implement and apply the estimation algorithms of interest in practical situations in an easy way., Comment: 46 pages, 8 figures, 4 tables, 24 algorithms with pseudo-codes

Kuiper's statistic is a good measure for the difference of ideal distribution and empirical distribution in the goodness-of-fit test. However, it is a challenging problem to solve the critical value and upper tail quantile, or simply Kuiper pair, of Kuiper's statistics due to the difficulties of solving the nonlinear equation and reasonable approximation of infinite series. In this work, the contributions lie in three perspectives: firstly, the second order approximation for the infinite series of the cumulative distribution of the critical value is used to achieve higher precision; secondly, the principles and fixed-point algorithms for solving the Kuiper pair are presented with details; finally, finally, a mistake about the critical value $c^\alpha_n$ for $(\alpha, n)=(0.01,30)$ in Kuiper's distribution table has been labeled and corrected where $n$ is the sample capacity and $\alpha$ is the upper tail quantile. The algorithms are verified and validated by comparing with the table provided by Kuiper. The methods and algorithms proposed are enlightening and worth of introducing to the college students, computer programmers, engineers, experimental psychologists and so on., Comment: 20 pages, 6 figures, 5 tables, code available on GitHub

Zernike circular polynomials (ZCP) play a significant role in optics engineering. The symbolic expressions for ZCP are valuable for theoretic analysis and engineering designs. However, there are still two problems which remain open: firstly, there is a lack of sufficient mathematical formulas of the ZCP for optics designers; secondly the formulas for inter-conversion of Noll's single index and Born-Wolf's double indices of ZCP are neither uniquely determinate nor satisfactory. An automatic method for generating symbolic expressions for ZCP is proposed based on five essential factors: the new theorems for converting the single/double indices of the ZCP, the robust and effective numeric algorithms for computing key parameters of ZCP, the symbolic algorithms for generating mathematical expressions of ZCP, and meta-programming \& \LaTeX{} programming for generating the table of ZCP. The theorems, method, algorithms and system architecture proposed are beneficial to both optics design process, optics software, computer-output typesetting in publishing industry as well as STEM education., Comment: This paper has proposed a method for creating mathematical table as demonstrated by generating Zenike circular polynomials. The code is available on GitHub site, please see: https://github.com/GrAbsRD/ZernikeSymbolicExpression

The problem-project-oriented STEM education plays a significant role in training students' ability of innovation. Although the conceive-design-implement-operate (CDIO) approach and the computational thinking (CT) are hot topics in recent decade, there are still two deficiencies: the CDIO approach and CT are discussed separately and a general framework of coping with complex STEM problems in system modeling and simulation is missing. In this paper, a collaborative strategy based on the CDIO and CT is proposed for solving complex STEM problems in system modeling and simulation with a general framework, in which the CDIO is about ``how to do", CT is about ``how to think", and the project means ``what to do". As an illustration, the problem of solving the period of mathematical pendulum (MP) is discussed in detail. The most challenging task involved in the problem is to compute the complete elliptic integral of the first kind (CEI-1). In the philosophy of STEM education, all problems have more than one solutions. For computing the CEI-1, four methods are discussed with a top-down strategy, which includes the infinite series method, arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method and Gauss-Legendre method. The algorithms involved can be utilized for R & D projects of interest and be reused according to the requirements encountered. The general framework for solving complex STEM problem in system modeling and simulation is worth recommending to the college students and instructors., Comment: 27 pages, 12 figures, 11 tables

Quaternion kinematical differential equation (QKDE) plays a key role in navigation, control and guidance systems. Although explicit symplectic geometric algorithms (ESGA) for this problem are available, there is a lack of a unified way for constructing high order symplectic difference schemes with configurable order parameter and the fractional interval sampling problem should be treated carefully. We present even order explicit symplectic geometric algorithms to solve the QKDE with diagonal Pad\'{e} approximation via a four-step strategy. Firstly, the Pad\'{e}-Cayley lemma is proved and used to simplify the symplectic Pad\'{e} approximation for the linear Hamiltonian system with infinitesimal symplectic structure. Secondly, both parallel and alternative iterative methods are proposed to construct the symplectic difference schemes with even order accuracy. Thirdly, the symplecity, orthogonality and invertibility of the single-step transition matrices are proved rigorously. Finally, the explicit symplectic geometric algorithms are designed for both the linear time-invariant and linear time-varying QKDE. The maximum absolute error for solving the QKDE is $\mathcal{O}((t_f-t_0)\tau^{2\ell})$ where $\tau$ is the time step, $\ell$ is the order parameter and $[t_0,t_f]$ is the time span. The linear time complexity and constant space complexity of computation as well as the simple algorithmic structure show that our algorithms are appropriate for real-time applications in aeronautics, astronautics, robotics and so on. The performance of the proposed algorithms are verified and validated by mathematical analysis and numerical simulation., Comment: 20 pages, 7 figures, 3 tables

The complete elliptic integral of the first kind (CEI-1) plays a significant role in mathematics, physics and engineering. There is no simple formula for its computation, thus numerical algorithms are essential for coping with the practical problems involved. The commercial implementations for the numerical solutions, such as the functions \lstinline|ellipticK| and \lstinline|EllipticK| provided by MATLAB and Mathematica respectively, are based on $\mathcal{K}_{\mathrm{cs}}(m)$ instead of the usual form $K(k)$ such that $\mathcal{K}_{\mathrm{cs}}(k^2) =K(k)$ and $m=k^2$. It is necessary to develop open source implementations for the computation of the CEI-1 in order to avoid potential risks of using commercial software and possible limitations due to the unknown factors. In this paper, the infinite series method, arithmetic-geometric mean (AGM) method, Gauss-Chebyshev method and Gauss-Legendre methods are discussed in details with a top-down strategy. The four key algorithms for computing CEI-1 are designed, verified, validated and tested, which can be utilized in R\& D and be reused properly. Numerical results show that our open source implementations based on $K(k)$ are equivalent to the commercial implementation based on $\mathcal{K}_{\mathrm{cs}}(m)$. The general algorithms for computing orthogonal polynomials developed are significant byproducts in the sense of STEM education and scientific computation., Comment: 22 pages, 11 figures, 5 tables

Zernike radial polynomials (ZRP) play a significant role in application areas such as optics design, imaging systems, and image processing systems. Currently, there are two kinds of numerical schemes for computing the ZRP automatically with computer programs: one is based on the definition in which the factorial operations may lead to the overflow problem and the high order derivatives are troublesome, and the other is based on recursion which is either unstable or with high computational complexity. In this paper, our emphasis is focused on exploring the balanced binary tree (BBT) schemes for computing the ZRP: firstly an elegant formulae for computation is established; secondly the recursive and iterative algorithms based-on BBT are proposed; thirdly the computational complexity of the algorithms are analyzed rigorously; finally the performance of BBT schemes by testing the running time is verified and validated. Theoretical analysis shows that the computational complexity of balanced binary tree recursive algorithm (BBRTA) and iterative algorithm are exponential and quadratic respectively, which coincides with the running time test very well. Experiments show that the time consumption is about $1\sim 10$ microseconds with different computation platforms for the balanced binary tree iterative algorithm (BBTIA), which is stable and efficient for real-time applications. In the sense of STEM education, the connection of the BBT and ZRP exhibits the beauty and applications of discrete mathematical structure behind the engineering problem, which is worthy of introducing to the college students, computer programmers and optics engineers., Comment: 17 pages, 4 figures