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The Hurst exponent is a significant metric for characterizing time sequences with long-term memory property and it arises in many fields such as physics, engineering, mathematics, statistics, economics, psychology, and so on. The available methods for estimating the Hurst exponent can be categorized into time-domain and spectrum-domain methods based on the representation of the time sequence, and into linear regression and Bayesian method based on parameter estimation techniques. Although there are various estimation methods for the Hurst exponent, there are still some disadvantages that should be overcome: firstly, the estimation methods are mathematics-oriented instead of engineering-oriented; secondly, the accuracy and effectiveness of the estimation algorithms are inadequately assessed; thirdly, the framework of classification for the estimation methods are insufficient; and lastly there is a lack of clear guidance for selecting proper estimation in practical problems involved in data analysis. The contributions of this paper lie in four aspects: 1) the optimal sequence partition method is proposed for designing the estimation algorithms for Hurst exponent; 2) the algorithmic pseudo-codes are adopted to describe the estimation algorithms, which improves the understandability and usability of the estimation methods and also reduces the difficulty of implementation with computer programming languages; 3) the performance assessment is carried for the typical estimation algorithms via the ideal time sequence with given Hurst exponent and the practical time sequence captured in applications; 4) the guidance for selecting proper algorithms for estimating the Hurst exponent is presented and discussed. It is expected that the systematic survey of available estimation algorithms could help the users to understand the principles and the assessment of the various estimation methods could help the users to select, implement and apply the estimation algorithms of interest in practical situations in an easy way.

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 Padé approximation via a four-step strategy. Firstly, the Padé-Cayley lemma is proved and used to simplify the symplectic Padé 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.

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.

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 1~10 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.

Efficient, robust and precise algorithms for linear quadratic regulator (LQR) and matrix differential Riccati equation (MDRE) are essential in optimal control. However, there are lack of good algorithms for time-varying LQR problem because of the difficulty of solving the nonlinear time-varying MDRE. In this paper, we proved that the $n$ -th order LQR problem is equivalent to $n$ parallel 1-dim Hamiltonian systems and proposed the explicit symplectic-precise iteration method (SPIM) for solving LQR and MDRE. The explicit symplectic-precise iteration algorithms (ESPIA) designed with SPIM have three typical merits: firstly, there are no accumulative errors in the sense of long-term time which inherits from symplectic difference scheme; secondly the stiffness problem due to the inverse of matrix is avoided by the precise iteration method; and finally the algorithmic structure of ESPIA is simple and no extra assumptions are required. Systematic analysis shows that the time complexity of the symplectic algorithms for the $n$ -th order LQR and MDRE is $\mathcal {O}(k_{max}n^{3})$ where $k_{max}$ is the iteration times specified by the time duration. Numerical examples and simulations are provided to validate the performance of the ESPIA.