An Analytic Expression for the Inverse Involute

This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.

Optimal sampling rates for approximating analytic functions from pointwise samples

We consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in M, and that any method with faster exponential convergence must also be exponentially ill conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith & Rivlin concerning the maximal behavior of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results—based on computing the maximal polynomial via a variant of the classical Remez algorithm—confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree N < M is possible only once M is sufficiently large for there to be a subset of N of the nodes that mimic the behavior of the $N$th set of Chebyshev nodes.

Optimal staggered-grid finite-difference schemes based on the minimax approximation method with the Remez algorithm

Finite-difference (FD) schemes, especially staggered-grid FD (SFD) schemes, have been widely implemented for wave extrapolation in numerical modeling, whereas the conventional approach to compute the SFD coefficients is based on the Taylor-series expansion (TE) method, which leads to unignorable great errors at large wavenumbers in the solution of wave equations. We have developed new optimal explicit SFD (ESFD) and implicit SFD (ISFD) schemes based on the minimax approximation (MA) method with a Remez algorithm to enhance the numerical modeling accuracy. Starting from the wavenumber dispersion relations, we derived the optimal ESFD and ISFD coefficients by using the MA method to construct the objective functions, and solve the objective functions with the Remez algorithm. We adopt the MA-based ESFD and ISFD coefficients to solve the spatial derivatives of the elastic-wave equations and perform numerical modeling. Numerical analyses indicated that the MA-based ESFD and ISFD schemes can overcome the disadvantages of conventional methods by improving the numerical accuracy at large wavenumbers. Numerical modeling examples determined that under the same discretizations, the MA-based ESFD and ISFD schemes lead to greater accuracy compared with the corresponding conventional ESFD or ISFD scheme, whereas under the same numerical precision, the shorter operator length can be adopted for the MA-based ESFD and ISFD schemes, so that the computation time is further decreased.

Design of IIR Filter using Remez Algorithm

In this paper, we present a numerical method for the equiripple approximation of Impulse Infinite Response digital filters. The proposed method is based on the formulation of a generalized eigenvalue problem by using Rational Remez Exchange algorithm. In this paper, conventional Remez algorithm is modified to get the ratio of weights in the different bands exactly. In Rational Remez, squared magnitude response of the IIR filter is approximated in the Chebyshev sense by solving for an eigenvalue problem, in which real maximum eigenvalue is chosen and corresponding to that eigenvectors are found, and from that optimal filter coefficients are obtained through few iterations with controlling the ratio of ripples. The design algorithm is computationally efficient because it not only retains the speed inherent in the Remez exchange algorithm but also simplifies the interpolation step.

Research of Chebyshev Equivalent Ripple Approximation Filter Based on DSP

Researched Chebyshev equiripple approximation FIR filter theory. In the algorithm process, we adopt direct remez algorithm, by a recursive formula to calculate the filter coefficients directly. Combined TMS320VC5502, we designed software and hardware system of Chebyshev equiripple band-pass filter to realize the voice signal filtering. In hardware system TLC320AC01 complete A/D and D/A converter, TMS320VC5502 complete data processing and filtering.