Root Finding Techniques Research Articles

Curve linearization and discretization for meshing composite parametric surfaces

AbstractIn the context of composite parametric surface meshing using a direct or an indirect (via the meshing of parametric domains) method, the discretization of interface curves joining patches plays a central part. In the case of an indirect method, the discretization of interface curves must be taken back to parametric spaces. This is usually done using root finding techniques for generally non‐linear parametric functions. This paper presents a new linearization technique to determine the above correspondence. In addition, this linearization of interface curves makes it possible to obtain their discretization without explicitly computing the arc length. This method has been implemented in a surface mesh generator and an example from Fluid Mechanics is given to illustrate the pertinence of the proposed approach. Copyright © 2004 John Wiley & Sons, Ltd.

Automated Method to Compute Orbital Reentry Trajectories with Heating Constraints

An algorithm is presented that is designed to compute reentry trajectories for unpowered lifting reusable launch vehicles. The algorithm uses self-contained trajectory simulation and root finding techniques to determine the appropriate control sequences for solving reentry problems. For orbital reentries, the solution process breaks the trajectory into two distinct parts. The first part begins where the deorbited vehicle first encounters substantial atmosphere and can exercise trajectory control through the manipulation of aerodynamic lift. This phase of the flight is governed by an analytical, constant heat-rate following, bank angle control law. The second and final part of the trajectory begins where heat-rate control is no longer desired. During this time trajectory control is used to meet terminal range and altitude targets and is governed by a linear bank angle control law. The planning algorithm determines the value of the individual trajectory control parameters that shape the reentry. The planned trajectory is then used by a profile following guidance algorithm during actual flight. Test results for a variety of orbital and suborbital missions are shown.

A Simple Least-Squares Design of M-D IIR Filters with Fixed Separable Denominator Based on Multivariate Division Algorithm

In this paper, we propose a simple algorithmic solution to the best approximation problem of finding the nearest multivariate rational function, with a fixed separable denominator polynomial, from a given multivariate polynomial, where the numerator polynomial is desired to minimize the integral of the squared error over the distinguished boundary of the unit polydisc. The proposed algorithm does not require any numerical integration or numerical root finding technique because this is realized based on the standard multivariate division algorithm. A simple observation of the proposed algorithm leads to an ideal membership problem characterizing the solution to the problem. A relation of this characterization and a multivariate generalization of the Walsh's Theorem is also discussed with another ideal membership problem derived by applying a corollary of the Hilbert Nullstellensatz to the Walsh's Theorem. Although the discussion to derive the latter ideal membership problem seems to be roundabout, such a characterization would be useful for further generalization, for example to some weighted least-squares approximation. Numerical examples demonstrate the practical applicability of the proposed method to design problems of multidimensional IIR filters.

Algebraic multidimensional phase unwrapping and zero distribution of complex polynomials-characterization of multivariate stable polynomials

We define the multidimensional unwrapped phase for any finite extent multidimensional signal that may have its zero on the distinguished boundary of the unit polydisc. By using this definition, we deduce that multivariate stable polynomials can be simply characterized in terms of the proposed unwrapped phase. A rigorous symbolic algebraic solution to the exact phase unwrapping problem for multidimensional finite extent signals is also proposed. This solution is based on a newly developed general Sturm sequence and does not need any numerical root finding or numerical integration technique. Furthermore, it is shown that the proposed algebraic phase unwrapping algorithm can be used to determine the exact zero distribution of any univariate complex polynomial without suffering the so-called singular case problem.

Accurate Computation of Weights in Classical Gauss–Christoffel Quadrature Rules

For many classical Gauss–Christoffel quadrature rules there does not exist a method which guarantees a uniform level of accuracy for the Gaussian quadrature weights at all quadrature nodes unless the nodes are known exactly. More disturbing, some algebraic expressions for these weights exhibit an excessive sensitivity to even the smallest perturbations in the node location. This sensitivity rapidly increases with high order quadrature rules. Current uses of very high order quadratures are common with the advent of more powerful computers, and a loss of accuracy in the weights has become a problem and must be addressed. A simple but efficient and general method for improving the accuracy of the computation of the quadrature weights is proposed. It ensures a high level of accuracy for these weights even though the nodes may carry a significant large error. In addition, a highly efficient root-finding iterative technique with superlinear converging rates for computing the nodes is developed. It uses solely the quadrature polynomials and their first derivatives. A comparison of this method with the eigenvalue method of Golub and Welsh implemented in most standard software libraries is made. The proposed method outperforms the latter from the point of view of both accuracy and efficiency. The Legendre, Lobatto, Radau, Hermite, and Laguerre quadrature rules are examined.

Controllability and motion planning for noncatastatic nonholonomic control systems

A motion planning problem for a class of noncatastatic nonholonomic control systems is considered. Under appropriate controllability assumptions we determine an open loop control function which steers the system from a specified initial state to a specified final state over a specified time interval. The method is based on combining second order averaging technique and a finite-dimensional root finding technique.

A Fixed Point Method for Finding Percentage Points

SUMMARY Iterative methods for finding accurate estimates of percentage points are usually based on numerical root finding techniques applied to the distribution function. In this paper we take a different approach by creating an auxiliary function, whose fixed point is shown to be the desired percentage point, and then applying Steffenson's acceleration technique to find the fixed point. For the auxiliary function used, the conditions for convergence of Steffensen's method are mild and are satisfied by many percentage point problems arising in practice. The method is tested by calculating four-place x2 percentage points and two-place F percentage points which agree with those in published tables.

Magnetohydrodynamic flow between a solid rotating disk and a porous stationary disk

In this paper we examine the flow of a conducting fluid between a solid rotating disk and a stationary porous disk with uniform injection of fluid through the porous disk in the presence of an axial magnetic field. The equations of motion are solved using least change secant update quasi-Newton and modern root finding techniques. The fluid motion depends on the crossflow Reynolds number, rotational Reynolds number and Hartmann number. The effects of the parameters on the flow field are presented graphically.

Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices

A technique is developed for the design of analysis filters in an M-channel maximally decimated, perfect reconstruction, finite-impulse-response quadrature mirror filter (FIR QMF) bank that has a lossless polyphase-component matrix E(z). The aim is to optimize the parameters characterizing E(z) until the sum of the stopband energies of the analysis filters is minimized. There are four novel elements in the procedure reported here. The first is a technique for efficient initialization of one of the M analysis filters, as a spectral factor of an Mth band filter. The factorization itself is done in an efficient manner using the eigenfilters approach, without the need for root-finding techniques. The second element is the initialization of the internal parameters which characterize E(z), based on the above spectral factor. The third element is a modified characterization, mostly free from rotation angles, of the FIR E(z). The fourth is the incorporation of symmetry among the analysis filters, so as to minimize the number of unknown parameters being optimized. The resulting design procedure always gives better filter responses than earlier ones (for a given filter length) and converges much faster.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical Solution of the Eigenvalue Problem for Symmetric Rationally Generated Toeplitz matrices

A numerical method is proposed for finding all eigenvalues of symmetric Toeplitz matrices $T_n = ( t_{j - i} )_{ij = 1}^n $, where the $\{ t_j \}$ are the coefficients in a Laurent expansion of a rational function. Matrices of this kind occur, for example, as covariance matrices of ARMA processes. The technique rests on a representation of the characteristic polynomial as $\det ( \lambda I_n - T_n ) = W_n G_{0n} G_{1n} $ in which $G_{0n} ( \lambda ) = 0$ for the eigenvalues of $T_n $ associated with symmetric eigenvectors, $G_{1n} ( \lambda ) = 0$ for those associated with skew-symmetric eigenvectors, both functions are free of extreme variations, and both can be computed with cost independent of n. It is proposed that root finding techniques be used to compute the zeros of $G_{0n} $ and $G_{1n} $. Numerical experiments indicate that the method may be useful.

A Fully Parallel Algorithm for the Symmetric Eigenvalue Problem

In this paper we present a parallel algorithm for the symmetric algebraic eigenvalue problem. The algorithm is based upon a divide and conquer scheme suggested by Cuppen for computing the eigensystem of a symmetric tridiagonal matrix. We extend this idea to obtain a parallel algorithm that retains a number of active parallel processes that is greater than or equal to the initial number throughout the course of the computation. We give a new deflation technique which together with a robust root finding technique will assure computation of an eigensystem to full accuracy in the residuals and in the orthogonality of eigenvectors. A brief analysis of the numerical properties and sensitivity to round off error is presented to indicate where numerical difficulties may occur. The algorithm is able to exploit parallelism at all levels of the computation and is well suited to a variety of architectures.Computational results are presented for several machines. These results are very encouraging with respect to both...

Application of a New Complex Root-Finding Technique to the Dispersion Relations for Elastic Waves in a Fluid-Loaded Plate

Straight-crested waves propagating freely in a fluid-loaded plate undergo attenuation as a consequence of radiation of sound into the fluid. If the two sides of the plate are loaded by different fluids, both antisymmetric and symmetric waves are present. To find the complex roots of the dispersion relations, a new numerical technique was developed. The basic algorithm indicates the location of a root relative to a line situated close to the root (determined by the real wave speed for an unloaded plate). By successively decreasing the spacing of two mutually perpendicular pairs of lines, the root is enclosed within a square of arbitrarily small size. This routine is applied to the above problem for exact and thick-plate theories. Exact theory predicts a noticeable effect on attenuation, if both wave types are included, for wavelengths small compared with the plate thickness. Thick-plate theory gives results close to exact theory for longer wavelengths but fails to reproduce the effect of including both wave types.

Characterization of equilibrium sets for bilinear systems with feedback control

Simple analytical methods for the characterization of the equilibrium sets of bilinear systems with linear feedback control are developed based on elimination theory. It is shown that the number of equilibrium points can be determined as a function of the feedback gains, without having to solve for the locations of the equilibrium points. Necessary and sufficient conditions for the origin to be the unique equilibrium point are also presented. These results provide the basis for computing the equilibrium points via a straightforward procedure, which eliminates the convergence, stability and uncertainty problems that can occur when iterative root finding techniques are used. Several illustrative examples are included. The methods developed in this paper can also be used to characterize the equilibrium sets of other classes of nonlinear systems such as quadratic systems.

Determination of Eigenvalues of Matrices Having Polynomial Elements

1. Summary. A method is described for obtaining the eigenvalues of matrices whose elements are polynomials in the eigenvalue parameter. The method is based on an iterative root-finding technique coupled with the use of estimates. The process converges with virtually arbitrary estimates while significant reduction of computing time is achieved with the use of good estimates. Only those eigenvalues of interest need be obtained if sufficiently good estimates are available. Accuracy is maintained through direct use of the original data. A subroutine employing this procedure has been applied to a large number of problems, the most elaborate involving the determination of the 32 eigenvalues of an 18th order matrix with quadratic elements.