Rational Fitting Research Articles

Lambert W function for applications in physics

The Lambert W(x) function and its possible applications in physics are presented. The actual numerical implementation in C++ consists of Halley’s and Fritsch’s iterations with initial approximations based on branch-point expansion, asymptotic series, rational fits, and continued-logarithm recursion. Program summaryProgram title: LambertWCatalogue identifier: AENC_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENC_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License version 3No. of lines in distributed program, including test data, etc.: 1335No. of bytes in distributed program, including test data, etc.: 25283Distribution format: tar.gzProgramming language: C++ (with suitable wrappers it can be called from C, Fortran etc.), the supplied command-line utility is suitable for other scripting languages like sh, csh, awk, perl etc.Computer: All systems with a C++ compiler.Operating system: All Unix flavors, Windows. It might work with others.RAM: Small memory footprint, less than 1 MBClassification: 1.1, 4.7, 11.3, 11.9.Nature of problem:Find fast and accurate numerical implementation for the Lambert W function.Solution method:Halley’s and Fritsch’s iterations with initial approximations based on branch-point expansion, asymptotic series, rational fits, and continued logarithm recursion.Additional comments:Distribution file contains the command-line utility lambert-w. Doxygen comments, included in the source files. Makefile.Running time:The tests provided take only a few seconds to run.

Fitting Algorithm of Parameterized Continued Fractions with Keeping Endpoints and its Application

In order to overcome fitting uncertainty, A nonlinear rational parameterized fitting algorithm with keeping endpoints was presented through inverse difference-continued fraction algorithm, the algorithm has property of keeping endpoints, parameterization of middle data points selected. The parameters obtained through minimum error function make fitting functions have much flexibility. Experimental results illustrate that the proposed algorithm may reduce degree of continued fraction interpolation function, numbers of iteration and avoiding non-existence on interpolation function of the continued fraction.The fitting values obtained are more accuracy, the computing efficiency is much more improved,the fitting errors are more stable, approximate effect is better, and the algorithm will be widely used in predict etc.

Phase-Domain Transmission-Line Modeling With Enforcement of Symmetry Via The Propagated Characteristic Admittance Matrix

Transmission-line models for electromagnetic transient simulation are usually based on the traveling-wave method. While recent developments have focused on phase-domain formulations with an emphasis on accuracy and passivity enforcement, no consideration has been given to ensure the symmetry of the model. In this paper, we introduce the symmetry preserving property (SPP) of the propagation matrix H as a way of enforcing a symmetrical model. This property is enforced by replacing the (unsymmetrical) propagation matrix H with the (symmetrical) propagated characteristic admittance matrix during the process of rational fitting. The propagation matrix H is recovered analytically for use in the final line model. We show a procedure for enforcement of SPP and the resulting model is implemented in an Electromagnetic Transients Program-like simulation environment. Although the computational efficiency decreases somewhat with the new approach, the symmetry of the model offers significant advantages. Besides being physically correct, it avoids confusion when performing sanity checking of complex network models.

Direct Rational Function Fitting Method for Accurate Evaluation of Sommerfeld Integrals in Stratified Media

The current work proposes a direct rational function fitting method, employing cylindrical waves alone, for the accurate evaluation of Sommerfeld integrals for planar multilayered structures. Three are the key points of the effort. 1) Until now, relative works require the extraction of the quasi-static/asymptotic terms, and branch cut/continuous wave contribution explicitly. In the current one, the explicit treatment of those terms is avoided. The proposed methodology is based on the direct fitting of the spectrum of the Green's function by rational functions. Thus, it provides the spatial Green's function solely in terms of cylindrical waves. 2) The effectiveness, robustness and accuracy improvement of the rational function fit rely upon the proper sampling of the spectrum. This accurate fitting is possible because instead of avoiding large variations of the spectral kernel, we introduce proper paths to include more variation, and thus more spectrum information, before we apply the modified VECTFIT algorithm. 3) Furthermore, proper weighting of the VECTFIT is proposed in order to guide the algorithm in providing increased accuracy in specific desired areas of the horizontal distance between the source and observation points. Armed with the above the direct rational function fitting method provides accurate results both for the near and far-field regions. Various examples, among them the correct treatment of a two branch case, are given that prove the excellent performance and robustness of the proposed approach.

A Parametric Approach to List Decoding of Reed-Solomon Codes Using Interpolation

In this paper, we present a minimal list decoding algorithm for Reed-Solomon (RS) codes. Minimal list decoding for a code C refers to list decoding with radius L, where L is the minimum of the distances between the received word r and any codeword in C. We consider the problem of determining the value of L as well as determining all the codewords at distance L. Our approach involves a parametrization of interpolating polynomials of a minimal Gröbner basis G . We present two efficient ways to compute G. We also show that so-called re-encoding can be used to further reduce the complexity. We then demonstrate how our parametric approach can be solved by a computationally feasible rational curve fitting solution from a recent paper by Wu. Besides, we present an algorithm to compute the minimum multiplicity as well as the optimal values of the parameters associated with this multiplicity, which results in overall savings in both memory and computation.

Model order reduction via eigen decomposition analysis

In this paper, we have proposed a novel model order reduction technique via rational transfer function fitting and eigenmode analysis considering residues. We define a constant as a key in the sorting algorithm as one of correlations in order to sort the order of eigenvalues. It is demonstrated that the accuracy via eigenmode analysis considering residues is improved. The proposed algorithm is a general method to match pole values with frequency domain poles for linear RC and RLC systems. Calculation of pole eigenvalues and eigen vectors can be done with more sophisticated analysis with the same level or smaller cost in the proposed algorithms in comparison to passive reduced order interconnect macromodeling algorithm (PRIMA). The experimental results show that our algorithm reduces up to 90% errors compared to the existing model order reduction algorithm, such as PRIMA, in wide frequency environment with the same number of poles in comparison.

Microwave small-signal modelling of FinFETs using multi-parameter rational fitting method

An effective approach based on a multi-parameter rational fitting technique is proposed to model the microwave small-signal response of active solid-state devices. The model is identified by fitting multi-bias scattering-parameter measurements and its analytical expression is implemented in a commercial microwave circuit simulator. The approach has been applied to the modelling of a silicon-based FinFET, and an excellent agreement is obtained between the measured data and model predictions.

On Passivity Characterization of Symmetric Rational Macromodels

This paper provides a theoretical framework for passivity characterization of symmetric rational macromodels. These may be obtained for linear and time-invariant reciprocal structures if structural symmetry is preserved during the rational fitting stage of the macromodel generation. Standard Hamiltonian-based methods can be used to characterize any passivity violation of such macromodels. Recent developments have suggested, however, that the same results may be obtained at a reduced computational cost, using so-called ¿half-size¿ passivity test matrices. In this paper, we generalize such results by providing a complete theoretical framework. In addition to imaginary Hamiltonian eigenvalues, we present a complete characterization of associated eigenvectors, allowing for precise localization of passivity violations. Since half-size matrices are also used for computing the eigenvectors, the overall computational cost is reduced up to a factor of eight. Several numerical examples validate and confirm the theoretical developments.

Rational Fitting of S-Parameter Frequency Samples With Maximum Absolute Error Control

Rational fitting techniques are often used for the macromodeling of linear systems from tabulated S-parameter frequency samples. This letter proposes a modified weighting scheme for the Vector Fitting algorithm that iteratively minimizes the maximum absolute error over the frequency range of interest, rather than the least-squares error. By considering the appropriate error measure in the fitting process, it is possible to obtain more accurate broadband macromodels without increasing the number of poles. The effectiveness of the approach is illustrated by several numerical examples.

Confinement effect of organic nanotubes toward green fluorescent protein (GFP) depending on the inner diameter size.

Transportation, release behavior, and stability of a green fluorescent protein (GFP, 3x4 nm) in self-assembled organic nanotubes with three different inner diameters (10, 20, and 80 nm) have been studied in terms of novel nanocontainers. Selective immobilization of a fluorescent acceptor dye on the inner surface enabled us to not only visualize the transportation of GFP in the nanochannels but to also detect release of the encapsulated GFP to the bulk solution in real time, based on fluorescence resonance energy transfer (FRET). Obtained diffusion constants and release rates of GFP markedly decreased as the inner diameter of the nanotubes was decreased. An endo-sensing procedure also clarified the dependence of the thermal and chemical stabilities of the GFP on the inner diameters. The GFP encapsulated in the 10 nm nanochannel showed strong resistance to heat and to a denaturant. On the other hand, the 20 nm nanochannel accelerated the denaturation of the encapsulated GFP compared with the rate of denaturation of the free GFP in bulk and the encapsulated GFP in the 80 nm nanochannels. The confinement effect based on rational fitting of the inner diameter to the size of GFP allowed us to store it stably and without denaturation under high temperatures and high denaturant concentrations.

A model equation for ultrashort optical pulses around the zero dispersion frequency

The nonlinear Schrodinger equation (NSE) based on the Taylor approximation of the material dispersion can become invalid for ultrashort and few-cycle optical pulses. Instead, we use a rational fit to the dispersion function around the zero dispersion frequency where the transition between anomalous and normal dispersion regimes occurs. This approach allows us to derive a simple non-envelope model for pulses propagating in time within a transparency window of a nonlinear dispersive medium with an instantaneous cubic nonlinearity. For this model we investigate integrals of motion and demonstrate that a uniformly moving non-envelope soliton does not exist. The only possible localized solution is the solitary breather with some intrinsic dynamics in the comoving frame. Classical envelope solitons oscillating in the comoving frame appear for a longer pulse for which the model is equivalent to the standard NSE. For an ultrashort pulse the model provides a natural bridge between the known non-envelope equations for the purely normal and anomalous dispersion regimes.

Admittance-Based Modeling of Transmission Lines by a Folded Line Equivalent

This paper describes a new transmission-line model for frequency-dependent modeling of untransposed overhead lines and underground cables. The nodal admittance matrix is decomposed into two blocks that, respectively, represent the open- and short-circuit conditions of a half-length line obtained by ldquofoldingrdquo about the middle. By subjecting these matrices to rational fitting with inverse magnitude weighting, one obtains a model where the eigenvalues of the associated nodal admittance matrix are effectively fitted with high relative accuracy. This is shown to overcome the error magnification problem that occurs with direct fitting of the nodal admittance matrix. In addition, the modeling process (fitting and passivity enforcement) becomes faster. We show that this folded line equivalent (FLE) is particularly suitable as a companion form for phase-domain traveling-wave-type models, to be used when the time step is selected shorter than the line travel time. In this situation, the required model order is low and so the FLE gives highly efficient time-domain simulations.

Rotational kinematic hardening model for three‐dimensional stress reversals in sand

AbstractA kinematic hardening mechanism has previously been proposed to capture the behavior of soil during large stress reversals in the triaxial plane. This mechanism is now extended to the principal stress space. It incorporates rotation and intersection of yield surfaces to achieve a consistent and physically rational fit with experimentally observed soil behavior during large three‐dimensional stress reversals. An existing elasto‐plastic model with isotropic hardening is used as the basic framework to which the rotational kinematic hardening mechanism has been added. The new combined model preserves the behavior of the isotropic hardening model under monotonic loading conditions, and the extension from isotropic to rotational kinematic hardening under three‐dimensional conditions is accomplished without introducing new material parameters. The framework of the model is presented here with some comparisons between theoretical and experimental directions of strain increment vectors to indicate the potential of the model. Copyright © 2008 John Wiley & Sons, Ltd.

New List Decoding Algorithms for Reed–Solomon and BCH Codes

In this paper, we devise a rational curve fitting algorithm and apply it to the list decoding of Reed-Solomon and Bose-Chaudhuri-Hocquenghen (BCH) codes. The resulting list decoding algorithms exhibit the following significant properties.

On the efficient computation of closed-form Green's functions in planar stratified media

The spatial-domain Green's functions for the vector and scalar potentials in planar stratified media are cast into closed forms via a two-level approximation of their spectral-domain counterparts. The proposed methodology begins with the approximation of the spectral-domain Green's functions over large values of the spectral variable by complex exponentials, and continues with the approximation of the remainder by rational functions. Finally, the closed-form Green's functions in terms of spherical and cylindrical waves are derived, making use of some well-known integral identities. A key-feature of the proposed approach is that although it does not call for an analytical extraction of the quasistatic terms and the surface wave poles, it provides the means for the accurate description of both the near-field and far-field physics. Moreover, the rational function spectrum fitting proposed here overcomes the problem of the spurious singular behavior of the spatial-domain Green's functions because of the use of Hankel functions. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

Passive rational fitting of a network transfer function from its real part

A methodology is presented for the rational function approximation of a passive network function from sampled values of its real part over the bandwidth of interest. The accuracy and validity of the proposed methodology are demonstrated through its application to the fitting of several broadband, multiport transfer functions. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

Accurate Approximation of Green's Functions in Planar Stratified Media in Terms of a Finite Sum of Spherical and Cylindrical Waves

A robust and computationally-expedient methodology is presented for accurate, closed-form approximation of the Green's functions used in the mixed-potential integral equation statement of the electromagnetic boundary value problem in planar stratified media. The proposed methodology is based on the fitting of the spectrum of the Green's function, after the extraction of the quasistatic part, making use of rational functions. The effectiveness and robustness of the proposed methodology rely upon the proper sampling of the spectrum in order to improve the accuracy of the rational function fit. The resulting closed-form approximation is in terms of both spherical and cylindrical waves. Thus, high accuracy is obtained in the approximation of the Green's function irrespective of the distance of the observation point from the source. The methodology is validated through its application to the approximation of the Green's function for a multi-layered, planar dielectric stack

Reconstruction of the optical potential from scattering data

We propose a method for reconstructing the optical potential (OP) from scattering data. The algorithm is a two-step, procedure. In the first step, the real part of the potential is determined analytically via solution of the Marchenko equation. At this point, we use a rational function fit of the corresponding unitary S matrix. In the second step, the imaginary part of the potential is determined via the phase equation of the variable phase approach. We assume that the real and imaginary parts of the OP are proportional (the Lax-type interaction). We use the phase equation to calculate the proportionality coefficient. A numerical algorithm is developed for a single and for coupled partial waves. The developed procedure is applied to analyze the ${}^{1}{S}_{0}\mathit{NN},{}^{3}{\mathit{SD}}_{1}\mathit{NN}$, and $P31\phantom{\rule{0.3em}{0ex}}{\ensuremath{\pi}}^{\ensuremath{-}}N$ data. For the NN states, we constructed partial potentials with forbidden states (Moscow potential). We examine the ${\ensuremath{\pi}}^{\ensuremath{-}}N$ and NN partial-wave analysis data and demonstrate that the Lax-type interaction may be valid for these systems at the considered energies.

Identification of a Multiphase Network Equivalent for Electromagnetic Transient Calculations Using Partitioned Frequency Response

This paper presents an algorithm for identifying a multiphase network equivalent for electromagnetic (EM) transient calculations. Conventional rational fitting methods intrinsically get into ill condition because the s/sup n/ terms in the rational function take a wide range of values which cannot be accurately treated within machine arithmetic accuracy. The proposed algorithm averts the ill conditioning by partitioning the given frequency response into sections along the frequency axis. Rational fitting is applied to each section of the frequency response to identify the poles, and then the corresponding residue matrices are obtained by a standard least-squares procedure using the entire frequency response. Adaptive weighting, column scaling, and iteration step adjustment are utilized to facilitate the rational fitting. The proposed identification algorithm is applied to obtain a reduced-order equivalent of a 500-kV power network, and the equivalent is used in a switching transient simulation to demonstrate its performance.

Evaluation of layered media Green's functions via rational function fitting

An efficient rational function fitting methodology, called VECTFIT, is utilized toward the closed-form evaluation of the Sommerfeld integrals associated with electromagnetic Green's functions in planar layered media. VECTFIT approximates the component of the spectrum of the Green's function that remains after the extraction of the primary source contribution and the quasistatic part with a rational function, thus enabling a robust and expedient closed-form evaluation of the Sommerfeld integral for electromagnetic potentials and associated field quantities.