Small Number Of Arithmetic Operations Research Articles

АЛГОРИТМ РЕШЕНИЯ ПРИВЕДЕННОГО ПОЛИНОМИАЛЬНОГО УРАВНЕНИЯ С ПОМОЩЬЮ НЕПРЕРЫВНЫХ ДРОБЕЙ

The article presents an algorithm where continued fractions are used to find the zeros of nthdegree polynomial. Our day there is a wide variety of methods and algorithms for solving suchtype problems. The proposed algorithm feature is its effective possibility using for large values ofn. Besides this algorithm can be applied for complex roots. Any real number can be represented asa continued fraction: finite or infinite. The main purpose of continuous fractions is that they allowus to find good approximations of real numbers in the ordinary fractions form in algebraic equationsand systems solution. The purpose of this work is algorithm with continued fractions for solvingreduced polynomial equation that contains both real and complex roots, estimation of thenumber of arithmetic steps in its numerical solution. The analytical expressions for polynomialequation solutions are given in this paper. The obtained analytical expressions represent the ratioof the Toeplitz determinants. A distinctive feature of these determinants is the presence of coefficientsof the solved algebraic equation as diagonal elements. A modified Rutishauser algorithmwas used to obtain a numerical solution. Complex roots can be found using the summation algorithmof divergent continued fractions. As an illustration of the proposed algorithm the results ofthe numerical solution for fifth degree polynomial equation are given. The advantage of the algorithmis the small number of arithmetic operations required, the possibility of considering highdegreepolynomials, and the small error of calculations.

Cabaret Difference Scheme with Improved Dispersion Properties

A difference scheme for the transfer problem, constructed as a linear combination of the Cabaret scheme and the central difference scheme, is proposed. The stability and dispersion properties of the scheme are studied. It is shown that the constructed scheme has the best dispersion properties for high-frequency harmonics at small Courant numbers compared with the known Cabaret scheme for the transport equation. A comparison is made of the errors of this scheme and the two-parameter third-order difference accuracy scheme based on the numerical experiments on the previously used test problem sets. It is shown that in the normal grid space L1 the developed scheme has smaller errors and also uses a more compact template (in calculating the inode the node values i – 1, i, and i + 1 are used), and the transition to the next time layer is carried out for a smaller number of arithmetic operations.

Digital Binary Phase-shift Keyed Signal Detector

We have developed the effective algorithm for detecting digital binary phase-shift keyed signals. This algorithm requires a small number of arithmetic operations over the signal period. It can be relatively easy implemented based on the modern programmable logic devices. It also provides high interference immunity by identifying signal presence when signal-to-noise ratio is much less that its working value in the receiving path. The introduced detector has intrinsic frequency selectivity and allows us to form the estimate of the noise level to realize the adaptive determination of decision threshold. In order to get confirmation of the detector operability and performance, we suggest the expressions for false alarm and missing probabilities. In addition, we have examine, both theoretically and experimentally, the influence of the detector parameters on its characteristics.

The unexpected beauty of modular bivariate quadratic functions

abstractModular bivariate quadratic functions (quadratic functions of two variables over the integers modulo q) can be used to procedurally generate grey-scale and colour textures that resemble ornamentation, skin, scales, feathers, textiles, and optical illusions. The latter appear to include distorted perspective, false 3D, and Fechner colours, the combination of which is disturbing to some viewers. This technique is particularly suited to parallel execution using a pixel shader since each pixel is computed independently by performing a small number of arithmetic operations on the pixel coordinates. A prototype browser-based procedural texture generator with an interface suitable for use by non-mathematicians such as designers and artists is described.

Rapid identification of pre-buckling states: A case of cylindrical shell

The problem to identify pre-buckling states for thin-walled shell corresponds to the problem to identify pre-bifurcation solutions (the inverse bifurcation problem) for von Karman equations that govern the structure. Typical solution sequences similar to those of post-bifurcation solutions observed along the bifurcation paths of the nonlinear boundary problem for von Karman equations are extracted to serve as precursors of bifurcation (tools to solve the problem). The method allows one to divide all operations required to solve the problem under study into two non-equal parts. The most time-consuming part (to trace bifurcation paths and cluster the respective solution) is performed off-line, while the part of the algorithm that is carried out on-line (the identification algorithm) requires a relatively small number of arithmetic operations. This allows development of the efficient system of rapid identification of pre-buckling states.

Poisson Model To Generate Isotope Distribution for Biomolecules.

We introduce a simplified computational algorithm for computing isotope distributions (relative abundances and masses) of biomolecules. The algorithm is based on Poisson approximation to binomial and multinomial distributions. It leads to a small number of arithmetic operations to compute isotope distributions of molecules. The approach uses three embedded loops to compute the isotope distributions, as compared with the eight embedded loops in exact calculations. The speed improvement is about 3-fold compared to the fast Fourier transformation-based isotope calculations, often termed as ultrafast isotope calculation. The approach naturally incorporates the determination of the masses of each molecular isotopomer. It is applicable to high mass accuracy and resolution mass spectrometry data. The application to tryptic peptides in a UniProt protein database revealed that the mass accuracy of the computed isotopomers is better than 1 ppm. Even better mass accuracy (below 1 ppm) is achievable when the method is paired with the exact calculations, which we term a hybrid approach. The algorithms have been implemented in a freely available C/C++ code.

Radix-2 Fast Algorithm for Computing Discrete Hartley Transform of Type III

In this brief, a new efficient radix-2 fast algorithm for the computation of type-III discrete Hartley transform of length that has a small arithmetic cost and is well suited for a very large-scale integration (VLSI) implementation is presented. This recursive method requires a small number of arithmetic operations compared with existing methods, has a regular and simple computational structure, and can be easily implemented. A small number of arithmetic operations are achieved with the proposed algorithm as compared with existing algorithms. Moreover, the regular and simple computational structure and the existing parallelism of the proposed algorithm offer the possibility of an efficient fast parallel VLSI implementation.

Explicit solutions to Poncelet’s porism

Solution to the following problem is considered: for given conics C and K and an integer N⩾3, determine whether there exists a closed N-sided polygon inscribed in C and circumscribed about K. The case of C and K being circles is considered in detail. Equations are proposed with a relatively small number of arithmetic operations – near log2N. Along the way, the following result is obtained: for circles with rational coefficients, the polygons can only have the following number of sides N=3, 4, 5, 6, 8, 10 and 12 (a subset of the Mazur’s set of integers for rational elliptic curves). The proposed solution may also be applied to determine whether a Hankel determinant of order N/2 having special form (used in the classical Cayley criterion) is equal to zero, and for related problems. Possible generalizations for ellipses and hyperbolas are also presented. Particularly, equations are proposed for parameters of concentric Poncelet’s ellipses (billiard case).

Simple non-iterative calibration for triaxial accelerometers

For high precision measurements, accelerometers need recalibration between different measurement occasions. In this paper, we derive a simple calibration method for triaxial accelerometers with orthogonal axes. Just like previously proposed iterative methods, we compute the calibration parameters (biases and gains) from measurements of the Earth's gravity for six different unknown orientations of the accelerometer. However, our method is non-iterative, so there are no complicated convergence issues depending on input parameters, round-off errors, etc. The main advantages of our method are that from just the accelerometer output voltages, it gives a complete knowledge of whether it is possible, with any method, to recover the accelerometer biases and gains from the output voltages, and when this is possible, we have a simple explicit formula for computing them with a smaller number of arithmetic operations than in previous iterative approaches. Moreover, we show that such successful recovery is guaranteed if the six calibration measurements deviate with angles smaller than some upper bound from a natural setup with two horizontal axes. We provide an estimate from below of this upper bound that, for instance, allows 5° deviations in arbitrary directions for the Colibrys SF3000L accelerometers in our lab. Similar robustness is also confirmed for even larger angles in Monte Carlo simulations of both our basic method and two different least-squares error extensions of it for more than six measurements. These simulations compare the sensitivities to noise and cross-axis interference. For instance, for 0.5% cross-axis interference, the basic method with six measurements, each with two horizontal axes, gave higher accuracy than allowing 10° deviation from horizontality and compensating with more measurements and least-squares fitting.

Fast recursive algorithms for 2-D discrete cosine transform

A new algorithm for computation of two-dimensional (2-D) type-III discrete cosine transform is presented. The algorithm is particularly suited to block size ( p 1∗2 m ) by ( p 2∗2 n ), where p 1 and p 2 are odd integers, and m and n are non-negative integers. It shows that the 2-D type-III DCT can be decomposed into cosine–cosine, cosine–sine, sine–cosine, sine–sine sequences, which can be further decomposed into similar sequences. The proposed algorithm provides the flexibility in choosing block size and has a simple indexing mapping scheme and a fairly regular computation structure. The algorithm also requires a smaller number of arithmetic operations for p 1= p 2=3.

Exact solutions for the problem of source location from measured time differences of arrival

Source location using the difference in time of arrival of a signal (TDOA) at an array of receivers is used in many fields including speaker location. Using three TDOA values from four noncollinear receivers one can, in principle, solve for the unknown source coordinates in terms of the receiver locations. However, the equations are nonlinear, and in practice signals are contaminated by noise. Practical systems often use multiple receivers for accuracy and robustness, and improved S/N, and solutions must be obtained via minimization. A new family of exact solutions, for the case of four receivers located in a plane, is presented. These solutions can be evaluated using a small number of arithmetic operations. The performance of these solutions for several practical situations is examined via simulation, and possible geometries of receiver locations suggested. For the case where multiple receivers (>4) are used, a new formulation is presented, that incorporates the present solutions, imposes additional constraints on source location, and enables use of a constrained L1 optimization procedure to achieve a robust estimate of the source location. The present estimator is compared with ones from the literature, and found to be robust and accurate, and more efficient. [Work partially supported by DARPA.]

Application of Miller's theorem to the stability analysis of numerical schemes; some useful tools for rapid inspection of discretisations in ocean modelling

An old theorem about the roots of a polynomial is resurfaced and examined, focusing on the practical question of determining stability criteria for numerical schemes. Is is demonstrated that the theorem can be useful both for analytical studies of the stability limits, as well as numerical searches for stability regions. It is particularly important that when deciding whether the scheme is stable or not, it is not necessary to search for the roots of the polynomial. Indeed, such a decision may be reached through a finite and small number of arithmetic operations and verifications of inequalities. In addition to the analysis of the theorem, some practical conditions for polynomial of different orders are presented, as well as some useful tips on how simply necessary or sufficient stability conditions can be obtained.

An Interior Random Vector Algorithm for MultiStage Stochastic Linear Programs

In this paper, an interior point algorithm for linear programs is adapted for solving multistage stochastic linear programs. The algorithm is based on Monteiro and Adler's path-following algorithm for deterministic linear programs. In practice, the complexity of the algorithm is linear with respect to the size of the sample space. The algorithm starts from a feasible solution of the problem and proceeds along a path of random vectors. The cubic polynomial complexity of the algorithm for deterministic linear programs is derived from the calculations of the Newton steps. In the algorithm developed in this paper, the probabilistic structure of the problem is taken into consideration while calculating a Newton step and the size of the sample space appears linearly in the complexity. The development of an algorithm that requires a relatively small number of arithmetic operations, in terms of the sample space size, allows the use of the algorithm for multistage stochastic linear programs with a very large number of scenarios.

An Algorithm for a Generalization of the Richardson Extrapolation Process

In this paper we present a recursive method, designated the $W^{(m)} $ algorithm, for implementing a generalization of the Richardson extrapolation process that has been ntroduced in [8]. Compared to the direct solution of the linear systems of equations defining the extrapolation procedure, this method requires a small number of arithmetic operations and very little storage. The technique is also applied to solve recursively the coefficient problem associated with the rational approximations obtained by applying the d-transformation of [6], [13] to power series. In the course of development a new recursive algorithm for implementing a very general extrapolation procedure is introduced, which is similar to that given in [2], [4] for solving the same problem. A FORTRAN program for the $W^{(m)} $algorithm is also appended.

Computationally efficient narrowband linear-phase FIR filters

A new approach is presented for synthesising and implementing narrowband linear-phase nonrecursive digital filters requiring a small number of arithmetic operations. The approach is based on transformations implemented by replacing subnetworks with transfer function (1+ Z−2)/2 in a prototype network by a subnetwork whose transfer function is a polynomial in (1+ Z−2)/2 but can be expressed in a form containing no multipliers. By appropriately designing the prototype network and selecting the sub-network, the resulting filter implementations require significantly less multiplications and additions per sample than conventional nonrecursive designs, at the expense of an increased filter order. Examples show that even a reduction from 1017 to 17 multipliers is possible. The new filters present also considerable advantages over nonrecursive filters composed of cascaded decimators and interpolators.