Set Of Simultaneous Linear Equations Research Articles

Successive Identification of Volterra Type Nonlinear Multi-Input-Output System and Its Application to Parameter Estimation

A simpler practical identification method of nonlinear system represented by discrete Volterra functional series is proposed. Discrete Volterra kernels, divided into group corresponding to each sampling time, are determined from a set of simultaneous linear equations. The sets of simultaneous linear equations are solved one by one, applying the solution of each step to solve the next set of equations having a larger number of unknowns. Thus, each identification procedure reduces the size of matrix to be dealt with and assures an accurate system identification with the combination of least squares estimation method in the presence of noise on output signals. Furthermore, as its application, a parameter estimation method using a part of the solution of linear equations is proposed.

General spectral transformations for digital filters

Constantinides [1] gave a method of transforming a prototype low-pass digital filter into a high-pass, bandpass, or bandstop filter by conformally mapping the complex variable Z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> . We generalize here his method to transformations from prototype low-pass filters to filters containing any number of pass- and stopbands. The computation procedure involves only the solution of a set of simultaneous linear equations, the number of which is related only to the number of band-edges and not to the order of the prototype filter.

Integral Equation Formulation for Ground-Water Flow

Advantages of boundary integral equation methods (BIEM) for the analysis of two-dimensional flows through porous media are numerous. The data required and the number of unknowns are significantly fewer than for other general, numerical methods such as finite differences of finite element methods. A BIEM based on the Cauchy integral equation is developed. This formulation has the advantage that the equation used for any unknown is a Fredholm equation of the second kind which results in well-conditioned equations and allows nodes to be closely spaced in the region of singularities. The method is extended to the analysis of multizone, anisotropic flows. Computational algorithms for solution of the equations by a simple iterative method and by direct solution of the full set of simultaneous linear equations are developed.

Determining a set of B-spline control vertices to generate an interpolating surface

It is desirable to be able to use the B-spline surface formulation to describe a surface initially defined by a completely specified network of points to be interpolated. An appropriate set of B-spline control vertices which will generate an interpolating surface can be determined by solving a set of simultaneous linear equations. These equations are derived, efficient algorithms are designed to solve them rapidly, and the implementation of this theory in a computer-aided geometric design system is presented.

Transient heat conduction problems in inhomogeneous media discretized by means of boundary-volume element

An integral equation formulation is presented for the transient heat conduction problems in inhomogeneous media. The material constants are assumed to be prescribed as arbitrary, continuous and differentiable functions of position vector. The governing integral equations are derived from the weighted residual statement of the problems in which the fundamental solution to the corresponding heat conduction problems in homogeneous media is used as the weight function. The whole domain of interest is discretized into a series of boundary-volume-time elements, and then a set of linear simultaneous equations are obtained. Their solutions yield the temperature in the whole domain as well as the heat flux on the boundary.

On a Numerical Method for Solution of the Mathieu-Hill Type Equations

This paper presents a computer-programmable numerical method for the solution of a class of linear, second order differential equations with periodic coefficients of the Mathieu-Hill type. The method is applicable only when the initial conditions are prescribed and the solution is not requiried to be periodic. The solution is facilitated by representing the coefficient functions as a sum of step functions over corresponding sub-intervals of the fundamental interval. During each sub-interval, the solution form is assumed to be that of the differential equations with “constant” coefficients. Constraint equations are derived from imposing the conditions of “compatibility” of response at the end nodes of the intermediate sub-intervals. This set of simultaneous linear equations is expressed in matrix form. The matrix of coefficients may be represented as a triangular one. This form greatly simplifies the solution process for simultaneous equations. The method is illustrated by its application to some specific problems.

Sound propagation in parallel sheared flows in ducts: the mode estimation problem

We consider the inviscid, linear propagation of discrete frequency sound in an infinitely long, two-dimensional duct, possibly with soft walls, carrying a steady parallel sheared flow. Completely explicit expressions are obtained for the amplitudes of the sound modes associated with the discrete wavenumber spectrum generated upstream and downstream of a specified source distribution. By comparison with a less explicit solution for this problem given by Swinbanks (1975) we obtain an explicit expression for the group velocity of a mode in the case of rigid duct walls. Explicit expressions for the mode amplitudes can also be obtained in the case that the pressure, axial pressure gradient and transverse velocity are specified at one axial station. In the case of the problem of expanding a pressure profile in terms of modes entirely of the downstream or upstream family it is shown that in general this results in the need to solve an infinite set of linear simultaneous equations. Two such sets can be developed, suggesting th at in fact non-trivial constraints exist on the kinds of initial pressure profiles that are amenable to expansion in such a partial set of eigenfunctions. A necessary condition for the ‘optimum impedance’ problem is also derived. The entire analysis is based on a new approach to non-classical eigenvalue problems that combines the elements of residue theory and Sturm-Liouville theory and that is first illustrated by consideration of a non-classical vibrating string problem.

Synthesising recursive functions with side effects

Automatic Program Synthesis involves the automatic generation of a program (or plan) to achieve a specific goal. This means that smaller pre-defined (or previously synthesised) program segments are combined or modified to achieve the total goal. To guarantee that the required goal is actually achieved, the interactions between these program segments must be identified and considered. This paper shows how the side effects of segments can be derived and constructively used in achieving the required goal, and in guaranteeing the correctness of the resulting program. A program synthesising system PROSYN using these principles will then be described. A trace of a sample program synthesis, which generates a program that solves a general set of linear simultaneous equations, is given in the appendix.

A recursive digital filter with simultaneous maximally flat magnitude and group delay at an arbitrary specified frequency

A design technique for a recursive digital filter with simultaneously maximally flat magnitude and group delay at an arbitary specified frequency is presented, It employs the power series expansion to formulate a set of simultaneous linear equations which can be solved with a standard computational algorithm.

Linear prediction analysis of speech based on a pole-zero representation.

Speech analysis and synthesis by linear prediction is based on the assumption that the short-time spectral envelope of speech can be represented by a number of poles. An all-pole representation does not provide an accurate description of speech spectra, particularly for nasals and nasalized sounds. This paper presents a method for characterizing speech in terms of the parameters of a pole-zero model. In this method, an impulse response representing the composite filtering action of the glottal wave, the vocal tract, the radiation, and the speech recording system is first constructed from the speech signal. This impulse response is obtained by performing several stages of all-pole LPC analysis. The pole-zero parameters are determined from the impulse response by solving a set of simultaneous linear equations. The method, being noniterative, is very suitable for automatic analysis of speech. The method has been applied to real speech data and the results show that the speech spectra derived from the pole-zero model agree very closely with the actual spectra derived by direct Fourier analysis.

Frequency response of 2-dimensional digital filters

A state-space representation of an elementary 2-dimensional digital network is presented. The problem of determining the frequency response starting from its state-space equation is reduced to solving two sets of linear simultaneous equations in real unknowns at each pair of frequencies. The efficiency and the computational aspects of the proposed method are also discussed.

A Model for Planning the Transition to Equilibrium of a University Budget

Five-year projections of university expense and income items are incorporated into a model requiring long-run financial equilibrium (LRFE) at the end of the planning period. LRFE means that both budget levels and growth rates are in balance. The “transition to equilibrium” model consists of a set of simultaneous linear equations that are solved for an estimate of the amount of budget base reductions needed to achieve LRFE five years later. The model was applied at Stanford University, the resulting $10.2 million budget adjustment target was accepted, and now (two years later) more than 85 percent of the needed changes have been implemented.

Reduced Green’s functions and coupled perturbed Hartree–Fock calculations. II. Application to the static dipole polarizability of the helium isoelectronic sequence

The reduced free-particle Green’s function formulation of the first-order coupled perturbed Hartree–Fock theory is applied to the calculation of the static dipole polarizability for the helium atom and for two-electron ions as a test of the formulation and to demonstrate a power-moment method of solution. The power-moment method reduces the first-order coupled perturbed Hartree–Fock (FO-CPHF) integral equation to a set of simultaneous linear equations tailored to fit the problem. The formalism facilitates a consistent treatment of three approximations to the coupled perturbed method, depending upon the potential from which the unoccupied states (appearing in the corrected Green’s function) are derived. In each case the occupied (hole) states are accurate series solutions to the zero-order Hartree–Fock equations. Results in agreement with previous calculations are reported for the helium isoelectronic sequence.

Dynamic compensation for state feedback decoupling of multivariable systems

A method of decoupling linear time-invariant multivariable systems, which cannot be decoupled by proportional state feedback, but are invertible, is presented in this paper. A design procedure is developed for pre-compensating such a system, so that the augmented system satisfies the condition of state feedback decoupling. An expression for evaluating the order of the compensator is derived, and an algorithm described to compute the parameters of the compensator by solving a set of simultaneous linear equations. The flexibility in the choice of compensator dynamics is illustrated.

RESIDUAL STATICS ANALYSIS AS A GENERAL LINEAR INVERSE PROBLEM

To estimate near‐surface time anomalies, it is commonly assumed that apparent seismic reflection times are comprised of the sum of “surface‐consistent” source and receiver static terms, “subsurface‐consistent” structure and residual normal moveout (RNMO) terms, and indeterminate noise. The model parameters (statics, RNMO, and structural terms) that, in a least‐squares sense, best satisfy traveltime observations in multifold seismic data are solutions to a set of linear simultaneous equations. Because these equations are ill conditioned and their solutions are known to be nonunique, conventional direct methods of solution are not applicable. Problems of this type which have both overdetermined and underconstrained aspects can be analyzed using the general linear inverse methodology. In this approach, observed time deviations are decomposed into linear combinations of orthogonal eigenvectors, each of which determines a related linear combination of model parameters. A property of this decomposition is that the uncertainty (standard deviation) in a model‐parameter eigenvector is functionally related to the uncertainty in its associated observation eigenvector. In particular, statics corrections having spatial wavelengths much shorter than a cable length have smaller uncertainties than do the observations themselves, whereas long‐wavelength corrections have much larger standard deviations and are thus poorly determined. In practice, iterative methods are commonly used to solve the large number of equations encountered for typical seismic profiles. Using the Gauss‐Seidel iterative formalism, we can know in advance how many iterations are required to obtain a given reduction of the original error for any wavelength contribution. Errors in shorter‐wavelength corrections converge rapidly to zero while a heavier price is exacted to compute longer‐wavelength corrections. However, because those longer‐wavelength corrections can be estimated only with large uncertainty, it is desirable to exclude them from the statics solution through judicious choice of the number of iteration cycles.

Piecewise Newton-Raphson load flow ߞAn exact method using ordered elimination

Based on the Diakoptic Equation, an algorithm is developed for block diagonalizing and solving any given set of linear simultaneous equations. This algorithm is then applied to block diagonalize and solve the Jacobian matrix and hence the load flow problem. The algorithm is perfectly general and can be appliedto any set of linear simultaneous equations, symmetrical or asymmetrical and hence can be applied to other fields as well. Applied to the load flow problem, it requires to consider a few more additional right hand side vectors to the Jacobian, and further an inter-subdivision matrix is to be formed and solved. No matrix inversion is required in this method and hence sparsity can be exploited to the maximum extent. Practical steps are given, so that the algorithm can be implimented by others.

Increment-decrement life tables: A comment

Schoens 1975 paper on the problem of constructing a set of k interrelated increment-decrement life tables presents explicit solutions only for 2 and 3 table models. These solutions are derived from a straightforward but complicated set of simultaneous linear equations; this complexity increases exponentially with larger values of k making computation economy desirable. A matrix solution also would be useful because it more clearly identifies the correspondence between single table and k table formulations. This article presents equations to do this and illustrates their use in a problem area multiregional life table construction. In this application increments are due to in-migration and decrements to out-migration. This application of Schoens method to multiregional life table construction eliminates an important restrictive assumption in Rogers (1975) procedure the assumed absence of multiple transitions during a unit age interval.

A practical method for the accurate analysis of high alloy steels by X‐ray emission

AbstractA method for the conversion of X‐ray intensities to composition is given. The method is based on the Lachance‐Traill equation which uses empirical inter‐element correction coefficients. The coefficients are treated as ‘sliding constants’ the values of which depend on the relative concentrations of the analyte and the interfering elements. Values for the coefficients are calculated over wide concentration ranges using the theoretical equations of Sherman. The calculated coefficients are adjusted empirically to achieve high accuracy.Only the measured X‐ray intensities from the sample and a single standard are required for an analysis. Final results are obtained by solving a set of simultaneous linear equations. Results of analyses of high alloy steels containing Cr, Ni, Mn, Mo, Cu, Si, Ti, Cb and Fe are given to illustrate the accuracy of the method.

The Anchor-Last Deployment Problem for Inextensible Mooring Lines

A lumped mass numerical model was developed which predicts the dynamic response of an inextensible mooring line during anchor-last deployment. The mooring line was modeled as a series of concentrated masses connected by massless inextensible links. A set of angles was used for displacement coordinates, and Lagrange’s Method was used to derive the equations of motion. The resulting formulation exhibited inertia coupling, which, for the predictor-corrector integration scheme used, required the solution of a set of linear simultaneous equations to determine the acceleration of each lumped mass. For the selected cases studied the results show that the maximum tension in the cable during deployment will not exceed twice the weight of the cable and anchor in water.

Green's-function method for crystal films and surfaces

The Green's-function method is adapted to the problem of an electron moving in a crystal film. The potential is taken as spherical in the atomic spheres and as dependent only on the coordinate normal to the film in the regions just outside the bounding atom planes. The appropriate new structure constants are defined. With the aid of the necessary logarithmic derivatives a rather small set of simultaneous linear equations is obtained from which eigenvalues and eigenfunctions can be calculated. In conjunction with the density functional formalism this procedure provides a convenient method for self-consistently calculating the electronic structure of films and surfaces. Comparison with earlier work by Pharisean, Scherer, Cooper, and Beeby is made.