Pivotal Method Research Articles

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn ≥ 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 n pivot steps before termination. However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications. In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.

Solution of boundary value problems for linear systems of ordinary differential equations by the method of directed orthogonalization

A VERSION of the orthogonal pivotal condensation method for solving boundary value problems for systems of linear ordinary differential equations is examined. During the orthogonalization the basis vectors are newly resolved at each step, using the eigenvectors and eigenvalues of the matrix of the system at the point of orthogonalization.

Block pivot methods for solving the complementarity problem

A family of algorithms which solve the complementarity problem (and certain generalized complementarity problems) is introduced. In these algorithms block pivots (multiple exchanges of basic and nonbasic variables) are permitted. The geometry of complementarity points is studied. The proof that these algorithms converge in a finite number of steps is based on certain elementary results from group theory, and does not rely on monotonicity arguments.

An indirect measure of perceived distance from oculomotor cues

The sensitivity of an indirect method of measuring perceived distance was compared in two experiments with the direct procedure of eliciting verbal reports of distance. Perceived distance was varied by varying the oculomotor cues to object distance. The indirect method, called the “adjustable pivot method,” uses an apparatus that physically moves the stimulus object laterally concomitantly with the lateral motion of the head. The magnitude and direction of this concomitant motion determines the distance of the point around which the direction of gaze to the object rotates (the pivot distance) as the head is moved. The pivot distance at which the object appears stationary with head movement measures the apparent distance of the object. Both types of measures were found to vary systematically with the oculomotor distance of the object for points of light (Experiment 1) and extended objects (Experiment 2). A previous study has shown that the adjustable pivot method avoids cognitive errors that can distort verbal reports of distance. The present study, by demonstrating the discriminative capability of this method under conditions in which differences in perceived distance were expected to occur, provides clear evidence that the adjustable pivot method is a sensitive and useful procedure for measuring perceived distance.

A variant of the pivotal condensation method for systems with connected boundary conditions

AN IMPLICIT difference scheme for the heat production equation with connected boundary conditions leads to the need to solve systems of equations with a matrix, which differs from a tridiagonal matrix by the presence of additional elements in some column. A method of solving the corresponding systems is proposed which uses double bordering combined with pivotal condensation, and its stability is demonstrated.

A Method for solving non-linear boundary value problems describing the minority carrier distribution in the base of a semiconductor structure

A COMPUTATIONAL algorithm is given for solving non-linear boundary value problems it uses the familiar methods of quasi-linearization, pivotal condensation, and continuation of the solution with respect to a parameter. The algorithm is applied in the study of a mathematical model of the carrier distribution in the base of a semiconductor structure, described by a non-linear boundary value problem for an ordinary second-order differential equation. Most numerical methods for solving non-linear boundary value problems, such as Newton's method [1–3], the finite-difference iterative method [4], and the method of quasi-linearization [5], are based on the construction of iterative processes. The continuous analogues of single-step iterative processes, described in [6], have come to be widely used in the investigation of various physical problems [7]. In particular, the study of certain models of the charge transport process in the quasi-neutral zone of a semiconductor structure reduces to the solution of a non-linear boundary value problem for an ordinary second-order differential equation. Attempts to solve this boundary value problem [8,9,10] have so far been limited to certain approximations, whereas modem requirements and technological possibilities have made it necessary to solve the initial (non-approximated) problem. It is in this connection that the present paper describes a computational algorithm for solving non-linear boundary value problems, based on the method of quasi-linearization, a discrete version of the method of continuing the solution with respect to a parameter [11,12], and the pivotal condensation method [13]. This algorithm makes it possible to increase substantially the domain of convergence of the relevant iterative procedures. The algorithm is described in general terms in Section 1, while in Section 2 it is applied to the solution of a concrete problem.

A comparison of the primal‐simplex and complementary pivot methods for linear programming

AbstractA comparison of the complementary pivot method of Lemke‐Howson and the more commonly used primal‐simplex method for solving linear programming problems in symmetric dual form has been made. In our tests the complementary pivot method shows a definite superiority over the simplex method both with regard to the number of iterations and computation time.

The pivotal condensation method for difference equations defined on a complex

The pivotal condensation method for difference equations defined on a complex

Stability of feedback systems with periodic parameters†

This paper deals with a frequency domain technique for stability analysis of linear feedback systems with periodically time-varying parameters. The approximate method of signal balance used by Clark et al. (1064) is generalized to yield better results in the case of a wide variety of systems including some TVP-coupled systems. For systems with TVP coupling, a novel and simple method of ‘pivotal points’ is suggested for quickly sketching the stability boundaries making use of the known stability of subsystems. The results of perturbation analysis (Rao 1969) are found to support the pivotal paint method.

Solution of a direct problem of heat exchange between a cooled turbine blade and a gas by the pivot method

Using the pivot method we solve a third-order differential equation that describes the heat exchange of a gas with a cooled turbine blade.

The pivotal condensation method for second-order systems

The pivotal condensation method for second-order systems

The stability of the pivotal condensation method for realizing difference schemes for equations of heat conduction and vibrations

IN this paper it is proved that if schemes with weights, approximating the first boundary value problem for the equations of heat conduction and vibrations, are stable with respect to the initial data, the pivotal condensation method, used to find the solution on the upper layer, is stable.

An improved two-layer scheme for sets of parabolic equations with constant coefficients

DIFFERENCE schemes of improved accuracy have been proposed for various types of partial differential equations; see [1, 2] for the references. An improved two-layer scheme was described in [2, 3] for sets of parabolic equations with constant coefficients and no mixed derivatives. Here, to obtain a priori estimates, fairly rigid conditions were imposed on the matrix of space operator coef- ficients. It is easy to construct a set of parabolic equations with constant coefficients which does not satisfy these conditions. We show in the present paper, by using Samarskii's general theory of stability of difference schemes [4, 5], that absolute stability, a priori estimates and convergence theorems can be proved for the improved schemes of [2, 3] without supplementary conditions on the space operator. It should be mentioned that the set of algebraic equations obtained may be solved by the matrix pivotal condensation method. In [6], Samarskii's method of regularisation was used to construct absolutely stable three-layer difference schemes of improved accuracy for sets of parabolic equations with constant coefficients and no mixed derìvatives, and their convergence in a uniform metric was proved in the case of a p-dimensional parallelepiped ( p = 2, 3). As distinct from the two-layer scheme of [2, 3], the sets of algebraic equations obtained in [6] may be solved by means of a uniform pivotal condensation algorithm.

The organization of the pivot method for an implicit scheme

A DETAILED investigation of an implicit difference scheme is given in [1] for the solution of a gas-dynamic problem. Below we give an economic variant of the pivot method for this scheme. To simplify the description only the case of two-dimensional flows is considered.

Evaluating Determinants by Pivotal Condensation

As far as the author is aware, there does not seem to be currently available in the literature a proof of the little-publicised pivotal condensation method of evaluating determinants. The purpose of this note is to supply a simple proof of this.

Discontinuous solutions in problems of nonlinear heat conduction

We propose a method of constructing discontinuous solutions for nonlinear problems in the theory of heat conduction. The method is a modification of the pivot methods for the solution of linear boundary-value problems. As an illustration we present numerical solutions of two problems.

Free axially-symmetric vibrations of thin elastic shells of revolution

THE problem is reduced to the search for eigenvalues and eigenfunctions of a set of ordinary linear differential equations with variable coefficients. In papers devoted to the numerical solution of this problem considerable difficulties have been noted, due to the presence of rapidly increasing solutions [1]. In this investigation a numerical method of solving the problem is suggested in which the difficulties indicated have been successfully overcome by using A.A. Abramov's pivotal condensation method [2]. The results of the numerical calculations are compared with solutions based on the asymptotic integration of differential equations [3, 4].

Disintegration of axisymmetric body of revolution made of a material of complex chemical composition in a high-enthalpy airflow

We present a numerical solution of the problem of the disintegration in a high enthalpy airflow of an axisymmetric body of revolution made of a plastic of complex chemical composition. The system of partial differential equations for the laminar multicomponent boundary layer is solved by the pivotal method. The use of an experimental kinetic condition for the disintegration in place of the vapor pressure curve, which is usually used for single-component subliming materials, permits the use of a single technique for calculating both coking and noncoking materials of complex composition. Calculated is the composition of the products of combustion of materials consisting of the chemical elements, H, C, N, O. As an example the results are presented of the calculation of the disintegration of a sphere made from a material similar to textolite in composition.

II. A solution of the integro-differential equation for the slope of the jet

In an earlier paper with the same general title (Spence 1956, referred to as I), a mathematical model was developed to discuss the flow past a two-dimensional wing at incidence a in a steady incompressible stream, with a jet of momentum coefficient C j emerging from the trailing edge at an angular deflexion r to the chordline. In linearized approximation it was shown that the slope of the jet is given by a certain singular integro-differential equation, and numerical solutions for the equation were obtained by a pivotal points method. A coordinate transformation has now been found (Spence 1959) which makes the equation independent of the jet strength for small values of 14 C j = u, say, yielding a simpler equation solved by Lighthill (1959) using Mellin transforms (and by Stewartson (1959) and the present author by other methods). In this paper the expansion of the slope function is continued in ascending powers of u and In u multiplied by functions of x found by solving, by Lighthill’s method, a series of closely-related inhomogeneous equations. From these, expansions of the lift derivatives with respect to a and r are found as To this order the expressions agree closely with the numerical results found earlier, the discrepancy at u = 1 being less than 4 %.

A modified Aitken Pivotal Condensation Method for partial regression and multiple correlation

In this modification of the Aitken Pivotal Condensation Method for obtaining partial regression weights and multiple correlation coefficients, the unit matrices used by Aitken are eliminated, with a resultant decrease in the computational work. Through a simplified arrangement of successive matrices, each one is reduced by the same simple rule. Other modifications also contribute to ease of computation and facilitate checking.