Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Summary. In this paper we present an approach for the numerical solution of delay differential equations where , and , different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems (1). Received May 4, 1998 / Revised version received January 25, 1999 / Published online November 17, 1999     

Numerical solution of differential algebraic equations using a multiquadric approximation scheme

The objective of this paper is to solve differential algebraic equations using a multiquadric approximation scheme. Therefore, we present the notation and basic definitions of the Hessenberg forms of the differential algebraic equations. In addition, we present the properties of the proposed multiquadric approximation scheme and its advantages, which include using data points in arbitrary locations with arbitrary ordering. Moreover, error estimation and the run time of the method are also considered. Finally some experiments were performed to illustrate the high accuracy and efficiency of the proposed method, even when the data points are scattered and have a closed metric.     

Estimate of a domain containing the solution of a system of linear algebraic equations

An estimate of the deviation of the solution of linear algebraic equations from the solution of a frozen system, under sufficiently small variations of the system's coefficients, is given within the framework of linear error theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 227–228, 1979.     

Pin-pointing solution of ill-conditioned square systems of linear equations

A new method is proposed for an accurate solution of nearly singular systems of linear equations. The method uses the truncated singular value decomposition of the initial coefficient matrix at the first stage and the Gaussian elimination procedure for a well-conditioned reduced system of linear equations at the last stage. In contrast to various regularization techniques, the method does not require any additional physical information on the problem.     

Numerical solution to systems of delay integrodifferential algebraic equations

The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.     

Multivariate polynomial perturbations of algebraic equations

In this note we study multivariate perturbations of algebraic equations. In general, it is not possible to represent the perturbed solution as a Puiseux-type power series in a connected neighborhood. For the case of two perturbation parameters we provide a sufficient condition that guarantees such a representation. Then, we extend this result to the case of more than two perturbation parameters. We motivate our study by the perturbation analysis of a weighted random walk on the Web Graph. In an instance of the latter the stationary distribution of the weighted random walk, the so-called Weighted PageRank, may depend on two (or more) perturbation parameters in a manner that illustrates our theoretical development.     

Investigation and solution of a pair of linear matrix equations

Translated from Matematicheskie Zametki, Vol. 57, No. 2, pp. 300–303, February, 1995.     

Numerical solution of separable nonlinear equations with a singular matrix at the solution

Numerical Algorithms - We present a numerical method for solving the separable nonlinear equation A(y)z + b(y) =?0, where A(y) is an m × N matrix and b(y) is a vector, with y ∈Rn...     

On the solution of systems of linear algebraic equations with symbolic elements

We introduce some general concepts and propositions relating to symbolic computation on a computer. We analyze the suitability of the known numerical methods for solving systems of linear algebraic equations with symbolic elements. We propose efficient algorithms for solving dense systems and certain sparse systems by branching continued fractions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 29–35.     

Solution of linear equations with coefficient matrix in band form

An algorithm is given for the solution of simultaneous linear equations having a band type coefficient matrix. The algorithm utilizes a suitable partitioning of the coefficient matrix. Some particular cases are described in which the use of the algorithm is advisable.This research was supported in part by the National Aeronautics and Space Administration Grant No. NGR-33-015-013.     

On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time

The cost of solving an initial value problem for index-1 differential algebraic equations to accuracy ɛ is polynomial in ln(1/ɛ). This cost is obtained for an algorithm based on the Taylor series method for solving differential algebraic equations developed by Pryce. This result extends a recent result by Corless for solutions of ordinary differential equations. The results of the standard theory of information-based complexity give exponential cost for solving ordinary differential equations, being based on a different model.     

Numerical solution of Sylvester matrix equations in the self-adjoint case

Algorithms of the Bartels-Stewart type for numerically solving Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are self-adjoint. The superiority of the modified algorithms over the original ones is illustrated by numerical results.     

Numerical solution of random differential equations: A mean square approach

This paper deals with the construction of numerical solutions of random initial value differential problems by means of a random Euler difference scheme whose mean square convergence is proved based on conditions expressed in terms of the mean square behavior of the right-hand side of the underlying random differential equation. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced and illustrative examples and possibilities are included. Expectation and variance of the approximating process are computed.     

Numerical solution of Sylvester matrix equations with normal coefficients

Algorithms of the Bartels–Stewart type for the numerical solution of Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are normal. The superiority of the modified algorithms over the original ones is illustrated by numerical results.     

On the solution of certain algebraic equations

This note deals with the problem of determining the roots of simple algebrac equations by constructing polynomial equations that have the same roots.     

Zeros of a random algebraic polynomial with coefficient means in geometric progression

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients {a_j}ⁿ⁻¹_j=0 of the polynomial T(x)=a₀+a₁x+a₂x²+?+a_n−1xⁿ⁻¹ are normally distributed, with mean E(a_j)=μ^j+1, where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of |μ|<1 and |μ|>1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0.     

Solution of polynomial equations in the field of algebraic numbers

A method of solving polynomial equations in a ring D[x] is described, where D is an arbitrary order of field ?(ω) and ω is an algebraic integer number.