Evaluation of Errors at Numerical Integration of Ordinary Differential Equations

In this paper the authors present a new theory of error evaluation at numerical integration.The classical theories of error evaluation based on the Taylor expansion inform us only of the order of magnitude of errors and are not capable of clarifying the exact value and the nature of errors.It is known the numerical integration of the set of linear differential equations can be reduced into the solution of the set of the corresponding difference equations and the difference equations can be solved by means of the matrix technique and the z-transform.With these facts in mind authors developed a new theory which clarifies the nature errors and gives criteria for selecting adequate time intervals that keep the errors for computation within the allowable limits.Though the round-off errors are not discussed in this paper it is shown by the example that the round off errors are much smaller than the truncation errors discussed in the paper.

In this paper,the authors present a new theory of error evaluation at numerical integration.The classical theories of error evaluation based on the Taylor expansion inform us only of the order of magnitude of errors,and are not capable of clarifying the exact value and the nature of errors.It is known the numerical integration of the set of linear differential equations can be reduced into the solution of the set of the corresponding difference equations,and the difference equations can be solved by means of the matrix technique and the z-transform.With these facts in mind,authors developed a new theory,which clarifies the nature errors and gives criteria for selecting adequate time intervals that keep the errors for computation within the allowable limits.Though the round-off errors are not discussed in this paper,it is shown by the example that the round off errors are much smaller than the truncation errors discussed in the paper.