Optimal Difference Formulas for an Approximate Solution of the Cauchy Problem

The key properties of numerical methods for solving ordinary differential equations are determined by their accuracy and stability. The step size is selected based on the accuracy of the numerical solution. In this paper, we will consider difference methods for the approximate solution of first-order ordinary differential equations. Here, we will find the square of the norm of the error functional of difference formulas. To obtain optimal coefficients, we will construct and analyze systems of linear algebraic equations. By solving this system, we will find the optimal coefficients of the difference formulas for specific spaces, and here we will calculate the square of the norm of the error functionals of the optimal difference formulas.