On the instability of symmetric formulas for numerical differentiation and integration

The stability of the order of symmetric formulas for derivative approximation used in finite-difference methods for solving differential equations is analyzed. The stability of the order of a numerical differentiation formula with respect to the shift of the application point (at which this formula is applied) is defined. The conditions of numerical experiments determining the behavior of the order of the simplest symmetric approximation formulas for the first and second derivatives in the case of point shifts are described, and some numerical results are presented. The instability of the maximum order of these formulas is shown in examples. A family of rectangular quadrature rules is examined in a similar manner, and the instability of the second order of the quadrature midpoint formula is demonstrated.