Error Analysis

The error of the truncated Newton series is

$$\varepsilon =\left\Vert f\left[ x_{0},\ldots ,x_{N-1},\widehat{O}\right] \prod _{j=0}^{N-1}\widehat{Q}_{j}\right\Vert .$$

The divided difference term in this error cannot be evaluated by repeated application of the operator $\widehat{O}$ as is desired for computational implementation. So the term to minimise is the product term which only depends on the interpolation points $x_{i}$. If the number of these points is equal to the number of eigenstates of the operator, this minimization is equivalent to the diagonalization of the operator so the error will converge to zero as this number rises.

There are two approaches to the minimization:

1. Uniform Approach: The error is minimised as an operator - with respect to the whole Hilbert space of states.
2. Non-Uniform Approach: The norm of the product term applied to a certain wave function is minimised (which is usually the initial state).