*2.2. Divided Difference*

Divided difference operator for multivariable function *F* (see [4,5,20]) is a mapping [·, · ; *F*] : *D* × *D* ⊂ R*m* × R*m* → *L*(R*m*) which is defined as

$$[\mathbf{x}, \mathbf{y}; F](\mathbf{x} - \mathbf{y}) = F(\mathbf{x}) - F(\mathbf{y}), \forall \,(\mathbf{x}, \mathbf{y}) \in \mathbb{R}^m. \tag{9}$$

If *F* is differentiable, we can also define first order divided difference as (see [4,20])

$$\mathbb{E}\left[\mathbf{x} + h, \mathbf{x}; \mathbb{F}\right] = \int\_0^1 F'(\mathbf{x} + th) \, dt,\,\forall \left(\mathbf{x}, h\right) \in \mathbb{R}^m. \tag{10}$$

This also implies that

$$[\mathbf{x}, \mathbf{x}; F] = F'(\mathbf{x}).\tag{11}$$

It can be seen that the divided difference operator [*x*, *y* ; *F*] is an *m* × *m* matrix and the definitions (9) and (10) are equivalent (for details see [20]). For computational purpose the following definition (see [5]), is used

$$[\mathbf{x}, y; F]\_{ij} = \frac{f\_i(\mathbf{x}\_1, \dots, \mathbf{x}\_j, y\_{j+1}, \dots, y\_m) - f\_i(\mathbf{x}\_1, \dots, \mathbf{x}\_{j-1}, y\_j, \dots, y\_m)}{\mathbf{x}\_j - y\_j}, \quad 1 \le i, j \le m. \tag{12}$$
