3.6. The local inverse operation of the derivative

The inverse operation to the finite differences derivative, at a given point, is the summation with weights 2χð Þ v;Δ

$$\sum\_{j=n}^{m} 2\chi(v,\Delta)(\mathcal{D}g)\_{j} = \sum\_{j=n}^{m} \left(\mathcal{g}\_{j+1} - \mathcal{g}\_{j-1}\right) = \mathcal{g}\_{m+1} + \mathcal{g}\_{m} - \mathcal{g}\_{n} - \mathcal{g}\_{n-1}.\tag{46}$$

ð Þ Dg hð Þ <sup>j</sup>

 � h qj 

4. The commutator between coordinate and derivative

ð Þ <sup>D</sup><sup>q</sup> <sup>j</sup> <sup>¼</sup> qjþ<sup>1</sup> � qj�<sup>1</sup>

Hence, the finite differences derivative of the product qg qð Þ is

expected. An equivalent expression is

<sup>χ</sup>ð Þ <sup>v</sup>;<sup>Δ</sup> <sup>≈</sup>

<sup>χ</sup> <sup>v</sup>; h qjþ<sup>1</sup>

derivative of h qð Þ with respect to q

tive of q,

i.e.,

operator.

≔

ghqjþ<sup>1</sup>

<sup>2</sup><sup>χ</sup> <sup>v</sup>; h qjþ<sup>1</sup>

is a finite differences derivative of g hð Þ with respect to h, and the second factor approaches the

h qjþ<sup>1</sup> � h qj

Thus, we will recover the usual chain rule for continuous variable functions in the limit Δ ! 0.

Let us determine the commutator, from a local point of view first, between the coordinate—the points of the partition P Nð Þ—and the finite differences derivative. We begin with the deriva-

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup>

ð Þ <sup>D</sup>qg <sup>j</sup> <sup>¼</sup> qjþ<sup>1</sup>ð Þ <sup>D</sup><sup>g</sup> <sup>j</sup> <sup>þ</sup> gj�<sup>1</sup>ð Þ <sup>D</sup><sup>q</sup> <sup>j</sup> <sup>¼</sup> qjþ<sup>1</sup>ð Þ <sup>D</sup><sup>g</sup> <sup>j</sup> <sup>þ</sup> gj�<sup>1</sup>

ð Þ <sup>D</sup>cqg <sup>j</sup> � qjþ<sup>1</sup>ð Þ <sup>D</sup>cg <sup>j</sup> <sup>¼</sup> gj�<sup>1</sup>

ð Þ <sup>D</sup>qg <sup>j</sup> � qj�<sup>1</sup>ð Þ <sup>D</sup><sup>g</sup> <sup>j</sup> <sup>¼</sup> gjþ<sup>1</sup>

This is the finite differences version of the commutator between the coordinate q and the finite differences derivative D. This equality will become the identity operator in the small Δ limit, as

This is the finite differences version of the commutator between coordinate and derivative; the right hand side of this equality becomes gj in the small Δ limit, i.e., it becomes the identity

<sup>χ</sup>ð Þ <sup>v</sup>; <sup>Δ</sup> <sup>≈</sup> <sup>1</sup> � <sup>v</sup><sup>2</sup>

Δ

Δ

<sup>6</sup> <sup>Δ</sup><sup>2</sup>

Δ

<sup>χ</sup>ð Þ <sup>v</sup>;<sup>Δ</sup> : (53)

<sup>χ</sup>ð Þ <sup>v</sup>;<sup>Δ</sup> : (54)

� ghqj�<sup>1</sup>

� h qj

(49)

Matrices Which are Discrete Versions of Linear Operations

http://dx.doi.org/10.5772/intechopen.74356

31

<sup>Δ</sup> <sup>þ</sup> <sup>O</sup> <sup>Δ</sup><sup>2</sup> : (50)

: (51)

<sup>χ</sup>ð Þ <sup>v</sup>;<sup>Δ</sup> , (52)

<sup>þ</sup> <sup>O</sup> <sup>Δ</sup>h<sup>2</sup>

This equality is the equivalent to the usual result for continuous functions, Ð <sup>x</sup> <sup>a</sup> dy dg y ð Þ ð Þ=dy ¼ g xð Þ� g að Þ. Note that the inverse at the local level is a bit different from the expressions obtained by means of the inverse matrix S (see below) of the derivative matrix D. When dealing with matrices there are no boundary terms to worry about.
