1. Introduction

We are interested on matrices which are a local, as well as a global, exact discrete representation of operations on functions of continuous variable, so that there is congruency between the continuous and the discrete operations and properties of functions. Usual finite difference methods [1–4] become exact only in the limit of zero separation between the points of the mesh. Here, we are interested in having exact representations of operations and functions for finite separation between mesh points.

The difference between our method and the usual finite differences method is the quantity that appears in the denominator of the definition of derivative. The appropriate choice of that denominator makes possible that the finite differences expressions for the derivative gives the exact results for the exponential function. We concentrate on the derivative operation, and we define a matrix which represents the exact finite difference derivation on a local and a global

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

scale. The inverse of this matrix is just the integration operation. These are interesting subjects by itself, but they are also of interest in the quantum physics realm [5–7].

We start our study with a result about the determinant of D<sup>N</sup> � λIN,

α � 1=z 100 … 000 �1 α 1 0 … 000 0 �1 α 1 … 000 0 0 �1 α … 000  

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

Matrices Which are Discrete Versions of Linear Operations

 

 

: (6)

(4)

23

(5)

0 00 … α 100 0 00 … �1 α 1 0 0 00 … 0 �1 α 1 0 00 … 0 0 �1 α þ z

<sup>A</sup><sup>N</sup>�<sup>1</sup>ð Þþ <sup>α</sup> <sup>A</sup><sup>N</sup>�<sup>2</sup>ð Þ <sup>α</sup> ,

α 1 0 … 000 �1 α 1 … 000 0 �1 α … 000

0 0 … α 100 0 0 … �1 α 1 0 0 0 … 0 �1 α 1 0 0 … 0 0 �1 α þ z

> α 1 0 … 0 0 �1 α 1 … 0 0 0 �1 α … 0 0

> 0 0 … α 1 0 0 0 … �1 α 1 0 0 … 0 �1 α

Strikingly, we recognize the determinant Bjð Þ α as the Fibonacci polynomial of index j þ 1

¼ ð Þ α þ z B<sup>j</sup>�<sup>1</sup>ð Þþ α B<sup>j</sup>�<sup>2</sup>ð Þ α ,

∣D<sup>N</sup> � λIN∣ ¼ ∣D<sup>N</sup> þ αIN∣

 

¼

⋮

<sup>¼</sup> <sup>α</sup> � <sup>1</sup> z

> 

Ajð Þ α ≔

⋮

Bjð Þ¼ α

[10, 11], i.e., Bjð Þ α =F<sup>j</sup>þ<sup>1</sup>ð Þ α . Fibonacci polynomials are defined as

⋮

 

where λ ¼ �α,

and

In this chapter, we will consider only the case of the derivative and the integration of the exponential function.
