Solving recurrence systems

Overview of Recurrence Systems

A recurrence system is a way to define a sequence where each term after the first depends on the terms before it.
These systems are used in fields such as mathematics, computer science, and engineering to model problems that follow a repetitive process.
A recurrence relation may be linear or non-linear, and homogeneous or non-homogeneous, depending on the nature of the associated terms.

Procedure for Solving Recurrence Systems

The objective of solving a recurrence system is to find an explicit formula, i.e., a closed-form expression for the sequence terms that does not involve recursion.
The characteristic equation is a common method for solving linear homogeneous recurrence relations. This involves using auxiliary equations where the roots help determine the general solution.
For non-homogeneous recurrence relations, a particular solution needs to be found which is then combined with the homogeneous solution to obtain the general solution.

Characteristic Equation and Roots

The characteristic equation is a key part of solving linear homogeneous recurrence relations. The roots of this equation provide valuable insight into the form of the general solution.
Depending on whether the roots are real and distinct, real and repeated, or complex, the general solution takes on different forms.
If the roots are real and distinct, the general solution is of the form an^m, where ‘a’ is the root, ‘n’ is the term number, and ‘m’ corresponds to the root’s multiplicity.
If the roots are real and repeated, the general solution is n^m*(an+b), where ‘a’ is the root, ‘n’ is the term number, ‘b’ is a constant, and ‘m’ is the number of times the root is repeated.
If the roots are complex, the general solution is a combination of sine and cosine functions.

Method of Undetermined Coefficients

For non-homogeneous recurrence relations, the method of undetermined coefficients can be used to find the general solution.
This involves guessing the form of the particular solution based on the type of non-homogeneity present in the recurrence relation, plugging this form into the recurrence relation, and then solving for the undetermined coefficients.
The complete solution is the sum of the homogeneous solution and the particular solution.

Initial Conditions and Specific Solutions

The initial conditions are the specific known terms of the sequence which help determine the exact values of the constants in the general solution.
Once the initial conditions are substituted into the general solution, a specific solution for the recurrence relation is obtained.
Through this specific solution, any term of the sequence can be calculated without the need for knowing the terms before it.

Recurrence Relations in Problem Solving

Recurrence relations provide an essential tool in problem-solving, particularly for problems that exhibit a repetitive or cyclic nature.
They allow the translation of such problems into mathematical expressions, enabling more accurate solutions.
Mastery in solving recurrence relations is pivotal in further studies in mathematics, including areas like difference equations, dynamical systems, and mathematical modelling.