Recurrence relations

Recurrence Relations Overview

  • A recurrence relation is an equation that recursively defines a sequence, so each term is defined as a function of its preceding terms.

  • First-order linear recurrence relations are simple relations where each term in the sequence is determined from the preceding term by adding a constant or multiplying by a constant.

  • A second-order linear recurrence relation is slightly more complex, with each term being formed from the two preceding terms.

Main Types of Recurrence Relations

  • Linear Homogeneous Recurrence Relations: This is the simplest form of a recurrence relation where each term is a linear combination of its previous terms.

  • Linear Non-Homogeneous Recurrence Relations: This is a recurrence relation in which the right side of the equation contains a term independent of the sequence.

  • Non-Linear Recurrence Relations: In these relations, each term is a non-linear combination of its previous terms.

Solving Recurrence Relations

  • To solve a first order linear homogeneous recurrence relation, you can simply substitute the first term and simplify the resultant algebraic equation to find the nth term of the sequence.

  • To solve second order linear homogeneous recurrence relations, you will likely have to use the characteristic equation.

  • In many cases, advanced techniques such as generating functions or the method of undetermined coefficients may be required to find solutions for complicated recurrence relations.

Applications of Recurrence Relations

  • Recurrence relations are widely applied in a number of fields including computer science (for algorithms), mathematics, physics, and engineering.

  • Particularly, they are important in discrete mathematics, with applications in counting, discrete probability, and graph theory.

  • One of the most famous recurrence relations is the Fibonacci sequence, which has wide applications in areas such as computation, data storage, and network protocols.