Solving recurrence systems

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.