Linear recurrence = closed form?

Thread starter mikeph
Start date Apr 5, 2013
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Closed Form Linear Recurrence

In summary, a linear recurrence is a mathematical sequence where each term is a linear combination of the previous terms. A closed form expression is a mathematical equation that can be used to directly calculate the value of any term in a sequence. To convert a linear recurrence into a closed form expression, various techniques such as substitution, generating functions, or matrix methods can be used. The advantages of using a closed form expression include quicker and more efficient calculations, as well as a more general representation of the sequence. Linear recurrence and closed form expressions are commonly used in fields such as mathematics, computer science, physics, engineering, and economics to understand and model natural phenomena.

Apr 5, 2013

mikeph

Hi

I've got a recurrence relation: x_n = a*x_(n-1) + b;

I think I'm going mad trying to figure out a closed form, eg. x(n) = ? the nth iteration

Is there a trick or something I'm missing?

Thanks

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Apr 5, 2013

jbunniii

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Try writing out what you get for ##x_1##, ##x_2##, ##x_3## and see what the pattern is. Then take an educated guess and try to prove it by induction.

1. What is a linear recurrence?

A linear recurrence is a mathematical sequence where each term is a linear combination of the previous terms. This means that the next term is calculated by multiplying the previous terms by a constant and adding them together.

2. What is a closed form expression?

A closed form expression is a mathematical equation that can be used to directly calculate the value of any term in a sequence, without having to calculate all of the previous terms.

3. How do you convert a linear recurrence into a closed form expression?

To convert a linear recurrence into a closed form expression, you can use various techniques such as substitution, generating functions, or matrix methods. These methods involve manipulating the recurrence relation to eliminate the dependence on previous terms and solve for the desired term.

4. What are the advantages of using a closed form expression instead of a recurrence relation?

Closed form expressions allow for a quicker and more efficient way to calculate the value of a term in a sequence, as they do not require the computation of all the previous terms. They also provide a more general representation of the sequence, making it easier to analyze and understand its behavior.

5. In what fields of science is the concept of linear recurrence and closed form expression commonly used?

Linear recurrence and closed form expressions are commonly used in fields such as mathematics, computer science, physics, engineering, and economics. They are also essential in understanding and modelling various natural phenomena, including population growth, chemical reactions, and signal processing.