Reversing recurrence relationships

In summary, the conversation discusses the idea of numerically evaluating a recursive relation in reverse as a valuable approach when dealing with inaccurate or undiscoverable boundary conditions. The example of a Legendre polynomial is used to illustrate this concept. The speaker also mentions their struggle with understanding the difference between evaluating in the forward and reverse directions, and how writing programs to compare has provided some intuition. Ultimately, the speaker seeks confirmation that they are not missing anything else in this approach.
  • #1
ognik
643
2
A couple of times I have come across the suggestion that numerically evaluating a recursive relation in reverse can be a valuable approach. I can see this where, for example, the boundary conditions at one 'end' are inaccurate or undiscoverable. However, while the arithmetic of manipulating such equations seems simple, I wonder if I am missing something?
One example is a Legendre polynomial, given by (l+1)Pl+1 + lPl - (2l+1)xPl=0
Should I evaluate this in the 'forward' direction, by solving for Pl+1, and in the reverse direction by solving for Pl-1? I am also struggling for some intuition as to what the difference(s) may be?
 
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  • #2
After writing a few programs to compare, I am satisfied that when a numeric method has clear '3 points', then I can evaluate this in the 'forward' direction, by solving for Pl+1, and in the reverse direction by solving for Pl-1. In some other methods where steps are only visible as +(some step size, like h), then we can 'reverse' direction by using -h. This was all intuitively obvious, I just wanted confirmation I wasn't missing anything else ...
 

FAQ: Reversing recurrence relationships

What is a recurrence relationship?

A recurrence relationship, also known as a recurrence relation, is a mathematical equation or formula that describes a sequence of values where each term is defined in terms of the previous terms.

Why is it important to reverse a recurrence relationship?

Reversing a recurrence relationship is important because it allows us to find the initial conditions or starting values of a sequence when given the final value. This can be useful in solving various mathematical problems.

What are the steps involved in reversing a recurrence relationship?

The steps involved in reversing a recurrence relationship are as follows:

  1. Identify the original recurrence relationship
  2. Rearrange the equation to isolate the previous term on one side
  3. Substitute the final value into the equation
  4. Solve for the previous term to find the initial condition

Can all recurrence relationships be reversed?

No, not all recurrence relationships can be reversed. Some may require additional information or may not have a unique solution. It is important to carefully analyze the given recurrence relationship before attempting to reverse it.

What are some real-world applications of reversing recurrence relationships?

Reversing recurrence relationships can be applied in various fields such as finance, engineering, and computer science. For example, it can be used to calculate the initial investment needed to achieve a desired future value or to determine the starting values of a computer program that generates a sequence of numbers.

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