Recursive Relationships

[BL]

Can someone please explain the concept of recursive relationships? I don't think I fully understand when I should use them.
Thanks!

[Gandalf]

They're just a way of moving from one period to the next, usually a simpler process than starting from scratch.

For example, in course FM if you knew the value of s_immed_angle_12 and the interest rate, you could get s_immed_13 as s_12 * (1+i) + 1. On a financial calculator, it would easy enough to just calculate s_13 directly, but with a four-function calculator the recursive relationship is much easier.

For exam conditions, it's critical to know the recursive relationships since they might not give you all the data to do the direct calculation. (Returning to the FM example, suppose instead of saying "here's s_12 and i, calculate s_13", they said here's "s_n and i, calculate s_(n+1)". The recursive formula becomes much better. While you could do that FM one without recursion (first use the financial calculator to tell you want n must be), there can be situations of FM, and even more on M, where the recursive method is the only way with the info given.

[mlschop]

use them if you are given insurances/annuities/expectations for age x and x+1 (or x+n, for an n-step recursion).

if you see Ax and Ax+1, or Ax and Ax+2, you are more likely than not going to use recursion...

u should be happy when this happens, actually, since recursive formulas make sense if you take the time to look at them and decipher their meaning

[Pseudolus]

See this thread. Sorry.

[carrytheCrøss]

Originally Posted by Pseudolus

See this thread. Sorry.

[Mathsucks]

Originally Posted by Pseudolus

See this thread. Sorry.

[bholle]

Originally Posted by Pseudolus

See this thread. Sorry.

Very nice.

[Captain Nemo]

Originally Posted by BL

Can someone please explain the concept of recursive relationships? I don't think I fully understand when I should use them.
Thanks!

Recursive relationships allow for self-reference. Usually, they are as Gandalf and mlschop described, where there is a defined "root" and "branches" that come from them; of course, Pseudolus has the example I was thinking as soon as I saw your post

However, there can be a mix. For example:

An angler catches a fish which is 10 inches long plus half its own length. how long is it?

A simple algebra problem (x = 10 + 0.5*x), but the "recursive" piece is that the fish's length is actually used to define... the fish's length. And indeed, can be used to determine itself using an iterative process.

"Well, then the fish is 10 inches long. But then we add half its length, so it's 10 + 5 inches long. But if it's 10 inches plus half of 15 inches, it's 17.5 inches long..." which converges nicely to 20 inches, the correct answer.

The factorial function can be defined (on the non-negative integers) recursively as follows:

0! = 1.
n! = n*(n-1)!

While in some sense the second equation is not particularly illuminating, on an exam, if you were given that 9! was 362,880, I imagine that you'd prefer to use the recursive definition to determine that 10! = 3,628,800 rather than starting from scratch and multiplying out 1*2*3*4*5*6*7*8*9*10.

[altalingua]

Can you give an example of a recursive relationship involving Ax and Ax+1?

Originally Posted by mlschop

use them if you are given insurances/annuities/expectations for age x and x+1 (or x+n, for an n-step recursion).

if you see Ax and Ax+1, or Ax and Ax+2, you are more likely than not going to use recursion...

u should be happy when this happens, actually, since recursive formulas make sense if you take the time to look at them and decipher their meaning

[Gandalf]

If you know any formula involving both Ax and Ax+1, it's almost certainly a recursive relationship. If it expresses one in terms of the other (using some of v,i,p,q as well), it definitely is recursive.