# Understanding what equations mean

#### Otis

##### Senior Member
… in terms of i and j …
For a start, let's agree that all symbols {i, j, n} represent Natural numbers (i.e., {1, 2, 3, …}).

Symbols i and j are variables (i.e., they each take on more than one value) and symbol n is a parameter (i.e., a constant; however, its value may change when starting a new application of the equation).

The constant n limits how large j can be. Symbol j starts at 1 in value and increments up, to take on number n as its final value. For example, if n is set at 10, then j takes on the values {1,2,3,4,5,6,7,8,9,10} -- one at a time.

That is what the line 1 ≤ j ≤ n tells us.

r() and v() are called "Function Notation". Are you familiar with function notation?

Last edited:

##### Full Member
We don't have a definition for any function r(x). We don't have a definition for function v(j), either.

Also, for the purposes of discussion, let's limit i and say: 1 ≤ i ≤ n

The equation assigns a value to r(i), for each i input. The value for r(i) is the smallest value in a set of numbers. This set is defined by:

{ 1 + r(i - v(j)) }, for each j from 1 through n.

So we see that each value r(i) is defined in terms of r(i-v(j)). That is, the values of r(i-v(j)) must be listed first (for each j), before r(i) can be determined.

Each value in that list is increased by 1, and the smallest number appearing in the resulting set is assigned to r(i).

Let r(x) = 2x

In other words, function r outputs its input doubled.

Let v(j) = 1 - j/10

In other words, function v outputs the difference between 1 and one-tenth its input.

Let n = 3

In other words, variables i and j will each take on values {1,2,3}. Variable j will loop through its values three times (once for each i); variable i will loop through its values only once.

Code:
[FONT=courier new]
i   j   v(j)   i - v(j)   r(i - v(j))   {1 + r(i-v(j))}   r(i) = MIN{}
----------------------------------------------------------------------
1   1   0.9      0.1          0.2            {1.2         r(1) = 1.2
1   2   0.8      0.2          0.4             1.4
1   3   0.7      0.3          0.6             1.6}

2   1   0.9      1.1          2.2            {3.2         r(2) = 3.2
2   2   0.8      1.2          2.4             3.4
2   3   0.7      1.3          2.6             3.6}

3   1   0.9      2.1          4.2            {5.2         r(3) = 5.2
3   2   0.8      2.2          4.4             5.4
3   3   0.7      2.3          4.6             5.6}[/FONT]
Again, that example is contrived; it's meant only to show how the pieces of the equation go together. :cool:

thanks very much!!