J Labs

Iteration and the Power Operator

(1 of 10) ITERATION

The repeated application of a function, such as doubling or halving, is called iteration. For example:

(2 of 10) POWER OPERATOR

The power operator ^: can be used to produce any specified number of iterations. For example:

(3 of 10) POWER OPERATOR (ctd)

We will illustrate the use of the power operator on other functions, including the square (^&2), subtotals (+/), and permutations or anagrams (k&A.) .

Further information on any of these expressions may be found by pressing the function key F1 to display the vocabulary, and then clicking the mouse on the desired item:

(4 of 10) POWER OPERATOR (ctd)

The larger powers in the table ^&2 ^: i i appeared in "scientific" or exponential form. Such large numbers may be produced in "extended precision" as illustrated below:

(5 of 10) BINOMIAL COEFFICIENTS

A table of binomial coefficients is often presented as a triangle (Pascals triangle) by suppressing the zeros that result from the number of ways that n elements can be chosen from a lesser number of items. Thus:

```1```
```1 1```
```1 2 1```
```1 3 3 1```
```1 4 6 4 1```

A row of these coefficients can be simply obtained from the preceding row, as illustrated below:

(6 of 10) BINOMIAL COEFFICIENTS (ctd)

A function for this process may be defined and iterated as follows:

(7 of 10) BINOMIAL COEFFICIENTS (ctd)

The operations of matrix algebra can be applied to this matrix of binomial coefficients in useful and interesting ways, as developed in a companion lab. For example, the inverse yields a table of alternating binomial coefficients:

(8 of 10) NEGATIVE POWERS

Negative powers of a function yield powers of its inverse. For example:

(9 of 10) LIMITS

An iteration may approach a limiting value such that further applications of the function produce no discernible change. The cosine applied to 1 has such a property:

(10 of 10) LIMITS (ctd)

Since _ denotes infinity, the expression Cos ^: _ signifies an infinite number of iterations, but terminates automatically when two successive results agree. Consequently, the result y agrees with Cos y, and is therefore a solution of the equation expressed by the final line.

End of Lab