# Vocabulary/icapdot

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`I. y`Indices

Rank 1 *-- operates on lists of y, producing a list of variable length for each one --*
WHY IS THIS IMPORTANT?

The indexes of every `1` in the Boolean list `y`

I. 0 0 1 0 1 0 2 4 I. 3 1 4 > 1 5 9 NB. It's true only in position 0 0

### Common uses

1. Convert a Boolean *mask* to the list of indexes where the mask is 1

] y=: i:3 _3 _2 _1 0 1 2 3 ] cond=: y<0 1 1 1 0 0 0 0 cond # i.#y 0 1 2

2. For example, comparisons produce Boolean lists, but `{` and `}` needs lists of indexes.

To replace all atoms in `y` satisfying the Boolean condition (`y<0`) with `0`

y=: i:3 0 (I. y<0) } y NB. y<0 creates a mask, which I. converts to indexes 0 0 0 0 1 2 3

### More Information

1. ` I. y` is the same as (`y # i. # y`)

2. If an atom of `y` is greater than 1, the corresponding index is repeated.

follows from 1.

`x I. y`Interval Index

Rank Infinity *-- operates on x and y as a whole, by items of x --*
WHY IS THIS IMPORTANT?

`x` must be **in sorted order**, and thus defines (`1+#x`) intervals,
where each interval but the last **ends** on an item of `x` (inclusive) and the last interval ends at infinity.

`x I. y` gives the *index of the interval* in which `y` lies.

This is also called the *insert-before point* of `y`: the smallest index of `x` **before which** `y` could be inserted while keeping the array in order.

If `x` is in ascending order, `x I. y` is the number of atoms of `x` that are less than `y`. If `x` is in decending order, `x I. y` is the number of atoms of `x` that are greater than `y`.

x=: 0 1 2 4 8 16 32 NB. x represents the 8 intervals: NB. [__,0], (0,1], (1,2], (2,4], NB. (4,8], (8,16], (16,32], (32,_] x I. _100 NB. _100 is in (__,0] -- which has index 0 in x 0 x I. 0.9 NB. 0.9 is in (0,1] -- which has index 1 in x 1 x I. 3.9 NB. 3.9 is in (2,4] -- which has index 3 in x 3 x I. 16 NB. 16 is in (8,16] -- which has index 5 in x (exact match) 5 x I. 17 NB. 17 is in (16,32] -- which has index 6 in x 6 x I. 32 NB. 32 is in (16,32] -- which has index 6 in x 6 x I. 33 NB. 33 is in (33,_] -- which has index 7 in x 7 x I. 15 16 17 NB. 3 search terms at once 5 5 6

### Common uses

1. Find which section of a piecewise-linear function applies to a given `y`

xpieces =: 0 1 2 NB. point,slope for the interval ENDING at each x value. There is one more NB. point,slope than x positions, because the last interval ends at _ ]pointslope =: _2 ]\ 0 0 0 1 1 2 3 3 0 0 0 1 1 2 3 3 xpieces I. 0.5 5 NB. Which interval applies to these values? 1 3

### More Information

1. `x I. y` performs *binary search* in `x`.
**It it vital that the items of x be in sorted order**, or the search will give erroneous results.

2. `x` must be in sorted order, but it can be either ascending or descending order!
If all the items of `x` are identical, they are assumed to be in "ascending" order.

3. `x I. y` is **not** a member of the i.-family, but it does use the concept of **internal rank**.
The rank used is the rank of items of `x`.

4. The comparisons during execution of `x I. y` are exact (i.e. intolerant).

5. For arguments other than numeric lists, the rules for sorting define the proper insertion point of `y`.