# Numerical tools


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Continue from [Rearranging and transforming
arrays](06_rearranging-arrays.ipynb).

## Bit operations

### `ᵇ` (Bitwise)

In binary, the digit positions have values 1, 2, 4, and so on, counting
from the right. The digits are 0 or 1, called bits. Six is binary `110`
and three is `011`. Bitwise applies a Boolean function separately to
each bit. `∧` keeps a 1 only where both have a 1, giving binary `010`,
or two:

``` bpl
6∧ᵇ3
```

    2ₓ

``` bpl
6∨ᵇ3
```

    7ₓ

``` bpl
6≠ᵇ3
```

    5ₓ

## Mathematical tools

### `⍭` (Primes)

A prime is an integer greater than 1 divisible only by itself and 1.
`⍭n` gives the prime with `n` primes before it. Position 0 is the prime
2:

``` bpl
⍭⍳8ₓ
```

    [2 3 5 7 11 13 17 19]ₓ

On a prime, the inverse gives its position. It counts the primes
strictly below its argument, including when the argument is not itself
prime:

``` bpl
⍭⁻¹11
⍭⁻¹10
```

    4ₓ

    4ₓ

A left argument chooses another prime operation. Code 1 tests whether a
number is prime; code 2 gives the next prime:

``` bpl
1⍭1 2 3 4 5 6
```

    $f $t $t $f $t $f

``` bpl
2⍭10 11 12
```

    [11 13 13]ₓ

### `⨸` (Factors)

A number’s prime factors are primes whose product is that number.
Factors gives them in ascending order, with repetitions. Multiplying
them recovers the number:

``` bpl
⨸360ₓ
```

    [2 2 2 3 3 5]ₓ

``` bpl
×/⨸360ₓ
```

    360ₓ

Factors’ inverse multiplies the factor list back into the number.

``` bpl
⨸⁻¹[2 2 2 3 3 5]ₓ
```

    360ₓ

An exponent counts the copies of one prime factor in the product. A left
argument of `¯∞` gives a table of distinct factors above their
exponents:

``` bpl
¯∞⨸360ₓ
```

    [2 3 5
     3 2 1]ₓ

### `⌻` (Polynomials)

A polynomial is a sum of multiples of nonnegative integer powers, such
as 1 + 2x + 3x². The multipliers `1 2 3` are its coefficients, starting
with the constant term: the part without x. Fixing that left argument
makes a polynomial function:

``` bpl
p←[1 2 3]⌻
p[0 1 2]
```

    1 6 17

Derivative applies to this function too:

``` bpl
p∂2
```

    14

A root is an input where the polynomial is zero. With one argument, `⌻`
gives the coefficient of the highest power and the approximate roots.
For x² − 3x + 2, that coefficient is 1 and the roots are 1 and 2:

``` bpl
⌻2 ¯3 1
```

    [1 [1 2]]

The inverse rebuilds the coefficients from the highest-power coefficient
and roots.

``` bpl
⌻⁻¹[1 [1 2]]
```

    2 ¯3 1

### `⇄` (Optimize)

Optimize searches for the point where a function is least, starting from
its argument. The squared distance from 2 is least at 2:

``` bpl
cost←{(⍵-2)²}
cost⇄10
```

    2

With `"solve":$t`, it instead searches for a zero. This finds the
positive square root of 2, starting from 1:

``` bpl
["solve":$t] {⍵²-2}⇄ 1
```

    1.414213562

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