mech-math 0.3.3

Math library for the Mech language
Documentation 
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math/cbrt
===============================================================================

%% Cube Root Function

1. Usage
-------------------------------------------------------------------------------

```mech:disabled
Y := math/cbrt(X)
```

2. Description
-------------------------------------------------------------------------------

Computes the cube root of each element of `X`. Unlike the square root, the cube root is defined for all real numbers, including negative values. The result `Y` has the same shape as the input `X`.

3. Input
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `X`      | `float`, `[float]`       | Input value(s). Can be real or complex. If `X` is complex, `cbrt` returns complex results. |

4. Output
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Cube root of the input values. If `X` is real, `Y` is real. If `X` is complex, `Y` may have both real and imaginary parts. The shape of `Y` matches the shape of `X`. |

5. Examples  
-------------------------------------------------------------------------------

(a) Compute the cube root of a positive number

```mech:ex1
y := math/cbrt(8)
```

(b) Compute the cube root of a negative number

```mech:ex2
y := math/cbrt(-27)
```

(c) Compute cube root for a vector

```mech:ex3
x := [1, 8, 27]
y := math/cbrt(x)
```

(d) Compute cube root for a matrix

```mech:ex4
x := [1, -8; 27, -64]
y := math/cbrt(x)
```

6. Details
-------------------------------------------------------------------------------

The cube root function is defined as:

$$ cbrt(x) = x^{1/3}

For real numbers, the cube root is uniquely defined for all `x` (unlike the square root, which is undefined for negative reals). For negative values of `x`, the cube root is also negative.

For complex numbers, the cube root has three possible values (since complex roots are multi-valued). By convention, most implementations return the **principal cube root**, which has the smallest non-negative argument (angle in the complex plane).

This function can be applied element-wise to vectors and matrices, preserving the input shape.