mech-math 0.3.3

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

%% Ceiling Function

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

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

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

Computes the smallest integer greater than or equal to each element of `X`. 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. For complex inputs, `ceil` is applied separately to the real and imaginary parts. |

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

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Ceiling of the input values. The result is an integer-valued float. The shape of `Y` matches the shape of `X`. |

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

(a) Ceiling of a positive non-integer

```mech:ex1
y := math/ceil(2.3)
```

(b) Ceiling of a negative non-integer

```mech:ex2
y := math/ceil(-2.7)
```

(c) Ceiling of a vector

```mech:ex3
x := [1.2, 2.8, -3.1]
y := math/ceil(x)
```

(d) Ceiling of a matrix

```mech:ex4
x := [1.1, -2.5; 3.9, -4.2]
y := math/ceil(x)
```

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

The ceiling function maps a real number `x` to the smallest integer `n` such that

$$ n \geq x, \,\, n \in \mathbb{Z}

For example:

- `ceil(2.3) = 3`
- `ceil(-2.7) = -2`

This function is useful in rounding operations, discretization, and ensuring values are large enough for indexing or allocation. For arrays, the operation is applied element-wise.

For complex numbers, the ceiling is usually applied independently to the real and imaginary parts:

$$ ceil(a + bi) = ceil(a) + ceil(b)i