## Introduction

The trapezoidal rule is a numerical integration technique used to approximate the definite integral of a function. It approximates the area under a curve by dividing the area into a series of trapezoids. The trapezoidal rule is widely used in engineering, physics, and other fields that require numerical integration.

## Implementation

The OCaml code below implements the trapezoidal rule algorithm. The function `trapzd` takes four arguments: `f` is the function to be integrated, `a` and `b` are the lower and upper bounds of the integral, and `n` is the number of trapezoids to be used in the approximation.

``````let trapzd f a b n =
let error () =
let s =
Printf.sprintf
"trapzd requires n > 0 and a <= b whereas n = %i, a = %g, b = %g"
n
a
b
in
Owl_exception.INVALID_ARGUMENT s
in
Owl_exception.verify (n > 0 && a <= b) error;
if n = 1
then 0.5 *. (b -. a) *. (f a +. f b)
else (
let m = 2. ** float_of_int (n - 1) in
let d = (b -. a) /. m in
let x = ref (a +. (0.5 *. d)) in
let s = ref 0. in
for _i = 1 to int_of_float m do
x := !x +. d;
s := !s +. f !x
done;
(0.5 *. d *. (f a +. f b)) +. (!s *. d))
``````

The function first checks that `n` is greater than zero and that `a` is less than or equal to `b`. If either of these conditions is not met, an exception is raised.

If `n` is equal to 1, the function returns the area of a single trapezoid. Otherwise, the function computes the width of each trapezoid `d` and the midpoint of the first trapezoid `x`. It then iterates over the remaining trapezoids, computing the area of each and summing them together.

The final result is the sum of the areas of all the trapezoids.

## Step-by-step Explanation

1. Verify that `n` is greater than zero and that `a` is less than or equal to `b`. If either of these conditions is not met, raise an exception.
2. If `n` is equal to 1, return the area of a single trapezoid: `0.5 *. (b -. a) *. (f a +. f b)`
3. Compute the width of each trapezoid: `d = (b -. a) /. m`, where `m = 2. ** float_of_int (n - 1)`
4. Compute the midpoint of the first trapezoid: `x = ref (a +. (0.5 *. d))`
5. Initialize the sum of the areas of all the trapezoids to 0: `s = ref 0.`
6. Iterate over the remaining trapezoids, computing the area of each and summing them together:
1. Increment the midpoint by the width of the trapezoid: `x := !x +. d`
2. Compute the area of the trapezoid and add it to the sum: `s := !s +. f !x`
7. Compute the final result: `(0.5 *. d *. (f a +. f b)) +. (!s *. d)`

## Complexity Analysis

The trapezoidal rule algorithm has a time complexity of O(n), where n is the number of trapezoids used in the approximation. This is because the algorithm iterates over each trapezoid once, computing its area and summing them together.

The space complexity of the algorithm is O(1), because it only uses a constant amount of memory to store the variables `d`, `x`, and `s`.