## Introduction

Polynomial interpolation is a mathematical technique used to construct a polynomial function that passes through a given set of points. Given a set of n points (x_0, y_0), (x_1, y_1), …, (x_n-1, y_n-1), polynomial interpolation finds a unique polynomial of degree n-1 or less that passes through these points. This technique is widely used in various fields such as computer graphics, numerical analysis, and signal processing.

## Implementation

The following code is an implementation of the polynomial interpolation algorithm in OCaml from Owl library. The function `polint` takes three arguments - `xs`, `ys`, and `x`. The arrays `xs` and `ys` represent the x-coordinates and y-coordinates of the n points, respectively. The variable `x` represents the point at which we want to evaluate the polynomial. The function returns a tuple of two values - the interpolated value of the polynomial at `x` and the error estimate.

``````let polint xs ys x =
let n = Array.length xs in
let m = Array.length ys in
let error () =
let s =
Printf.sprintf
"polint requires that xs and ys have the same length, but xs length is %i \
whereas ys length is %i"
n
m
in
Owl_exception.INVALID_ARGUMENT s
in
Owl_exception.verify (m = n) error;
let c = Array.copy ys in
let d = Array.copy ys in
let ns = ref 0 in
let dy = ref 0. in
let dif = ref (abs_float (x -. xs.(0))) in
for i = 0 to n - 1 do
let dift = abs_float (x -. xs.(i)) in
if dift < !dif
then (
ns := i;
dif := dift)
done;
let y = ref ys.(!ns) in
ns := !ns - 1;
for m = 0 to n - 2 do
for i = 0 to n - m - 2 do
let ho = xs.(i) -. x in
let hp = xs.(i + m + 1) -. x in
let w = c.(i + 1) -. d.(i) in
let den = ho -. hp in
assert (den <> 0.);
let den = w /. den in
c.(i) <- ho *. den;
d.(i) <- hp *. den
done;
if (2 * !ns) + 2 < n - m - 1
then dy := c.(!ns + 1)
else (
dy := d.(!ns);
ns := !ns - 1);
y := !y +. !dy
done;
!y, !dy
``````

## Step-by-step Explanation

1. The function first checks if the lengths of the arrays `xs` and `ys` are the same. If they are not, an exception is raised.

2. Two arrays `c` and `d` are created, which are initialized to the values in the array `ys`.

3. The variable `ns` is initialized to 0, which will be used to keep track of the index of the closest point to `x`.

4. The variable `dif` is initialized to the absolute difference between `x` and the first point in the array `xs`.

5. A loop is executed over all the points in the array `xs`. For each point, the absolute difference between `x` and the point is computed. If this difference is less than the current value of `dif`, then the index of the point is stored in the variable `ns`, and the value of `dif` is updated to the new difference.

6. The value of `y` is initialized to the y-coordinate of the point in the array `ys` that is closest to `x`.

7. The variable `ns` is decremented by 1.

8. A loop is executed n-1 times. In each iteration, two nested loops are executed over the remaining points in the array `xs`. The variable `m` is used to keep track of the number of iterations.

9. In the inner loop, the variable `ho` is initialized to the difference between the current point and `x`, and the variable `hp` is initialized to the difference between the point m positions ahead and `x`.

10. The variable `w` is initialized to the difference between the y-coordinates of the point m positions ahead and the current point.

11. The variable `den` is initialized to the difference between `ho` and `hp`. If `den` is 0, an exception is raised.

12. The variable `den` is updated to `w/den`.

13. The value of `c` at the current index is updated to `ho*den`, and the value of `d` at the current index is updated to `hp*den`.

14. If the index of the closest point to `x` is less than `n-m-1` multiplied by 2, then the value of `dy` is updated to `c[ns+1]`. Otherwise, the value of `dy` is updated to `d[ns]`, and the index of the closest point to `x` is decremented by 1.

15. The value of `y` is updated to `y+dy`.

16. The function returns the tuple `(y, dy)`.

## Complexity Analysis

The polynomial interpolation algorithm has a time complexity of O(n^2), where n is the number of points. This is because the algorithm involves a nested loop over all the points in the input arrays. In the outer loop, the algorithm iterates n-1 times, and in the inner loop, it iterates over the remaining n-m-1 points. Therefore, the total number of iterations is given by the sum of the first n-1 positive integers, which is n(n-1)/2. Thus, the time complexity of the algorithm is O(n^2).

In terms of space complexity, the algorithm uses two arrays of size n to store the y-coordinates and two additional variables to store the interpolated value and the error estimate. Therefore, the space complexity of the algorithm is O(n).