Polynomial Interpolation
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_n1, y_n1), polynomial interpolation finds a unique polynomial of degree n1 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 xcoordinates and ycoordinates 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
Stepbystep Explanation

The function first checks if the lengths of the arrays
xs
andys
are the same. If they are not, an exception is raised. 
Two arrays
c
andd
are created, which are initialized to the values in the arrayys
. 
The variable
ns
is initialized to 0, which will be used to keep track of the index of the closest point tox
. 
The variable
dif
is initialized to the absolute difference betweenx
and the first point in the arrayxs
. 
A loop is executed over all the points in the array
xs
. For each point, the absolute difference betweenx
and the point is computed. If this difference is less than the current value ofdif
, then the index of the point is stored in the variablens
, and the value ofdif
is updated to the new difference. 
The value of
y
is initialized to the ycoordinate of the point in the arrayys
that is closest tox
. 
The variable
ns
is decremented by 1. 
A loop is executed n1 times. In each iteration, two nested loops are executed over the remaining points in the array
xs
. The variablem
is used to keep track of the number of iterations. 
In the inner loop, the variable
ho
is initialized to the difference between the current point andx
, and the variablehp
is initialized to the difference between the point m positions ahead andx
. 
The variable
w
is initialized to the difference between the ycoordinates of the point m positions ahead and the current point. 
The variable
den
is initialized to the difference betweenho
andhp
. Ifden
is 0, an exception is raised. 
The variable
den
is updated tow/den
. 
The value of
c
at the current index is updated toho*den
, and the value ofd
at the current index is updated tohp*den
. 
If the index of the closest point to
x
is less thannm1
multiplied by 2, then the value ofdy
is updated toc[ns+1]
. Otherwise, the value ofdy
is updated tod[ns]
, and the index of the closest point tox
is decremented by 1. 
The value of
y
is updated toy+dy
. 
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 n1 times, and in the inner loop, it iterates over the remaining nm1 points. Therefore, the total number of iterations is given by the sum of the first n1 positive integers, which is n(n1)/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 ycoordinates and two additional variables to store the interpolated value and the error estimate. Therefore, the space complexity of the algorithm is O(n).