## Introduction

The Romberg integration is an iterative method to approximate the definite integral of a function. The method is named after the German mathematician W. Romberg, who introduced it in 1955. It is a combination of the trapezoidal rule and Richardson extrapolation. The trapezoidal rule is used to estimate the integral, and Richardson extrapolation is used to improve the accuracy of the approximation.

## Implementation

The implementation of the Romberg integration algorithm is provided in the OCaml programming language. The function takes four parameters: the function to integrate, the lower and upper limits of integration, and two optional parameters for the maximum number of iterations and the desired accuracy. The function returns the estimated value of the definite integral.

``````let romberg ?(n = 20) ?(eps = 1e-6) f a b =
let s = Array.make (n + 1) 0. in
let h = Array.make (n + 2) 1. in
let rss = ref 0. in
let k = 5 in
(try
for i = 0 to n - 1 do
s.(i) <- trapzd f a b (i + 1);
if i >= k
then (
let s' = Array.sub s (i - k) k in
let h' = Array.sub h (i - k) k in
let ss, dss = Owl_maths_interpolate.polint h' s' 0. in
assert (abs_float dss > eps *. abs_float ss));
h.(i + 1) <- 0.25 *. h.(i)
done
with
| _ -> ());
``````

## Step-by-step Explanation

1. The function `romberg` takes four parameters: the function `f` to integrate, the lower limit of integration `a`, the upper limit of integration `b`, and two optional parameters `n` and `eps`.
2. The function initializes an array `s` of size `n+1` and an array `h` of size `n+2`.
3. The variable `rss` is a reference to a float value initialized to zero.
4. The variable `k` is set to 5.
5. The function uses a `try`-`with` block to handle any exceptions that may occur during the execution of the loop.
6. The loop runs from 0 to `n-1`.
7. In each iteration, the function `trapzd` is called to estimate the integral using the trapezoidal rule.
8. The result of the trapezoidal rule is stored in the array `s`.
9. If the current iteration is greater than or equal to `k`, the function performs Richardson extrapolation to improve the accuracy of the approximation.
10. The last `k` elements of the array `s` and `h` are used to perform the extrapolation.
11. The result of the extrapolation is stored in the variable `rss`.
12. The function checks if the error estimate is less than the desired accuracy. If not, it raises an assertion error.
13. The variable `h` is updated for the next iteration.
14. The loop ends, and the function returns the estimated value of the definite integral.

## Complexity Analysis

The Romberg integration algorithm has a time complexity of O(n^2), where n is the maximum number of iterations. This is because the algorithm requires the evaluation of the function f at 2^(i+1)-1 points in each iteration i. Therefore, the total number of function evaluations is approximately (n/2)^2.

The space complexity of the algorithm is O(n), which is the size of the arrays s and h used to store the intermediate results. However, the space complexity can be reduced to O(1) by reusing the same memory locations for the arrays s and h in each iteration.

The accuracy of the algorithm can be controlled by the parameter eps, which represents the desired relative error. The algorithm terminates when the relative error estimate is less than eps or when the maximum number of iterations is reached. The choice of the parameters n and eps depends on the specific problem and the desired level of accuracy.