Simpson's Methods

Before explaining what the Simpson's methods are used for and give an example, it is necessary to give the definition of numerical method.

A numerical method is a mathematical technique used for solving mathematical problems that cannot be solved or are difficult to solve analytically. To solve a problem analytically is to give an exact answer in the form of a mathematical expression.

In other words, a numerical method is an algorithm that converges to a solution that approximates to the exact answer. This solution is called numerical solution.

So, let's start with the main topic. Simpson's methods are used for approximating the value of an integral I( f ) of a function f(x) over an interval from a to b using quadratic (Simpson's 1/3 method) and cubic (Simpson's 3/8 method) polynomials. These methods are used when analytical integration is difficult or not possible, and when the integrand is given as a set of discrete points.

Simpson's 1/3 method uses a quadratic polynomial to approximate the integrand. We need three points to determine the coefficients of this polynomial. These points are x₁ = a, x₃ = b and x₂ = (a+b)/2

The name 1/3 in the method comes from the factor in the expression.

If you want a more accurate evaluation of the integral with this method, you can use the composite Simpson's 1/3 method in which you must divide the whole interval into n subintervals using an even number, because Simpson's 1/3 method needs three points for defining a quadratic polynomial, that means that this method applies two adjacent subintervals at a time.

Where, the subintervals n must be equally space. And h = (b-a)/n

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Simpson's 3/8 method uses a cubic polynomial to approximate the integrand. We need four points to determine the coefficients of this polynomial. These points are x₁ = a, x₂ = a+h, x₃ = a+2 h and x₄ = b

The name 3/8 in the method comes from the factor in the expression.

If you want a more accurate evaluation of the integral with this method, you can use the composite Simpson's 3/8 method in which you have to divide the whole interval into a number n of subintervals that is divisible by 3, because Simpson's 3/8 method need four points for constructing a cubic polynomial, that mean that this method applies three adjacent subintervals at a time.

Where, the subintervals n must be equally space. And h = (b-a)/n

These methods are applied in the real world to calculate areas, volumes, curve lengths and other problems related to integrals.

For example: the company ECO wants to drain and fill a polluted marsh (see the image below) that has a depth of 5 feet. The CEO of ECO wants to know how many cubic feet of land are needed to fill the area after draining the marsh.

To solve this problem I used the composite Simpson's 1/3 method.

fn simpson(a: f64, b: f64, n: i32) -> f64 {
    let mut y: f64 = funcion(a) + funcion(b);
    let mut x: f64 = a;
    let h: f64 = (b-a)/(n as f64);

    for i in 1..n {
      x = x + h;
      if i % 2 == 0{
        y= y + 2.0*funcion(x);
      }else{
        y= y + 4.0*funcion(x);
      }
    }

    return (b-a) * y / (3.0*(n as f64));
}

Solution: To calculate the volume of the marsh, we must first estimate the surface area using the composite Simpson's 1/3 method.

let mut resultado: i32 = simpson(vec![146, 122, 76, 54, 40, 30, 13], 20);

fn simpson(v: Vec<i32>, h: i32) -> i32 {
    let mut y: i32 = 0;
    let mut con: i32 = 0;
    let size: usize = v.len()-1;

    for i in v {
      if con == 0{
        y = y + i;
      }else if con == (size as i32) {
        y = y + i;
      }else{
        if con % 2 == 0{
          y= y + 2*i;
        }else{
          y= y + 4*i;
        }
      }
      con = con +1;
    }

    return h*y/3;
}

And finally multiply by 5.

resultado = resultado * 5;

Obtaining that the approximate volume is 40 500 cubic feet.

In conclusion, integration with numerical methods is a useful technique when we try to integrate a complicated function or if we only have tabulated data. With the Simpson's methods we can approximate a complex integral to the integral of a polynomial and obtain a solution that approximates the exact answer, even in some cases we can obtain the exact answer.

References

• Gilat A., Subramaniam, V. (2013). Numerical methods for engineers and scientists. Wiley. Third Edition.
• Heath M., (2002). Scientific Computing: An Introductory Survet. McGraw Hill. Second Edition.
• Chapra, S., Canale, R. (2011). Métodos Numéricos para ingenieros. McGraw Hill. Sexta edición.

Additional resources

• Numerical Methods repository in GitHub, by LindaEliza.