Math.Log vs multiplication complexity in terms of geometric average which is better?

Question

I want to find geometric average of data and performance does matters.
Which one should I pick between

1. Keep multiplication over single variable and take Nth-root at the end of calculation

   X = MUL(x[i])^(1/N)  
   

   Thus, O(N) x Multiplication Complexity + O(1) x Nth-root

2. Use logarithm

   X = e ^ { 1/N * SUM(log(x[i])) }  
   

   Thus, O(N) x Logarithm Complexity + O(1) x Nth-division + O(1) Exponential

3. Specialized algorithm for geometric average. Please tell me if there is.

Answer 1

I thought I would try to benchmark this and get a comparison, here is my attempt.

Comparing was difficult since the list of numbers needed to be large enough to make timing it reasonable, so N is large. In my test N = 50,000,000 elements.

However, multiplying lots of numbers together which are greater than 1 overflows the double storing the product. But multiplying together numbers less than 1 gives a total product which is very small, and dividing by the number of elements gives zero.

Just a couple more things: Make sure none of your elements are zero, and the Log approach doesn't work for negative elements.

(The multiply would work without overflow if C# had a BigDecimal class with an Nth root function.)

Anyway, in my code each element is between 1 and 1.00001

On the other hand, the log approach had no problems with overflows, or underflows.

Here's the code:

class Program
{
    static void Main(string[] args)
    {
        Console.WriteLine("Starting...");
        Console.WriteLine("");

        Stopwatch watch1 = new Stopwatch();
        Stopwatch watch2 = new Stopwatch();

        List<double> list = getList();

        double prod = 1;

        double mean1 = -1;

        for (int c = 0; c < 2; c++)
        {
            watch1.Reset();

            watch1.Start();

            prod = 1;

            foreach (double d in list)
            {
                prod *= d;
            }

            mean1 = Math.Pow(prod, 1.0 / (double)list.Count);

            watch1.Stop();

        }

        double mean2 = -1;

        for (int c = 0; c < 2; c++)
        {
            watch2.Reset();

            watch2.Start();

            double sum = 0;

            foreach (double d in list)
            {
                double logged = Math.Log(d, 2);
                sum += logged;
            }

            sum *= 1.0 / (double)list.Count;

            mean2 = Math.Pow(2.0, sum);

            watch2.Stop();

        }
        Console.WriteLine("First way gave: " + mean1);
        Console.WriteLine("Other way gave: " + mean2);

        Console.WriteLine("First way took: " + watch1.ElapsedMilliseconds + " milliseconds.");
        Console.WriteLine("Other way took: " + watch2.ElapsedMilliseconds + " milliseconds.");

        Console.WriteLine("");
        Console.WriteLine("Press enter to exit");
        Console.ReadLine();
    }

    private static List<double> getList()
    {
        List<double> result = new List<double>();

        Random rand = new Random();

        for (int i = 0; i < 50000000; i++)
        {
            result.Add( rand.NextDouble() / 100000.0 + 1);
        }

        return result;
    }
}

My computer output describes that both geometric means are the same, but that:

Multiply  way took: 466 milliseconds
Logarithm way took: 3245 milliseconds

So, the multiply appears to be faster.

But multiply is very problematic with overflow and underflow, so I would recommend the Log approach, unless you can guarantee the product won't overflow and that the product won't get too close to zero.