astrojhgu / adaptrapezoid_benchmark

benchmark languages that potentially suitable to scientific computing with the adaptive trapezoid numeric integration algorithm

A benchmark of different languages that are potentially suitable to scientific computing with the adaptive trapezoid integration algorithm.

Currently implemented languages: Rust, C++, Scala, Haskell, Python, C#, C

Directories

1. rs: Rust
2. cpp: C++
3. hs: Haskell
4. py: Python
5. scala: Scala
6. cs: C#
7. c: C

Benchmark results

Result.md.

Behcnmark methods for each language are listed in Benchcmd.md.

Description to the algorithm

The adaptive trapezoid quadrature method (i.e., the definite integration) works by dividing the integration interval iteratively (or in other words, recursively) and approximate the result by the summing areas of trapezoids of all the intervals.

The detailed algorithm is

1. Setting up the function F to be integrated, setting the tolerant value eps.
2. Initialize a set (more than 1) of initial ticks, which defines the initial intervals. The x values of the initial ticks should be in increasing order. Say a set of points with X's=x_i (i=0,1,2,..n-1), then the corresponding initial intervals are [x_0, x_1], [x_1, x_2], [x_2, x_3], ...[x_{n-2}, x_{n-1}]
3. The result of any interval [x_1, x_2] is calculated as I(F, x_1, x_2), which is defined in following algorithm.
4. Sum up the areas of each sub-intervals by using the (Neumaier Sum algorithm)[https://en.wikipedia.org/wiki/Kahan_summation_algorithm#Alternatives].

The definition of I(F, x_1, x_2) is

1. Calculate the diff=T(F, x_1, x_2)-T(F, x_1, (x_1+x_2)/2)-T(F, (x_1+x_2)/2, x_2), where T(F, a, b)=(F(a)+F(b))*(b-a)/2.
2. If diff<eps/W*(x_2-x_1), go to 3, otherwise go to 4 where W is the width of the whole initial integration interval.
3. return T(F, x_1, (x_1+x_2)/2)+T(F, (x_1+x_2)/2, x_2).
4. return I(F, x_1, (x_1+x_2)/2)+I(F, (x_1+x_2)/2, x_2).

The integrand function and the precise result

\int_0^{\sqrt{\pi/8}} \sin(x^2)dx=\sqrt{\pi/4}S(4).

The numeric result is 0.527038339761566009286263102166809763899326865179511011538

according to Mathematica alpha