README.md

Homework Assignment 13

Gauss quadrature is a clever procedure for numerically evaluating definite integrals. Gauss quadrature gets more accurate as the number (order) of quadrature points increases.

Your assignment is to create a parallel scheme with MPI.jl, using only Send, Isend, Recv, and Gather/Gather!/Gatherv! calls, that can be used to generate a table of the number of Gauss points and the corresponding value of the integral from 1 to 50. To compute the value of an integral with an integrand defined by f(x) we'll use the FastGaussQuadrature package as follows

using FastGaussQuadrature

ξ, w = gausslegendre(int_order)
x = (b - a) / 2 .* ξ .+ (a + b) / 2
value = sum(w .* f.(x)) * (b - a) / 2

where a and b are the lower and upper bounds of the definite integral respectively, value is the computed value of the integral, and int_order is the integration order.

You should not use Scatter to evenly distribute an array of the integration orders, because the higher orders are more computationally expensive and this method will cause most of the work to be done on the highest numbered rank, leaving the others unutilized; but, instead, try to design a program that will keep all ranks busy computing the integrals until they are all complete. One idea is to use a boss/worker model, where the rank 0 processor just serves to hand out work to other processors when they are not busy.

Complete the function generate_my_table(f::Function, limits::NTuple{2, Real}, N::Integer, comm::MPI.Comm) in assignment13.jl. Each rank that is assigned work should store a [int_order, value] pair as a row in the Vector{SVector{2, Float64}} that has already been allocated for you to push! updates to, where each row will be SVector[ int_order, value ].

generate_my_table takes a function defining the integral, the integration limits, the total number of integration points in the table, and an MPI communicator as arguments. See the tests if you need assistance in using the function.

When the script is executed with the following command in the Terminal application from the root of the assignment repository with

mpiexecjl --project=. -np 2 julia -e 'using assignment13; run_parallel(x -> x^2, (0, 1), 5) |> print'

it will print the table for the integrand $f(x) = x^2$ integrated from 0 to 1 to the screen for 5 orders of Guass integration.

If you haven't already done so, please run the following commands from the Julia REPL one time before starting your assignment to install the project aware version of mpiexec.

using MPI; MPI.install_mpiexecjl()

Testing

To see if your answer is correct, run the following command at the Terminal command line from the repository's root directory

julia --project=. -e "using Pkg; Pkg.test()"

the tests will run and report if passing or failing.