Added HW files for Maxime and Laura

run_LG_MCS.jl

@@ -0,0 +1,2 @@
+include("src/HWint_LG_MCS.jl")	# load our code
+HWint_LG_MCS.runall()

src/HWint_LG_MCS.jl

@@ -0,0 +1,142 @@
+
+module HWintegration
+
+	const A_SOL = 4
+	println("The analytical solution is $A_SOL")
+	# imports
+	using FastGaussQuadrature
+	using Roots
+	using Sobol
+	using Plots
+	using Distributions
+	using NullableArrays
+	using DataFrames
+	using DataStructures  # OrderedDict
+
+	# set random seed
+	srand(12345)
+
+	# demand function
+	function q(p)
+		2*p ^(-0.5)
+	end
+
+	# makes a plot for question 1
+	function plot_q1()
+	global dem = plot(q, 0.1, 5, title="Demand Function", xaxis=("Price"), yaxis=("Quantity"), labels = "q(p)")
+	hline!([2, 1], label = "", linestyle=:dash)
+	scatter!([1], [2], label = "", color = ["red"])
+	scatter!([4], [1], label = "", color = ["red"])
+	end
+
+	function question_1a(n)
+	nw = gausslegendre(n)
+	glx = 3/2*nw[1]+ 5/2
+	gly = [q(3/2*x + 5/2) for x in nw[1]]
+	int = 3/2*sum([q(3/2*x + 5/2) for x in nw[1]] .* nw[2])
+	global gl = scatter(glx, gly, labels = "Gauss-Legendre", title = "n = $n")
+	diff = A_SOL-int
+	println("The Gauss-Legendre method gives $int when using $n nodes. The error is $diff")
+	end
+
+	function question_1b(n)
+	mcx = rand(Uniform(1,4), n)
+	mcy = q.(mcx)
+	int = 3/n * sum([q(x) for x in mcx])
+	global mc = scatter(mcx, mcy, labels = "Monte-Carlo", title = "n = $n")
+	println("The Monte-Carlo method gives $int")
+	end
+
+	function question_1c(n)
+	s = SobolSeq(1)
+	qmcx = 3*[ hcat([next(s) for i = 1:n])[x][1] for x = 1:n]+1
+	qmcy = q.(qmcx)
+	int = 3/n * sum([q(x) for x in qmcx])
+	global qmc = scatter(qmcx, qmcy, labels = "Quasi Monte-Carlo", title = "n = $n")
+	println("The quasi-Monte-Carlo method gives $int")
+	end
+
+	# question 2
+
+	function question_2a()
+		function p(t = [0,0])
+			function eqm(x)
+				exp(t[1])* float(x)^(-1) + exp(t[2])*float(x)^(-0.5) - 2
+			end
+			fzero(eqm, [0,10000])
+		end
+		μ = [0,0]
+		Σ = [0.02   0.01 ; 0.01  0.02]
+		Ω = chol(Σ)
+		#rand(MvNormal(μ, Σ), n)
+		nodes = Any[]
+		push!(nodes,repeat(gausshermite(10)[1],inner=[1],outer=[10]))  # dim1
+		push!(nodes,repeat(gausshermite(10)[1],inner=[2],outer=[5]))  # dim2
+		#weights1 = kron(gausshermite(10)[2], gausshermite(10)[2])
+		dict = Dict(gausshermite(10)[1][i] => gausshermite(10)[2][i] for i in 1:10)
+		#df = hcat(DataFrame(weights=weights),DataFrame(nodes,[:dim1,:dim2]))
+		coor = [ [nodes[1][i], nodes[2][i]] for  i in 1: 100]
+		weights2 = [dict[coor[i][1]]*dict[coor[i][2]] for i in 1:length(coor)]
+		cofv = [Ω *sqrt(2)*coor[i]+ μ for i in 1:100]
+		expectation = (1/sqrt(π)  * sum(weights2 .* p.( cofv)))
+		variance = (1/sqrt(π)  * sum(weights2 .* ((p.(cofv)- expectation).^2)))
+		println("The Gauss-Hermite method for approximating integrals gives:")
+		println("For the expectation: $expectation")
+		println("For the variance: $variance")
+	end
+
+	function question_2b(n)
+		μ = [0,0]
+		Σ = [0.02   0.01 ; 0.01  0.02]
+		randompoints = rand(MvNormal(μ, Σ),  n)
+		randompoints = [[randompoints[1,i], randompoints[2,i]] for i in 1:n]
+		function p(t = [0,0])
+			function eqm(x)
+				exp(t[1])* float(x)^(-1) + exp(t[2])*float(x)^(-0.5) - 2
+			end
+			fzero(eqm, [0,10000])
+		end
+		expectation = 1/n * sum(p.( randompoints))
+		variance = 1/n * sum((p.(randompoints) - expectation).^2)
+		println("The Monte-Carlo method for approximating integrals gives:")
+		println("For the expectation: $expectation")
+		println("For the variance: $variance")
+	end
+
+	function question_2bonus()
+		println("We couldn't solve this question.")
+	end
+
+	# function to run all questions
+	function runall()
+		info("Running all of HWintegration")
+		info("Question 1:")
+		for n in (10,15,20)
+			info("Now showing results for n = $n")
+			info("Question 1.A - Gauss-Legendre:")
+			question_1a(n)
+			info("Question 1.B - Monte-Carlo")
+			question_1b(n)
+			info("Question 1.C - Quasi-Monte-Carlo")
+			question_1c(n)
+			println("")
+			plot_q1()
+			allplots = plot(dem, gl, mc, qmc, layout = (2,2))
+			display(allplots)
+			savefig("n=$n.png")
+			println("Plot for n = $n added to the current directory as a .png file.")
+		end
+		info("============================")
+		println("")
+		info("Question 2.A - Gauss-Hermite:")
+		question_2a()
+		info("Question 2.B - Monte-Carlo:")
+			for n in (10,15,20)
+			info("Now showing results for n = $n")
+			question_2b(n)
+		end
+		info("Bonus Question: Quasi Monte-Carlo:")
+		question_2bonus()
+	info("End of HWintegration")
+	end
+end