HW

Project.toml

@@ -3,6 +3,13 @@ uuid = "ce1ef771-b552-5ca6-80e4-7358b8c578e2"
 authors = ["Florian Oswald <florian.oswald@gmail.com>"]
 version = "0.1.0"
 
+[deps]
+ApproxFun = "28f2ccd6-bb30-5033-b560-165f7b14dc2f"
+Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
+FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+PyPlot = "d330b81b-6aea-500a-939a-2ce795aea3ee"
+
 [extras]
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 

src/HWfuncapp.jl

@@ -4,7 +4,7 @@ using FastGaussQuadrature # to get chebyshevnodes
 
 # you dont' have to use PyPlot. I did much of it in PyPlot, hence
 # you will see me often qualify code with `PyPlot.plot` or so.
-using PyPlot  
+using PyPlot
 import ApproXD: getBasis, BSpline
 using Distributions
 using ApproxFun
@@ -13,37 +13,77 @@ using Plots
 ChebyT(x,deg) = cos(acos(x)*deg)
 unitmap(x,lb,ub) = 2 .* (x .- lb) ./ (ub .- lb) .- 1	#[a,b] -> [-1,1]
 
-function q1(n)
+function q1(n = 15)
 	f(x) = x .+ 2x.^2 - exp.(-x)
+	deg = n-1
+	lb = -3
+	ub = 3
+	node = gausschebyshev(n)[1]
+	chebnode = 0.5 .* (lb .+ ub) .+ 0.5 .* (ub .- lb) .* node
+	y = map(f, chebnode)
+	basismatrix = [cos((n - i + 0.5) * (j - 1)* π / n) for i = 1:n, j = 1:n]
+	c = basismatrix \ y
+
+	n_new = 100
+	x_2 = range(lb, stop = ub, length = n_new)
+	base2 = [ChebyT(unitmap(x_2[i], lb, ub), j) for i in 1:n_new, j = 0:deg]
+	y2 = base2 * c
+	error = y2 - f(collect(-3:6/99:3))
+	Plots.plot(x_2, [f(collect(-3:6/99:3)) error], labels = ["true_val" "error"], title = "Q1", layout = 2)
+	scatter!(x_2, y2, labels = ["approx"])
 
 	# without using PyPlot, just erase the `PyPlot.` part
-	PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q1.png"))
-	return Dict(:error=>maximum(abs,err))
+	Plots.savefig(joinpath(dirname(@__FILE__),"..","q1.png"))
+	return Dict(:error=>maximum(abs,error))
 end
 
 function q2(b::Number)
 	@assert b > 0
 	# use ApproxFun.jl to do the same:
-
-	Plots.savefig(p,joinpath(dirname(@__FILE__),"..","q2.png"))
+	b = 4
+	n_new = 100
+	f(x) = x .+ 2x.^2 - exp.(-x)
+	f2 = Fun(f,-b..b)
+	x = range(-b, b, length = n_new)
+	y2 = f2.(x)
+	y = f.(x)
+	error = y - y2
+	Plots.plot(x, [y error], labels = ["true_val" "error"], title = "Q2", layout = 2)
+	scatter!(x, y2, labels = ["approx"])
+
+	Plots.savefig(joinpath(dirname(@__FILE__),"..","q2.png"))
 end
-
+using LinearAlgebra #for cumsum
 function q3(b::Number)
-
+	b = 10
+	x = Fun(identity, -b..b)
+	f = sin(x^2)
+	g = cos(x)
+	h = f - g
+	r = roots(h)
+	p = Plots.plot(h, label = "h", title = "Q3")
+	scatter!(r,h.(r),labels="roots h")
+	Plots.savefig(joinpath(dirname(@__FILE__),"..","q3.png"))
+	Plots.savefig(joinpath(p,dirname(@__FILE__),"..","q3.png"))
+	k1 = cumsum(h)
+	k = k1 + h(-b)
+	integral = norm(h-k)
 	# p is your plot
 	return (p,integral)
 end
 
 # optinal
-function q4()
-
+function q4(n)
+	x = range(0, 1, length = n)
+	x2 = collect(0:0.01:1)
+	fig = Plots.plot(x, [ChebyT(x2, deg) for deg in 1:9]) #acos not working
+	Plots.savefig(joinpath(p,dirname(@__FILE__),"..","q4.png"))
 	return fig
 end
 
-
 # I found having those useful for q5
 mutable struct ChebyType
-	f::Function # fuction to approximate 
+	f::Function # fuction to approximate
 	nodes::Union{Vector,LinRange} # evaluation points
 	basis::Matrix # basis evaluated at nodes
 	coefs::Vector # estimated coefficients
@@ -79,7 +119,7 @@ function q5(deg=(5,9,15),lb=-1.0,ub=1.0)
 
 	runge(x) = 1.0 ./ (1 .+ 25 .* x.^2)
 
-	
+
 	PyPlot.savefig(joinpath(dirname(@__FILE__),"..","q5.png"))
 
 end
@@ -113,6 +153,4 @@ function runall()
 	q7()
 end
 
-
-
 end # module

test/runtests.jl

@@ -3,8 +3,8 @@ using Test
 
 @testset "HWfuncapp.jl" begin
 	# test that q1 with 15 nodes has an error smaller than 1e-9
-
+    @test q1(15)[:error] < 1e-9
     # test that the integral of function h [-10,0] ≈ 1.46039789878568
-
+	@test q3(10) ≈ 1.46039789878568
     @test false  # by default fail
 end