rmt notebooks

Author: alanedelman

notebooks/circular.jl

@@ -47,12 +47,27 @@ end
 # ╔═╡ db8c16a0-79df-4cc0-9c69-a6bcfcd23a1c
 circular(3,2)
 
-# ╔═╡ f9f00c9d-633a-4734-838f-3ffdcef6efe0
-@bind β Slider([.5,1,4,10,100,1000,1e10]; default=1, show_value=true)
+# ╔═╡ 02db4122-8e3c-477c-b1ef-d9c236787910
+smallabs(U) = minimum(abs.(angle.(eigvals(Matrix(U)))))
+
+# ╔═╡ 14b4a088-ff74-43f3-b301-4057593c0447
+function circular2(n)
+	A = randn(n,n) .+ im * randn(n,n)
+	U = qr(A).Q .* cis.(2π*rand(n))
+end
+
+# ╔═╡ 148317c1-e48f-4d39-833c-b51d2c43e5fd
+function circular1(n) #β=1
+	A = randn(n,n) .+ im * randn(n,n)
+	U = qr(A).Q * transpose( Matrix(qr(A).Q ))
+end
 
 # ╔═╡ e4418523-234c-4faa-a29b-43484013b322
 @bind n Slider(2:17; default=3, show_value=true)
 
+# ╔═╡ f9f00c9d-633a-4734-838f-3ffdcef6efe0
+@bind β Slider([.5,1,4,10,50,100,1000,1e10]; default=1, show_value=true)
+
 # ╔═╡ 63cd9672-d3d2-4b2a-8076-f98b60dc11e6
 begin
   e = eigvals(circular(n,β))
@@ -64,36 +79,23 @@ begin
   plot!(cis.(θ),label=false)
 end
 
-# ╔═╡ 02db4122-8e3c-477c-b1ef-d9c236787910
-smallabs(U) = minimum(abs.(angle.(eigvals(Matrix(U)))))
+# ╔═╡ 5cd13d1b-9bb4-4a19-9e6a-07e214572639
+begin
 
-# ╔═╡ 14b4a088-ff74-43f3-b301-4057593c0447
-function circular2(n)
-	A = randn(n,n) .+ im * randn(n,n)
-	U = qr(A).Q .* cis.(2π*rand(n))
-end
+	t = 20
 
-# ╔═╡ 148317c1-e48f-4d39-833c-b51d2c43e5fd
-function circular1(n) #β=1
-	A = randn(n,n) .+ im * randn(n,n)
-	U = qr(A).Q * transpose( Matrix(qr(A).Q ))
+ #    m = [ smallabs(circular2(n)) for i=1:t]
+	# display((mean(m),var(m)))
+ #    stephist(m,normalize=true,label="dense")
+	mh = [ smallabs(circular(n,β)) for i=1:t]
+	#display((mean(mh),var(mh)))
+	
+	stephist(mh,normalize=true,label="Hessenberg")	
+	title!("β=$β n=$n t=$t")
 end
 
-# ╔═╡ 5cd13d1b-9bb4-4a19-9e6a-07e214572639
-# ╠═╡ disabled = true
-#=╠═╡
-let
-	n = 5
-	t = 50
-    m = [ smallabs(circular2(n)) for i=1:t]
-	display((mean(m),var(m)))
-    stephist(m,normalize=true,label="dense")
-	mh = [ smallabs(circular(n,2)) for i=1:t]
-	display((mean(mh),var(mh)))
-	title!("β=2 n=$n t=$t")
-	stephist!(mh,normalize=true,label="Hessenberg")	
-end
-  ╠═╡ =#
+# ╔═╡ a6be0162-84fd-408c-b3d6-d3cee3b58b61
+
 
 # ╔═╡ 63bf73bf-b312-49cc-a909-c043bfc80fd1
 circular1(5)
@@ -1369,14 +1371,15 @@ version = "1.4.1+1"
 # ╠═fab8681e-0be3-4d97-a0cb-2ad57472282a
 # ╠═5455f492-7c41-11ef-3315-19809ebaffc3
 # ╠═db8c16a0-79df-4cc0-9c69-a6bcfcd23a1c
-# ╠═f9f00c9d-633a-4734-838f-3ffdcef6efe0
-# ╠═e4418523-234c-4faa-a29b-43484013b322
 # ╠═63cd9672-d3d2-4b2a-8076-f98b60dc11e6
 # ╠═02db4122-8e3c-477c-b1ef-d9c236787910
 # ╠═14b4a088-ff74-43f3-b301-4057593c0447
 # ╠═148317c1-e48f-4d39-833c-b51d2c43e5fd
 # ╠═2443d72e-42f4-4f66-8170-04633f9b9031
+# ╠═e4418523-234c-4faa-a29b-43484013b322
+# ╠═f9f00c9d-633a-4734-838f-3ffdcef6efe0
 # ╠═5cd13d1b-9bb4-4a19-9e6a-07e214572639
+# ╠═a6be0162-84fd-408c-b3d6-d3cee3b58b61
 # ╠═63bf73bf-b312-49cc-a909-c043bfc80fd1
 # ╠═82ce98e9-37a9-4023-ac62-5c092a8fa932
 # ╠═5e4d9a97-db38-445e-8982-7813b9656546

notebooks/geometric_sequence

@@ -0,0 +1 @@
+a,r,n  refers implicitly to  \rightarrow a,ar,ar^2,\ldots,ar^{n-1}$$ 

notebooks/geometric_sequence.jl

@@ -0,0 +1,25 @@
+# g(a,r,n) = geometric_sequence
+# refers implicitly to  →    a,ar,ar²,…, arⁿ⁻¹
+
+# state(i,t) = iteration number and current term (i.e. t = arⁱ)
+# start = state(0,i)
+# next = (i,t) → (i+1, r*t)
+# done when i=n
+
+struct geometric_sequence
+    a :: Float64  # start
+    r :: Float64  # ratio
+    n :: Int   # number of terms
+end
+
+struct geometric_sequence_state 
+    i :: Int     # iteration number
+    t :: Float64 # current term  
+end
+
+iterate(g::geometric_sequence) =  geometric_sequence_state(0,g.a) # start
+
+# next should return term and state
+function iterate(g::geometric_sequence, s::geometric_sequence_state)
+    s.t, geometric_sequence_state(s.i+1, g.r * s.t )
+end