Update toeplitz.jl

Author: dlfivefifty

examples/toeplitz.jl

@@ -1,5 +1,15 @@
+####
+# This file shows how the ∞-dimensional QL decomposition can be used
+# for spectral theory of Toeplitz operators. We investigate the
+# QL decomposition on each of the examples in Trefethen & Embree.
+#
+
+
 using InfiniteLinearAlgebra, BandedMatrices, PyPlot
 
+###
+# Basic routines for plotting
+###
 
 function ℓ11(A,λ; kwds...) 
     try 
@@ -9,9 +19,11 @@ function ℓ11(A,λ; kwds...)
     end
 end
 
-function findsecond(λ) 
-    j = sortperm(λ)[end-1]
-    λ[j], j
+function branch(k)
+    function(λ)
+        j = sortperm(λ)[end-k+1]
+        λ[j], j
+    end
 end
 
 qlplot(A::AbstractMatrix; kwds...) = qlplot(BandedMatrix(A); kwds...)
@@ -39,7 +51,7 @@ end
 ###
 
 A = BandedMatrix(1 => Fill(2im,∞), 2 => Fill(-1,∞), 3 => Fill(2,∞), -2 => Fill(-4,∞), -3 => Fill(-2im,∞))
-clf();qlplot(A; x=range(-10,7; length=100), y=range(-7,8;length=100)); symbolplot(A; color=:black); title("Trefethen & Embree")
+clf(); qlplot(A; x=range(-10,7; length=100), y=range(-7,8;length=100)); symbolplot(A; color=:black); title("Trefethen & Embree")
 clf(); qlplot(transpose(A); x=range(-10,7; length=200), y=range(-7,8;length=200)); symbolplot(A; color=:black); title("Trefethen & Embree, transpose")
 
 ###
@@ -52,17 +64,49 @@ clf(); qlplot(A; branch=findsecond, x=range(-2,3; length=100), y=range(-2.5,2.5;
 clf(); qlplot(transpose(A); x=range(-2,3; length=100), y=range(-2.5,2.5;length=100)); symbolplot(A; color=:black); title("Limacon, transpose")
 
 
-ql(A-(0.5+0.000001im)*I; branch=findsecond)
-
 ### 
 # bull-head
 ###
 
 A = BandedMatrix(-3 => Fill(7/10,∞), -2 => Fill(1,∞), 1 => Fill(2im,∞))
-clf();qlplot(A; x=range(-10,7; length=100), y=range(-7,8;length=100)); symbolplot(A; color=:black); title("Bull-head")
+clf(); qlplot(A; x=range(-10,7; length=100), y=range(-7,8;length=100)); symbolplot(A; color=:black); title("Bull-head")
 clf(); qlplot(transpose(A); x=range(-10,7; length=100), y=range(-7,8;length=100)); symbolplot(A; color=:black); title("Bull-head, transpose")
 
-
 ###
 # Grcar
 ###
+
+A = BandedMatrix(-3 => Fill(1,∞), -2 => Fill(1,∞), -1 => Fill(1,∞), 0 => Fill(1,∞), 1 => Fill(-1,∞))
+clf(); qlplot(A; x=range(-4,5; length=100), y=range(-6,5;length=100)); symbolplot(A; color=:black); title("Grcar")
+clf(); qlplot(A; branch=branch(2), x=range(-4,5; length=100), y=range(-6,5;length=100)); symbolplot(A; color=:black); title("Grcar, branch 2")
+clf(); qlplot(A; branch=branch(3), x=range(-4,5; length=100), y=range(-6,5;length=100)); symbolplot(A; color=:black); title("Grcar, branch 3")
+clf(); qlplot(transpose(A); x=range(-4,5; length=100), y=range(-6,5;length=100)); symbolplot(A; color=:black); title("Grcar, transpose")
+
+###
+# Triangle
+###
+
+A = BandedMatrix(-2 => Fill(1/4,∞), 1 => Fill(1,∞))
+clf(); qlplot(A; x=range(-2,2; length=100), y=range(-2,2;length=100)); symbolplot(A; color=:black); title("Triangle")
+clf(); qlplot(transpose(A); x=range(-2,2; length=100), y=range(-2,2;length=100)); symbolplot(A; color=:black); title("Triangle, transpose")
+clf(); qlplot(transpose(A); branch=branch(2), x=range(-2,2; length=100), y=range(-2,2;length=100)); symbolplot(A; color=:black); title("Triangle, transpose, branch 2")
+
+###
+# Whale
+###
+
+A = BandedMatrix(-4 => Fill(im,∞), -3 => Fill(4,∞), -2 => Fill(3+im,∞), -1 => Fill(10,∞), 
+                    1 => Fill(1,∞), 2 => Fill(im,∞), 3 => Fill(-(3+2im),∞), 4=>Fill(-1,∞))
+
+clf(); qlplot(A; x=range(-15,20; length=100), y=range(-20,20;length=100)); symbolplot(A; color=:black); title("Whale")
+clf(); qlplot(transpose(A); x=range(-15,20; length=100), y=range(-20,20;length=100)); symbolplot(A; color=:black); title("Whale, transpose")
+
+###
+# Butterfly
+###
+
+A = BandedMatrix(-2 => Fill(1,∞), -1 => Fill(-im,∞), 1 => Fill(im,∞), 2 => Fill(-1,∞))
+clf(); qlplot(A; x=range(-3,3; length=100), y=range(-3,3;length=100)); symbolplot(A; color=:black); title("Butterfly")
+clf(); qlplot(A; branch=branch(2), x=range(-3,3; length=100), y=range(-3,3;length=100)); symbolplot(A; color=:black); title("Butterfly, branch 2")
+clf(); qlplot(transpose(A); x=range(-3,3; length=100), y=range(-3,3;length=100)); symbolplot(A; color=:black); title("Butterfly")
+clf(); qlplot(transpose(A); branch=branch(2), x=range(-3,3; length=100), y=range(-3,3;length=100)); symbolplot(A; color=:black); title("Butterfly")