starting code for computing Lebesgue constant

Lebesgue2D/DerivLebesgue2D.m

@@ -0,0 +1,58 @@
+
+function maxleb = Lebesgue2D(N, r, s, Nsamples, Nlevels)
+
+  V = Vandermonde2D(N, r, s);
+
+[Dr,Ds] = Dmatrices2D(N, r, s, V);
+maxleb = norm(Dr);
+ return
+
+  lcoeffs = inv(V);
+
+  M = Nsamples;
+
+  % randomly sample lebesgue function
+  L1 = rand(M,1);
+  L2 = rand(M,1);
+  L3 = 1-L1-L2;
+  ids = find(L3>0);
+  L1 = L1(ids);
+  L2 = L2(ids);
+  L3 = L3(ids);
+
+  h = 1;
+
+  for cnt=1:Nlevels
+    f = derivlebfn2d(N, lcoeffs, L1, L2, L3);
+
+    [fsort, ids] = sort(-f);
+    ids = ids(1:(M/5));
+
+    L1 = L1(ids);
+    L2 = L2(ids);
+    L3 = L3(ids);
+
+    % now perturb again
+    h = h/2;
+    L1new = L1*ones(1, 20) + h*(2*rand(length(L1),20)-1);
+    L2new = L2*ones(1, 20) + h*(2*rand(length(L2),20)-1);
+    L3new = 1-L1new-L2new;
+
+    L1 = [L1(1:100);L1new(:)];
+    L2 = [L2(1:100);L2new(:)];
+    L3 = [L3(1:100);L3new(:)];
+
+    ids = find( (L1+L2+L3)<=1  & L1>=0 & L2>=0 & L3>=0);
+    L1 = L1(ids);
+    L2 = L2(ids);
+    L3 = L3(ids);
+
+  end
+
+
+  f = derivlebfn2d(N, lcoeffs, L1, L2, L3);
+
+  [fsort, ids] = sort(-f);
+  ids = ids(1);
+  maxleb = f(ids(1))
+

Lebesgue2D/Lebesgue2D.m

@@ -0,0 +1,54 @@
+
+function maxleb = Lebesgue2D(N, r, s, Nsamples, Nlevels)
+
+  V = Vandermonde2D(N, r, s);
+
+  lcoeffs = inv(V);
+
+  M = Nsamples;
+
+  % randomly sample lebesgue function
+  L1 = rand(M,1);
+  L2 = rand(M,1);
+  L3 = 1-L1-L2;
+  ids = find(L3>0);
+  L1 = L1(ids);
+  L2 = L2(ids);
+  L3 = L3(ids);
+
+  h = 1;
+
+  for cnt=1:Nlevels
+    f = lebfn2d(N, lcoeffs, L1, L2, L3);
+
+    [fsort, ids] = sort(-f);
+    ids = ids(1:(M/5));
+
+    L1 = L1(ids);
+    L2 = L2(ids);
+    L3 = L3(ids);
+
+    % now perturb again
+    h = h/2;
+    L1new = L1*ones(1, 20) + h*(2*rand(length(L1),20)-1);
+    L2new = L2*ones(1, 20) + h*(2*rand(length(L2),20)-1);
+    L3new = 1-L1new-L2new;
+
+    L1 = [L1(1:100);L1new(:)];
+    L2 = [L2(1:100);L2new(:)];
+    L3 = [L3(1:100);L3new(:)];
+
+    ids = find( (L1+L2+L3)<=1  & L1>=0 & L2>=0 & L3>=0);
+    L1 = L1(ids);
+    L2 = L2(ids);
+    L3 = L3(ids);
+
+  end
+
+
+  f = lebfn2d(N, lcoeffs, L1, L2, L3);
+
+  [fsort, ids] = sort(-f);
+  ids = ids(1);
+  maxleb = f(ids(1))
+

Lebesgue2D/lebfn2d.m

@@ -0,0 +1,15 @@
+
+function f = lebfn2d(N, lcoeffs, L1, L2, L3) 
+
+ X = -1*L2 +  1*L3  + -1*L1;
+ Y = -1*L2 + -1*L3  +  1*L1;
+
+ vdm  = Vandermonde2D(N, X, Y);
+
+ fn = vdm*lcoeffs;
+
+ f = sum(abs(fn'));     
+
+
+
+