% sing_diff_mat_01.m - analyzing singularity of the
% differentiation matrix
% Setting up the problem for a specific value of N;
N = 2.^5;
N = 8;
h = 2*pi/N; x = -pi + (1:N)'*h;
u = exp(sin(x)); uprime = cos(x).*u;
% Construct sparse second-order differentiation matrix;
e = ones(N,1);
D = sparse(1:N,[2:N 1],e/2.0,N,N);
D = (D-D')/h;
D
full(D)
% Eigenvalues and eigenvectors of the original differentiation matrix;
[V, E] = eig(full(D));
[e, i] = min(abs(diag(E)));
diag(E)
V1 = V(:,i);
V2 = V(:,i+1);
figure(1); plot( (1:N), V1, (1:N), V2 );
null(full(D))
size(null(full(D)),2)
% return;
% Removing the singularity - assume that f_1 and f_2 have to be specified
D1 = D;
D1(1:2,:) = 0.0;
D1(1,1) = 1.0;
D1(2,2) = 1.0;
% The new condition number, eigenvalues and eigenvectors;
condest(D1)
[V, E] = eig(full(D1));
diag(E)
% Dependence of the conditioning of the desingularized matrix on N;
Nvec = 2.^(3:10);
for N = Nvec
h = 2*pi/N; x = -pi + (1:N)'*h;
u = exp(sin(x)); uprime = cos(x).*u;
% Construct sparse second-order differentiation matrix:
e = ones(N,1);
D = sparse(1:N,[2:N 1],e/2.0,N,N);
D = (D-D')/h;
D1 = D;
D1(1:2,:) = 0.0;
D1(1,1) = 1.0;
D1(2,2) = 1.0;
K = condest(D1);
LL = abs(svd(full(D1)));
S1 = min(LL);
S2 = max(LL);
figure(2);
loglog(N,K,'x',N,S1,'o',N,S2,'s'), hold on
end
hold off
legend('cond number','min sval','max sval')