$\circ$First-order differentiation matrix. Accurate eigenvalue approximations can be obtained even when continuum boundary conditions are not exactly represented in the discrete formulation. Repeat the above discretization $x_k=kh$, $k=1,\dots ,m$, $h=1/(m+1)$ for the first-order differentiation operator ${\partial }_x$ with eigenvalues $\xi$ associated with eigenfunctions $e^{\xi x}$

The derivative may be approximated by

The eigenproblem

differs from the discretization (1)

hence $\sin (\xi h)/h$

Figure 1. Comparison of eigenvalues of second-order differentiation matrix $𝑫$ (blue circles) with those

∴ 

m=8; o=ones(m-1,1)[:,1]; D=diagm(-1 => -o, 1 => o)

∴ 

function ErrorλD(m)
  h=π/m; o=ones(m-1,1)[:,1]; D=diagm(-1 => o, 1 => o)-2*I;
  λ=sort(eigvals(D),rev=true);
  ε=norm(λ.+4*sin.(collect(1:m).*h/2).^2)/norm(λ)
end;

∴ 

ε=ErrorλD.(20:20:100)';

∴ 

print(round.(ε; digits=3))

∴