n = 1000000;
A = SparseArray[{
      {1, 1} -> 1. n,
      {-1, -1} -> 1. n,
      Band[{2, 1}] -> -1. n,
      Band[{1, 1}] -> 2. n,
      Band[{1, 2}] -> -1. n
      },
     {n + 1, n + 1}, 0.
     ]; // AbsoluteTiming // First
M = SparseArray[{
      {1, 1} -> 1./(3 n),
      {-1, -1} -> 1./(3 n),
      Band[{2, 1}] -> 1./(6 n),
      Band[{1, 1}] -> 2./(3 n),
      Band[{1, 2}] -> 1./(6 n)
      },
     {n + 1, n + 1}, 0.
     ]; // AbsoluteTiming // First
{[Lambda], U} = Eigensystem[{A, M}, -10, Method -> {"Arnoldi", "Shift" -> 1.}]; //
   AbsoluteTiming // First