3. Fix N > 2, and let h = i. Consider the (N 1) X [N 1] matrix given by N 2 1 0 [1 1 2 1 L = 0 0 ' 1 2 1 0 0 1 2 (a) Make a rough prediction on the locations of the eigenvalues of L based on lGershgorin disks. (b) With or := %, check that the eigenpairs of L are given by, for 1 S j 5 N 1 A3; = 2(cos[jo:) 1), Vi = (sina), sin(2jo:), . . . ,sin((N 1)joc))T . That is to say, for every 1 g j E N 1, check that ij : Ajvj. (c) The matrix L appears in Example 12 of the notes, reducing a heat conduction problem to the ODE via semidiscretization in space du ,11 _ = _ L a 112 u\" for an unknown vector function u(t) = (11103), . . . ,uN_1[t)]). Fix a timestep At and write (i) Forward Euler and (ii) Heun schemes for the above ODE. In each case, you should obtain an iterative system um\") : Ant\") and using part (b), give all the eigenvalues of APE and AHBW. (d) Assuming h is xed, for each scheme in (c), give conditions on At so that all eigenvalues of A are of modulus less than 1. (this is the 'stability condition' for the numerical solution to remain bounded at all times)