The following content has been provided by the University of Erlangen, Nürnberg.

Okay, we are now in the process of discussing the iterative methods.

I recognise it's a little bit brief,

a little bit brief on the transparencies,

so I may make some amendments on the blackboard.

make some amendments on the blackboard. The idea is to come step by step to more

new and more efficient methods but of course we need some sort of a measure

and as a measure we take our standard problem that is the Laplasti discrete

a five-point stencil discretization of the Laplacian

for the Laplacian with Dirichlet boundary condition equals to zero and

the dimension should be that our overall dimension M which is the dimension of

the set of equations we would like to solve is n minus 1 squared so that is

not hundred percent consistent with a notation we had in the beginning so that

means we have n plus 1 points in each of the two spatial directions so we have n

minus 1 interior points so this is the amount of interior discretization points

which we have so and the point is with a Laplacian with a discretization

matrix which comes out we can explicitly write down the eigenvalues so we can

compute everything what we need to assess the accuracy of a method we can

compute the condition number everything what we need so the eigenvalues I I gave

them to you the eigenvalues of a h first of the discretization matrix is 2 times

2 minus cosine k pi over n minus cosine l pi over n so this is this n from here

and the L is running and the K is running between 1 and n minus 1 so we

have n minus 1 squared we have n m single eigenvalues corresponding eigenvectors

which I'm not now not going to write down the eigenvectors are just discrete

sine curves which get more and more higher more frequent and as we are in

two dimensions it's just the product of the one dimensional sine curves and

that's also the reason why we can so easily write down here the the eigenvalues

and the eigenfunction or eigenvectors so if I would now write down the eigenvectors

of course you could just check that these are the eigenvectors to these

eigenvalues multiply the matrix with the vectors and see factor times vector

comes out so okay that's the first thing and if we now talk we start with a

Jacobi method as the most simple method and the Jacobi method has an iteration

matrix M let me also call it m h to indicate that it's really the question

how the convergence behavior depends on the age that we really want to see what

happens in the situation of the age gets smaller the dimension of the set of

equations becomes large the m h in this case is just what is it we have minus

the inverse of the diagonal matrix but the diagonal matrix only has force so we

look at the version where we have multiplied already with the h square

factor so we have minus 1 over 4 that is just the multiplication with the

diagonal matrix and then we have here a times the diagonal the tag and those 4

times the identity so what what we remain what we have here is the identity minus

1 over 4 times a h and of course and this will always be the the case as the

matrices which appear as iteration matrix will be matrix polo polynomials of the

original matrix as we have it here we can directly compute the eigenvalues we

know if we do a matrix polynomial instead of the matrix then this matrix

polynomial is the same eigenvectors and the eigenvalues are the

polynomial are the the values which is the polynomial gives us applied to the

eigenvalues so what we have to have here we have to look at 1 minus 1 fourth of

those values here and okay let's just write it down

so we have 1 minus 1 fourth so this part goes away and then we get instead of a