Applying Gershgorins theorem to study the convergence of iterative linear solvers

Applying Gershgorin s theorem to study the convergence of iterative linear solvers 

As a demonstration of the useMness of Gershgorin s theorem, we generate a convergence criterion for the Jacobi iterative method of solving Ax = b. This example is typical of the use of eigenvalues in numerical analysis, and also shows why the questions of eigenvector basis set existence raised in the next section are of such importance. 

We develop here a simple alternative to Gaussian elimination for solving a linear system Ax = b. It is based on forming an initial guess of the solution, and iteratively refining it to form a sequence x, ... that hopefully converges to a solution i.e.. 

Under what conditions must file Jacobi mefliod converge To answer this question, let us define the error vector at iteration k as 

We want hmyt co II = 0. We obtain a rule for the transformation of the error vector at each iteration by subtracting (3.69) fi om (3.70), 

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