### review of CG method:II

#### by Yi Zhang

Now let’s examine the convergence property of steepest descent method. For this purpose, define the error against exact solution at each iteration step

By iteration scheme there is

Since at every step matrix would be multiplied, and the iterative property of a matrix is largely determined by its eigenvalues, one’s first reaction should be look at the decomposition of the vector space under discussion into its standard forms by

where is the orthonormal basis of eigenvector space of . By this defitino, there are

In which is eigenvalue corresponding to . One can see if the **normal/spectral condition number** , i.e. all the eigenvalues of are same, the error at next step immediately drops to zero. Geometricly this simply means contours are spheres and the steepest descent direction is toward the centre. Generally, there is

where the norm is energy norm induced by

and (subscript emphasizes the dependence on each step) is determined by spectral condition number and **slope** :

Picture below shows the dependence of to two variables. One can see that the slowest convergence happens when both condition number and slope are large. is referred to as ** ill-conditioned** when is large.