Past Computational Mathematics and Applications Seminar

We show that Sorensen's (1992) implicitly restarted Arnoldi method

(IRAM) (including its block extension) is non-stationary simultaneous

iteration in disguise. By using the geometric convergence theory for

non-stationary simultaneous iteration due to Watkins and Elsner (1991)

we prove that an implicitly restarted Arnoldi method can achieve a

super-linear rate of convergence to the dominant invariant subspace of

a matrix. We conclude with some numerical results the demonstrate the

efficiency of IRAM.

It has been known for some while now that every radial basis function

in common usage for multi-dimensional interpolation has associated with

it a uniquely defined Hilbert space, in which the radial basis function

is the `minimal norm interpolant'. This space is usually constructed by

utilising the positive definite nature of the radial function, but such

constructions turn out to be difficult to handle. We will present a

direct way of constructing the spaces, and show how to prove extension

theorems in such spaces. These extension theorems are at the heart of

improved error estimates in the $L_p$-norm.