How to make the Lanczos algorithm converge slowly

How to Make the Lanczos Algorithm Converge Slowly

The Paige style Lanczos algorithm is an iterative method for finding a few eigenvalues of large sparse symmetric matrices. Some beautiful relationships among the elements of the eigenvectors of a symmetric tridiagonal matrix are used to derive a perverse starting vector which delays convergence as long as possible. Why i such slow convergence is never seen in practice is also examined.

How Slowly Can Quadrature Formulas Converge?

Let {Q„)denote a sequenceof quadrature formulas, Q„(j) m Yfj-iW^fix^), such that ß„(/) -> P0 j(x) dx for all / G CTO, 1], Let 0 < e < \ and a sequence (aX_j.be given, where a, ä si ^ a, 5 • • • , and where a„ —> 0 as n —* c°. Then there exists a function / G CTO, l]and a sequence |nt-)"=i suchthat |/(x)| g 2(7,71(1 4e)|, and such that n,Kx)dx Q„k(1) = ak,k = 1,2, 3, ••• .

How to Make Brucker's Algorithm Polynomial

How to Make the Quantum Adiabatic Algorithm Fail

The quantum adiabatic algorithm is a Hamiltonian based quantum algorithm designed to find the minimum of a classical cost function whose domain has size N . We show that poor choices for the Hamiltonian can guarantee that the algorithm will not find the minimum if the run time grows more slowly than √ N . These poor choices are nonlocal and wash out any structure in the cost function to be mini...

Nested Lanczos : Implicitly Restarting a Lanczos Algorithm Nested Lanczos : Implicitly Restarting a Lanczos Algorithm

In this text, we present a generalisation of the idea of the Implicitly Restarted Arnoldi method to the nonsymmetric Lanczos algorithm, using the two-sided Gram-Schmidt process or using a full Lanczos tridi-agonalisation. The Implicitly Restarted Lanczos method can be combined with an implicit lter. It can also be used in case of breakdown and ooers an alternative for look-ahead.