Eigenvalue problems arise in many fields of physics and engineering science for example in structural engineering from prediction of dynamic stability of structures or vibrations of a fluid in a closed cavity. Their solution consumes a large amount of memory and CPU time if more than very few (1-5) vibrational modes are desired. Both make the problem a natural candidate for parallel processing....

A solution to get the problem off, have you found it? Really? What kind of solution do you resolve the problem? From what sources? Well, there are so many questions that we utter every day. No matter how you will get the solution, it will mean better. You can take the reference from some books. And the the lanczos method evolution and application is one book that we really recommend you to read...

Resolution matrices are useful in seismic tomography because they allow us to evaluate the information content of reconstructed images. Techniques based on the multiplicity of equivalent exact formulas that may be used to define the resolution matrices have been used previously by the author to design algorithms that avoid the need for any singular value decomposition of the ray-path matrix. An...

Lanczos bidiagonalization is a competitive method for computing a partial singular value decomposition of a large sparse matrix, that is, when only a subset of the singular values and corresponding singular vectors are required. However, a straightforward implementation of the algorithm has the problem of loss of orthogonality between computed Lanczos vectors, and some reorthogonalization techn...

Low-rank approximation of large and/or sparse matrices is important in many applications, and the singular value decomposition (SVD) gives the best low-rank approximations with respect to unitarily-invariant norms. In this paper we show that good low-rank approximations can be directly obtained from the Lanczos bidiagonalization process applied to the given matrix without computing any SVD. We ...

In papers[1, 2], the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov sub...

The Lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem. There is a well-established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs. These bounds suggest at least linear convergence in terms of the number of Lanczos steps, assuming there are gaps between individual eigenvalues. In practice, o...