Two method for computation of the spectra certain infinite graphs are suggested. The first one can be viewed as a reversed Gram--Schmidt orthogonalization procedure. It relies heavily on spectral theory Jacobi matrices. second is related to Schur complement block A number examples including with tails, chains cycles and ladders worked out in detail.

We present a new batch learning algorithm for text classification in the vector space of document representations. The algorithm uses ellipsoid separation [3] in the feature space which leads to a semidefinite program. An approximation of the latent semantic feature extraction approach using Gram-Schmidt orthogonalization [2] is used for the feature extraction. Preliminary results demonstrate s...

We prove a determinantal formula for quantities related to the problem of enumeration of (semi-) meanders, namely the topologically inequivalent planar configurations of non-self-intersecting loops crossing a given (half-) line through a given number of points. This is done by the explicit Gram-Schmidt orthogonalization of certain bases of subspaces of the Temperley-Lieb algebra.

Eigen values evaluation is an integral but computation-intensive part for many image and signal processing applications. Modified Gram-Schmidt Orthogonalization (MGSO) is an efficient method for evaluating the Eigen values in face recognition algorithms. MGSO applies normalization of vectors in its iterative orthogonal process and its accuracy depends on the accuracy of normalization. Using sof...

The advent of supercomputers with hierarchical memory systems has imposed the use of block algorithms for the linear algebra algorithms. Although block algorithms may result in impressive improvements in performance, their numerical properties are quite different from their scalar counterpart and deserve an in-depth study. In this paper, the numerical stability ofblock Gram-Schmidt orthogonaliz...

Connectionist models of memory storage have been studied for many years, and aim to provide insight into potential mechanisms of memory storage by the brain. A problem faced by these systems is that as the number of items to be stored increases across a finite set of neurons/synapses, the cumulative changes in synaptic weight eventually lead to a sudden and dramatic loss of the stored informati...

We present a novel distributed QR factorization algorithm for orthogonalizing a set of vectors in a decentralized wireless sensor network. The algorithm is based on the classical Gram-Schmidt orthogonalization with all projections and inner products reformulated in a recursive manner. In contrast to existing distributed orthogonalization algorithms, all elements of the resulting matrices Q and ...

Orthogonalization methods play a key role in many iterative methods. In this paper, we establish new properties for the modified Gram-Schmidt algorithm. We show why the modified Gram-Schmidt algorithm generates a well-conditioned set of vectors. This result holds under the assumption that the initial matrix is not “too ill-conditioned” in a way that is quantified. As a consequence we show that ...

In this note, we consider the modified Gram-Schmidt algorithm with reorthogonalization applied on a numerical nonsingular matrix, we explain why the resulting set of vectors is orthogonal up to the machine precision level. To establish this result, we show that a certain L-criterion is necessarily verified after the second reorthogonalization step, then we prove that this L-criterion implies th...