A new method for automatic indexing and retrieval is described. The approach is to take advantage of implicit higher-order structure in the association of terms with documents (“semantic structure”) in order to improve the detection of relevant documents on the basis of terms found in queries. The particular technique used is singular-value decomposition, in which a large term by document matrix is decomposed into a set of ca. 100 or-thogonal factors from which the original matrix can be approximated by linear combination. Documents are represented by ca. 100 item vectors of factor weights. Queries are represented as pseudo-document vectors formed from weighted combinations of terms, and documents with supra-threshold cosine values are re-turned. initial tests find this completely automatic method for retrieval to be promising.

Low rank approximation of large and/or sparse matrices is important in many applications. We show that good low rank matrix approximations can be directly obtained from the Lanczos bidiagonalization process without computing singular value decomposition. We also demonstrate that a so-called one-sided reorthogonalization process can be used to maintain adequate level of orthogonality among the Lanczos vectors and produce accurate low rank approximations. This technique reduces the computational cost of the Lanczos bidiagonalization process. We illustrate the efficiency and applicability of our algorithm using numerical examples from several applications areas.

Abstract. We discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix. Similar to the Jacobi–Davidson method for the eigenvalue problem, we compute in each step a correction by (approximately) solving a correction equation. We give a few variants of this Jacobi–Davidson SVD (JDSVD) method with their theoretical properties. It is shown that the JDSVD can be seen as an accelerated (inexact) Newton scheme. We experimentally compare the method with some other iterative SVD methods. Key words. Jacobi–Davidson, singular value decomposition (SVD), singular values, singular vectors, norm, augmented matrix, correction equation, (inexact) accelerated Newton, improving singular values AMS subject classifications. 65F15 (65F35) PII. S1064827500372973

This report describes BLZPACK (an acronym for Block _Lanczos Package), an imple- mentation of the block Lanczos algorithm intended for the solution of eigenproblems involving real, sparse, symmetric matrices. The package works in an interactive way, so the matrices of the target problem are not passed as arguments for the interface subprogram.

A matrix-free algorithm, IRLANB, for the efficient computation of the smallest singular triplets of large and possibly sparse matrices is described. Key characteristics of the approach are its use of Lanczos bidiagonalization, implicit restarting, and harmonic Ritz values. The algorithm also uses a deflation stategy that can be applied directly on Lanczos bidiagonalization. A refinenement postprocessing phase is applied on the converged singular vectors. The computational costs of the above techniques are kept small as they make direct use of the bidiagonal form obtained in the course of the Lanczos factorization. Several numerical experiments with the method are presented that illustrate its effectiveness and indicate that it performs well compared to existing codes.

A new method for automatic indexing and retrieval is described. The approach is to take advantage of implicit higher-order structure in the association of terms with documents ("semantic structure") in order to improve the detection of relevant documents on the basis of terms found in queries. The particular technique used is singular-value decomposition, in which a large term by document matrix is decomposed into a set of ca 100 orthogonal factors from which the original matrix can be approximated by linear combination. Documents are represented by ca 100 item vectors of factor weights. Queries are represented as pseudo-document vectors formed from weighted combinations of terms, and documents with supra-threshold cosine values are returned. Initial tests find this completely automatic method for retrieval to be promising. Deerwester - 1 - 1.