A new preconditioner for symmetric positive definite systems is proposed, analyzed, and tested. The preconditioner, compressed incomplete modified Gram–Schmidt (CIMGS), is based on an incomplete orthogonal factorization. CIMGS is robust both theoretically and empirically, existing (in exact arithmetic) for any full rank matrix. Numerically it is more robust than an incomplete Cholesky factoriza...

The polynomial interpolation in one dimensional space R is an important method to approximate the functions. The Lagrange and Newton methods are two well known types of interpolations. In this work, we describe the semi inherited interpolation for approximating the values of a function. In this case, the interpolation matrix has the semi inherited LU factorization.

Tensors are higher order generalizations of matrices to model multi-aspect data, e.g., a set of purchase records with the schema (user id, product id, timestamp, feedback). Tensor factorization is a powerful technique for generating a model from a tensor, just like matrix factorization generates a model from a matrix, but with higher accuracy and richer information as more attributes are availa...

The LU factorization is an important numerical algorithm for solving system of linear equations in science and engineering and is characteristic of many dense linear algebra computations. It has even become the de facto numerical algorithm implemented within the LINPACK benchmark to rank the most powerful supercomputers in the world, collected in the TOP500 website. In this context, the challen...

2 Analytic factorization of q-difference operators 7 2.1 The Newton polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Admissible q-difference operators . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Analytic factorization of admissible q-difference operators . . . . . . . . . . . 9 2.4 A digression on formal factorization of q-difference operators . . . . . ...

2 Integral Domains 12 2.1 Factorization in Integral Domains . . . . . . . . . . . . . . . . 12 2.2 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fermat’s Two Squares Theorem . . . . . . . . . . . . . . . . . 17 2.5 Maximal Ideals and Prime Ideals . . . . . . . . . . . . . . . . 20 2.6 Unique Fact...

Pietsch Factorization and Grothendieck Factorization are the two landmark theorems in modern functional analysis. They were first introduced to the numerical linear algebra community by the work of Joel A. Tropp in the column subset selection problem, which seeks to extract from a matrix a column submatrix that has lower spectral norm. Despite their broad application in functional analysis, the...