A note on the error analysis of classical Gram-Schmidt

An error analysis result is given for classical Gram--Schmidt factorization of a full rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular. The work presented here shows that the computed $R$ satisfies \normal{R}=\normal{A}+E where $E$ is an appropriately small backward error, but only if the diagonals of $R$ are computed in a manner similar to Cholesky factorization of the normal equations matrix. A similar result is stated in [Giraud at al, Numer. Math. 101(1):87--100,2005]. However, for that result to hold, the diagonals of $R$ must be computed in the manner recommended in this work.Comment: 12 pages This v2. v1 (from 2006) has not the biliographical reference set (at all). This is the only modification between v1 and v2. If you want to quote this paper, please quote the version published in Numerische Mathemati

Numerical hyperinterpolation over nonstandard planar regions

We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes

Fast solving of Weighted Pairing Least-Squares systems

This paper presents a generalization of the "weighted least-squares" (WLS), named "weighted pairing least-squares" (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods, suitable for solving full rank systems as well as rank deficient systems, are studied. Computational experiments clearly show that the best method, in terms of speed, accuracy, and numerical stability, is based on a special {1, 2, 3}-inverse, whose computation reduces to a very simple generalization of the usual "Cholesky factorization-backward substitution" method for solving linear systems

A DEIM Induced CUR Factorization

We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a low rank approximate decomposition of the form $A \approx C U R$, where $C$ and $R$ are subsets of the columns and rows of $A$, and $U$ is constructed to make $CUR$ a good approximation. Given a low-rank singular value decomposition $A \approx V S W^T$, the DEIM procedure uses $V$ and $W$ to select the columns and rows of $A$ that form $C$ and $R$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of $V$ and $W$. For large-scale problems, $V$ and $W$ can be approximated using an incremental QR algorithm that makes one pass through $A$. Numerical examples illustrate the favorable performance of the DEIM-CUR method, compared to CUR approximations based on leverage scores

Polynomial fitting and interpolation on circular sections

We construct Weakly Admissible polynomial Meshes (WAMs) on circular sections, such as symmetric and asymmetric circular sectors, circular segments, zones, lenses and lunes. The construction resorts to recent results on subperiodic trigonometric interpolation. The paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

POLYNOMIAL ALGORITHM FOR QUADRATIC FORMS CLASSIFICATION USING GAUSSIAN ELIMINATION

Existing methods for quadratic forms classification have exponential time complexity or use approximation that weaken the result reliability. We develop an algorithm that improves the best case of quadratic form classification in constant time and is polynomial in the worst case. In addition, new strategies were used to guarantee the results reliability, by representing rational numbers as a fraction of integers and to identify linear combinations that are linearly independent using Gaussian Elimination.DOI: 10.36558/rsc.v11i3.739