1. Preliminaries Theorem A-1. (Theorem 3.1, (Chang, 2012)) Let A ∈ Rm×n be of full column rank with QR factorization A = QR, ∆A be a perturbation in A, and A + ∆A = (Q + ∆Q)(R + ∆R) be the QR-factorization of A + ∆A. Let PA and PA⊥ be the orthogonal projectors onto the range of A and the orthogonal complement of the range of A, respectively. LetQ⊥ be an orthonormal matrix such that matrix [Q,Q⊥...

The basic idea is that rather than doing a QR decomposition of A, do a QR decomposition on ΠA, where Π is a FJLT (or some other, e.g., data-aware subspace embedding), and then use the R̃−1 from QR on the subproblem as a preconditioner for an iterative algorithm on the original problem. We saw that if ΠA = Q̃R̃ then κ ( AR̃−1 ) = κ (SU). If we sample “enough,” i.e., Ω (d log(d)/ ), then this conditi...

Quick Response (QR) Codes helps us in encoding the data in an efficient manner. The data capacity is limited according to the various data formats used. For increasing the data capacity, data to be encoded can first be compressed using any of the data compression techniques. Then, the data can be encoded. This paper suggests a technique for data compression which in turn helped to increase the ...

The paper describes a parallel feed-forward neural network training algorithm based on the QR decomposition with the use of the Givens rotation. The beginning brings a brief mathematical background on Givens rotation matrices and elimination step. Then the error criterion and its necessary transformations for the QR decomposition are presented. The paper’s core holds an essential explanation to...

Für unterschiedlichste Ansprüche bietet die Schaumann aus dem österreichischen Taufkirchen erstklassiges Tierfutter an. Damit im Stall nichts verwechselt werden kann, alle Futtersäcke noch vor der Befüllung mit zahlreichen Produktinformationen sowie ein- und zweidimensionalen Codes gekennzeichnet.

This work aims to study of the stability of two generalizations of the functional equation f (pr,qs)+ f (ps,qr) = f (p,q) f (r,s) , namely (i) f (pr,qs)+g(ps,qr) = h(p,q)h(r,s) , and (ii) f (pr,qs)+ g(ps,qr) = h(p,q)k(r,s) for all p,q,r,s ∈ G , where G is a commutative semigroup. Thus this work is a continuation of our earlier works [15] and [16], and the functional equations studied here arise...

Recently a generalization of Francis’s implicitly shifted QR algorithm was proposed, notably widening the class of matrices admitting low-cost implicit QR steps. This unifying framework covered the methods and theory for Hessenberg and inverse Hessenberg matrices and furnished also new, single-shifted, QR-type methods for, e.g., CMVmatrices. Convergence of this approach was only suggested by nu...

The adoption of a new technology is often affected by its perceived utility and ease of use, both of which could vary by age due to cognitive differences. A recently invented technology, the quick response (QR) code, enables smartphone users to access content on their mobile devices by scanning two-dimensional barcodes. In this paper, we examine awareness, knowledge, and usage of QR codes acros...

This guideline for care of children with craniosynostosis was developed by a national working group with representatives of 11 matrix societies of specialties and the national patients’ society. All medical aspects of care for nonsyndromic and syndromic craniosynostosis are included, as well as the social and psychologic impact for the patient and their parents. Managerial aspects are incorpora...