Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices

It is well known that if a matrix $A\in\mathbb C^{n\times n}$ solves the matrix equation $f(A,A^H)=0$, where $f(x, y)$ is a linear bivariate polynomial, then $A$ is normal; $A$ and $A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of $A$ is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.Comment: 13 pages, 3 figure

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos--type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+U^H. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank--structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank--structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration

A CMV--based eigensolver for companion matrices

In this paper we present a novel matrix method for polynomial rootfinding. By exploiting the properties of the QR eigenvalue algorithm applied to a suitable CMV-like form of a companion matrix we design a fast and computationally simple structured QR iteration.Comment: 14 pages, 4 figure

THE ROLE OF HLA TYPING IN RHEUMATIC DISEASES

Association between HLA-DR4 and rheumatoid arthritis (RA) has been known for 4 decades, and amino acid sites within HLA-DRB1 (11/13, 71, 74) are highly associated with RA. HLA is not useful for diagnosis or prognosis, but it may help predict severe and erosive disease. Since 90% of patients with ankylosing spondylitis (AS) and 50-70% of other spondyloarthritis (SpA) patients are HLA-B*27 positive, HLA is a stronghold of diagnostic algorithms. Genetic predisposition to juvenile idiopathic arthritis (JIA) is mainly due to HLA class II, and to a lesser extent to HLA class I. Although HLA plays a role in rheumatic disorders, its clinical relevance is not homogeneous. When classical biomarkers are lacking or in complex cases, HLA typing may provide support for the management of patients

L'uso dell'eritropoietina nelle anemizzazioni dei pazienti critici

Uso dell'eritropoietina nelle anemizzazioni del paziente critico come sostituto alla trasfusione di sangue allogenico. Discussione sul miglioramento della concentrazione emoglobinica,incremento ematocrito,diminuzione sacche di sangue,richio infettivo,diminuzione giorni degenza in UTI e decremento mortalita', rispetto ad un gruppo controllo non trattato con EPO

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos-type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+UH. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank-structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank-structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration. © 2014 Elsevier Inc. All rights reserved