Fast linear algebra is stable

In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.Comment: 26 pages; final version; to appear in Numerische Mathemati

International consensus statement on nomenclature and classification of the congenital bicuspid aortic valve and its aortopathy, for clinical, surgical, interventional and research purposes

This International Consensus Classification and Nomenclature for the congenital bicuspid aortic valve condition recognizes 3 types of bicuspid valves: 1. The fused type (right-left cusp fusion, right-non-coronary cusp fusion and left-non-coronary cusp fusion phenotypes); 2. The 2-sinus type (latero-lateral and antero-posterior phenotypes); and 3. The partial-fusion (forme fruste) type. The presence of raphe and the symmetry of the fused type phenotypes are critical aspects to describe. The International Consensus also recognizes 3 types of bicuspid valve-associated aortopathy: 1. The ascending phenotype; 2. The root phenotype; and 3. Extended phenotypes.Cardiolog