# Roy Mathias

• SIAM J. Matrix Analysis Applications
• 2002
Aggressive early deflation is a QR algorithm deflation strategy that takes advantage of matrix perturbations outside of the subdiagonal entries of the Hessenberg QR iterate. It identifies and deflates converged eigenvalues long before the classic small-subdiagonal strategy would. The new deflation strategy enhances the performance of conventional(More)
abstract In this note we consider an iterative algorithm for moving a triangular matrix toward diagonality. The algorithm is related to algorithms for reening rank-revealing triangular decompositions and in a variant form to the QR algorithm. It is shown to converge if there is a suucient gap in the singular values of the matrix, and the analysis provides a(More)
Let f be a not necessarily analytic function and let A(t) be a family of n n matrices depending on the parameter t. Conditions for the existence of the rst and higher derivatives of f(A(t)) are presented together with formulae that represent these derivatives as a submatrix of f(B) where B is a larger block Toeplitz matrix. This block matrix representation(More)
• SIAM J. Matrix Analysis Applications
• 2002
This paper presents a small-bulge multishift variation of the multishift QR algorithm that avoids the phenomenon of shift blurring, which retards convergence and limits the number of simultaneous shifts. It replaces the large diagonal bulge in the multishift QR sweep with a chain of many small bulges. The small-bulge multishift QR sweep admits nearly any(More)
Demmel and Veseli c showed that, subject to a minor proviso, Jacobi's method computes the eigenvalues and eigenvectors of a positive deenite matrix more accurately than methods that rst tridiagonalize the matrix. We extend their analysis and there by 1. We remove the minor proviso in their results and thus guarantee the accuracy of Jacobi's method. 2. We(More)
Let A = " M R R N # and ~ A = " M 0 0 N # be Hermitian matrices. Stronger and more general O(kRk 2) bounds relating the eigen-values of A and ~ A are proved using a Schur complement technique. These results extend to singular values and to eigenvalues of non-Hermitian matrices. (1) be Hermitian matrices. Since jjA ? ~ Ajj = jjRjj one can bound the diierence(More)
BACKGROUND Ozone (O3), a common air pollutant, induces exacerbation of asthma and chronic obstructive pulmonary disease. Pulmonary surfactant protein (SP)-D modulates immune and inflammatory responses in the lung. We have shown previously that SP-D plays a protective role in a mouse model of allergic airway inflammation. Here we studied the role and(More)
Let H and H + H be positive deenite matrices. It was shown by Barlow and Demmel, and Demmel and Veseli c that if one takes a component-wise approach one can prove much stronger bounds on i (H)== i (H++H) and the components of the eigenvectors of H and H++H than by using the standard norm-wise perturbation theory. Here a uniied approach is presented that(More)
• Numerische Mathematik
• 1999
We use a simple matrix splitting technique to give an elementary new proof of the Lidskii-Mirsky-Wielandt Theorem and to obtain a multiplicative analog of the Lidskii-Mirsky-Wielandt Theorem, which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of Hermitian matrices and singular values of general matrices. We(More)
Let M n (F) denote the space of matrices over the eld F. Given A2 M n (F) deene jAj (A A) 1=2 and U(A) AjAj ?1 assuming A is nonsingular. Let 1 (A) 2 (A) n (A) 0 denote the ordered singular values of A. We obtain majorization results relating the singular values of U(A + A) ? U(A) and those of A and A. In particular we show that if A; A2 M n (R) and 1 ((A)(More)