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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)

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)

Let H =

It is shown that when parallel preex is used to compute the leading principal minors of a tridiagonal matrix T within a bisection algorithm to compute the eigenvalues of T the relative error in the computed eigenvalues can be as great as 3 , where is machine precision and is the condition number for the problem of computing the eigenvalues of T. An ideal… (More)

Although there is consensus that localized Ca(2+) elevations known as Ca(2+) puffs and sparks arise from the cooperative activity of intracellular Ca(2+) channels, the precise relationship between single-channel kinetics and the collective phenomena of stochastic Ca(2+) excitability is not well understood. Here we present a formalism by which mathematical… (More)

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)

We consider the problem of computing U k+1 = Q k U k+1 ; U 0 given in nite precision (M = machine precision) where U 0 and the Q i are known to be unitary. The problem is that ^ U k , the computed product may not be unitary, so one applies an O(n 2) orthogonalizing step after each multiplication to (a) prevent ^ U k from drifting too far from the set of… (More)

Denote by W (T), r(T) and T the numerical range, the numerical radius and the spectral norm of a complex matrix T. Let (A, B) be a pair of Hermitian matrices. It is shown that if 0 ∈ W (A + iB) then d(A, B) = inf{|µ| : µ / ∈ W (A + iB)}

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