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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)
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)
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)
We propose a definition for geometric mean of k positive (semi) definite matrices. We show that our definition is the only one in the literature that has the properties that one would expect from a geometric mean, and that our geometric mean generalizes many inequalities satisfied by the geometric mean of two positive semidefinite matrices. We prove some(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)
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)
  • Roy Mathias, Supfk Jaj ? Jbj K 1: Ka ? Bk, Supfk Jajb ? Bjaj, Bak
  • 1997
Let M n be the space of n n complex matrices and let k k 1 denote the spectral norm. Given matrices A = a ij ] and B = b ij ] of the same size we deene their Hadamard product to be A B = a ij b ij ]. We deene the Hadamard operator norm of A 2 M n by j j jAj j j 1 = maxfkA Bk 1 : kBk 1 1g: We show that j j jAj j j 1 = tr jAj=n (1) if and only if jAj I = jA j(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)
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)