Quadratic Matrix Polynomial Research Articles

Probabilistic factorization of a quadratic matrix polynomial

A purely algebraic result. We begin by stating the following theorem. Theorem. Let E be a finite set, and letdenote the set of real E × E matrices with non-negative off-diagonal elements and with non-positive row sums. Let A be a symmetric element of, and let V be a diagonal real E × E matrix. Then there exists a unique pair (H+, H−) of elements ofsuch thatI denoting the identity E × E matrix, and the superscript T signifying transpose. It is an immediate consequence that

On the Stationary Values of a Second-Degree Polynomial on the Unit Sphere

Previous article Next article On the Stationary Values of a Second-Degree Polynomial on the Unit SphereGeorge E. Forsythe and Gene H. GolubGeorge E. Forsythe and Gene H. Golubhttps://doi.org/10.1137/0113073PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. V. Balakrishnan, An operator theoretic formulation of a class of control problems and a steepest descent method of solution, J. SIAM Control Ser. A, 1 (1963), 109–127 (1963) MR0155711 0201.48502 Google Scholar[2] Philip Davis, An application of doubly orthogonal functions to a problem of approximation in two regions, Trans. Amer. Math. Soc., 72 (1952), 104–137 MR0046434 0046.08101 CrossrefISIGoogle Scholar[3] P. R. Garabedian, Partial differential equations, John Wiley & Sons Inc., New York, 1964xii+672, Chap. 3. MR0162045 0124.30501 Google Scholar[4] Donald Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math., 11 (1963), 431–441 10.1137/0111030 MR0153071 0112.10505 LinkISIGoogle Scholar[5] David D. Morrison, Methods for nonlinear least squares problems and convergence proofs, Proc. Jet Propulsion Laboratory Seminar, 1960, 1–9 Google Scholar[6] David L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach., 9 (1962), 84–97 MR0134481 0108.29902 CrossrefISIGoogle Scholar[7] S. Twomey, On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature, J. Assoc. Comput. Mach., 10 (1963), 97–101 MR0148249 0125.36102 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Solving the Trust-Region Subproblem By a Generalized Eigenvalue ProblemSatoru Adachi, Satoru Iwata, Yuji Nakatsukasa, and Akiko TakedaSIAM Journal on Optimization, Vol. 27, No. 1 | 22 February 2017AbstractPDF (618 KB)Computing the Signed Distance Between Overlapping EllipsoidsSIAM Journal on Optimization, Vol. 25, No. 4 | 19 November 2015AbstractPDF (4448 KB)Why Lagrange Multipliers with Extreme Magnitudes Give Extrema of Definite Hermitian Forms on Quadric SurfacesSIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 3 | 13 November 2009AbstractPDF (854 KB)A Krylov Subspace Method for Quadratic Matrix Polynomials with Application to Constrained Least Squares ProblemsSIAM Journal on Matrix Analysis and Applications, Vol. 25, No. 2 | 31 July 2006AbstractPDF (364 KB)N Roots of the Secular Equation in O(N) OperationsSIAM Journal on Matrix Analysis and Applications, Vol. 24, No. 2 | 31 July 2006AbstractPDF (232 KB)The Procrustes Problem for Orthogonal Stiefel MatricesSIAM Journal on Scientific Computing, Vol. 21, No. 4 | 25 July 2006AbstractPDF (344 KB)A D.C. Optimization Algorithm for Solving the Trust-Region SubproblemSIAM Journal on Optimization, Vol. 8, No. 2 | 31 July 2006AbstractPDF (534 KB)On Some Properties of Quadratic Programs with a Convex Quadratic ConstraintSIAM Journal on Optimization, Vol. 8, No. 1 | 31 July 2006AbstractPDF (271 KB)Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue PerturbationsSIAM Journal on Optimization, Vol. 5, No. 2 | 13 July 2006AbstractPDF (2667 KB)Trust Region Problems and Nonsymmetric Eigenvalue PerturbationsSIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 3 | 31 July 2006AbstractPDF (1918 KB)Orthogonal Rotation to a Partially Specified TargetSIAM Journal on Scientific and Statistical Computing, Vol. 4, No. 2 | 14 July 2006AbstractPDF (380 KB)Newton’s Method with a Model Trust Region ModificationSIAM Journal on Numerical Analysis, Vol. 19, No. 2 | 17 July 2006AbstractPDF (2105 KB)Some Modified Matrix Eigenvalue ProblemsSIAM Review, Vol. 15, No. 2 | 2 August 2006AbstractPDF (1109 KB)A Note on a Theorem of Forsythe and GolubSIAM Journal on Applied Mathematics, Vol. 23, No. 3 | 12 July 2006AbstractPDF (303 KB)The Maximum and Minimum of a Positive Definite Quadratic Polynomial on a Sphere are Convex Functions of the RadiusSIAM Journal on Applied Mathematics, Vol. 19, No. 3 | 12 July 2006AbstractPDF (279 KB)On the Stationary Values of a Fourth-Degree Polynomial on the Unit SphereSIAM Journal on Applied Mathematics, Vol. 18, No. 3 | 1 August 2006AbstractPDF (305 KB) Volume 13, Issue 4| 1965Journal of the Society for Industrial and Applied Mathematics913-1107 History Submitted:19 March 1965Published online:13 July 2006 InformationCopyright © 1965 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0113073Article page range:pp. 1050-1068ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics