Journal ArticleDOI
Numerical solution of a secular equation
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In this article, a method for the solution of a secular equation, arising in modified symmetric eigenvalue problems and in several other areas, is proposed for the root-finding problem.Abstract:
A method is proposed for the solution of a secular equation, arising in modified symmetric eigenvalue problems and in several other areas. This equation has singularities which make the application of standard root-finding methods difficult. In order to solve the equation, a class of transformations of variables is considered, which transform the equation into one for which Newton's method converges from any point in a certain given interval. In addition, the form of the transformed equation suggests a convergence accelerating modification of Newton's method. The same ideas are applied to the secant method and numerical results are presented.read more
Citations
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An introduction to pseudo-linear algebra
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Multiscale Scientific Computation: Review 2001
TL;DR: A wide range of multiscale computational methods is described, emphasizing main ideas and inter-relations between various fields, including top-efficiency multigrid methods in fluid dynamics; inverse PDE problems and data assimilation; feedback optimal control.
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Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations
TL;DR: This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations.
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Solving secular and polynomial equations: A multiprecision algorithm
Dario Andrea Bini,Leonardo Robol +1 more
TL;DR: The concept and the properties of root-neighborhoods from polynomials to secular functions, provide perturbation results of the roots, obtain an effective stop condition for the EA iteration and guaranteed a posteriori error bounds are extended.
References
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Journal ArticleDOI
Smoothing by spline functions. II
TL;DR: In this paper, the authors generalize the results of [4] and modify the algorithm presented there to obtain a better rate of convergence, which is the same as in this paper.
Journal ArticleDOI
Computational Methods of Linear Algebra
Journal ArticleDOI
Some modified matrix eigenvalue problems
TL;DR: In this paper, the problem of finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one is considered.
Journal ArticleDOI
Rank-one modification of the symmetric eigenproblem
TL;DR: An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eIGensystem and the explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues.