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  • 1. Besana, Gian Mario
    et al.
    Di Rocco, Sandra
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hauenstein, Jonathan D.
    Sommese, Andrew J.
    Wampler, Charles W.
    Cell decomposition of almost smooth real algebraic surfaces2013In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 63, no 4, p. 645-678Article in journal (Refereed)
    Abstract [en]

    Let Z be a two dimensional irreducible complex component of the solution set of a system of polynomial equations with real coefficients in N complex variables. This work presents a new numerical algorithm, based on homotopy continuation methods, that begins with a numerical witness set for Z and produces a decomposition into 2-cells of any almost smooth real algebraic surface contained in Z. Each 2-cell (a face) has a generic interior point and a boundary consisting of 1-cells (edges). Similarly, the 1-cells have a generic interior point and a vertex at each end. Each 1-cell and each 2-cell has an associated homotopy for moving the generic interior point to any other point in the interior of the cell, defining an invertible map from the parameter space of the homotopy to the cell. This work draws on previous results for the curve case. Once the cell decomposition is in hand, one can sample the 2-cells and 1-cells to any resolution, limited only by the computational resources available.

  • 2.
    Hanke, Michael
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Lamour, R.
    Consistent initialization for nonlinear index-2 differential-algebraic equation: large sparse systems in MATLAB2003In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 32, no 1, p. 67-85Article in journal (Refereed)
    Abstract [en]

    An important component of any initial-value solver for higher-index differential-algebraic equations consists in the computation of consistent initial values. In a recent paper [5], an algorithm is proposed which is applicable to a very general class of index-2 systems. Unfortunately, the computational expense is rather high. We present a modification of this approach, which gives rise to a MATLAB implementation capable of handling systems of moderate dimension (several thousands of unknowns). The algorithm is illustrated by examples.

  • 3.
    Jarlebring, Elias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Koskela, Antti
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Mele, Giampaolo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Disguised and new quasi-Newton methods for nonlinear eigenvalue problems2018In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 79, no 1, p. 311-335Article in journal (Refereed)
    Abstract [en]

    In this paper, we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type M(λ)v = 0, where (Formula presented.) is a holomorphic function. We investigate which types of approximations of the Jacobian matrix lead to competitive algorithms, and provide convergence theory. The convergence analysis is based on theory for quasi-Newton methods and Keldysh’s theorem for NEPs. We derive new algorithms and also show that several well-established methods for NEPs can be interpreted as quasi-Newton methods, and thereby, we provide insight to their convergence behavior. In particular, we establish quasi-Newton interpretations of Neumaier’s residual inverse iteration and Ruhe’s method of successive linear problems.

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