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Iterative Solution of Nonlinear Equations in Several Variables

1971-04-01
E. I. , J. M. Ortega , Werner C. Rheinboldt

<i>Generalized Hypergeometric Functions</i>

1967-03-01
L. J. Slater , Werner C. Rheinboldt
1. The Gauss Function 2. The Generalized Gauss Function 3. Basic Hypergeometric Functions 4. Hypergeometric Integrals 5. Basic Hypergeometric Integrals 6. Bilateral Series 7. Basic Bilateral Series 8. Appell Series … 1. The Gauss Function 2. The Generalized Gauss Function 3. Basic Hypergeometric Functions 4. Hypergeometric Integrals 5. Basic Hypergeometric Integrals 6. Bilateral Series 7. Basic Bilateral Series 8. Appell Series 9. Basic Appell Series.

An adaptive continuation process for solving systems of nonlinear equations

1978-01-01
Werner C. Rheinboldt
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Numerical analysis of parametrized nonlinear equations

1986-04-01
Werner C. Rheinboldt

A locally parameterized continuation process

1983-06-01
Werner C. Rheinboldt , John Burkardt
article Free AccessA locally parameterized continuation process Authors: Werner C. Rheinboldt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational … article Free AccessA locally parameterized continuation process Authors: Werner C. Rheinboldt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile , John V. Burkardt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 9Issue 2pp 215–235https://doi.org/10.1145/357456.357460Published:01 June 1983Publication History 195citation923DownloadsMetricsTotal Citations195Total Downloads923Last 12 Months29Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
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A Unified Convergence Theory for a Class of Iterative Processes

1968-03-01
Werner C. Rheinboldt
Unified theory for convergence results based on nonlinear estimates for iteration function and on majorizing sequences concept Unified theory for convergence results based on nonlinear estimates for iteration function and on majorizing sequences concept

Differential-algebraic systems as differential equations on manifolds

1984-01-01
Werner C. Rheinboldt
Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in … Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.
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On P- and S-functions and related classes of n-dimensional nonlinear mappings

1973-01-01
Jorge J. Morè , Werner C. Rheinboldt
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Solution Fields of Nonlinear Equations and Continuation Methods

1980-04-01
Werner C. Rheinboldt
Contihuation methods are considered here in the broad sense as algorithms for the computational analysis of specified parts of the solution field of equations of the form $Fx = b$ … Contihuation methods are considered here in the broad sense as algorithms for the computational analysis of specified parts of the solution field of equations of the form $Fx = b$ , where $F:R^m \to R^n $ is a given mapping and $m > n$. Such problems arise, for instance, in structural mechanics and then usually $m - n$ of the variables $x_i $ are designated as parameters. For the case $m = n + 1$ an existence theory for the regular curves of the solution field is developed here. Then approximate solutions are considered and shown to be solutions of certain perturbed problems. These results are used to prove that for the continuation methods with Eider-predictor and Newton-corrector a particular steplength algorithm is guaranteed to trace any regular solution of the field. Some numerical aspects of the procedure are discussed and a numerical example is included to illustrate the effectiveness of the approach.

Theoretical and numerical analysis of differential-algebraic equations

2002-01-01
Patrick J. Rabier , Werner C. Rheinboldt

A Mesh-Independence Principle for Operator Equations and Their Discretizations

1986-02-01
Eugene L. Allgower , Klaus Böhmer , F. A. Potra +1 more
The mesh-independence principle asserts that, when Newton’s method is applied to a nonlinear equation between some Banach spaces as well as to some finite-dimensional discretization of that equation, then the … The mesh-independence principle asserts that, when Newton’s method is applied to a nonlinear equation between some Banach spaces as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration and, as a consequence, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved only for certain classes of boundary value problems. In this paper a proof is presented for a general class of operator equations and discretizations. It covers the earlier results and extends them well beyond the cases that have been considered before.

Monotone Iterations for Nonlinear Equations with Application to Gauss-Seidel Methods

1967-06-01
James M. Ortega , Werner C. Rheinboldt
Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods
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Numerical analysis of continuation methods for nonlinear structural problems

1981-06-01
Werner C. Rheinboldt

On Steplength Algorithms for a Class of Continuation Methods

1981-10-01
C. den Heijer , Werner C. Rheinboldt
The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + … The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + 1} \to R^n $, for given $b \in R^n $. While these methods are similar in structure to those used for ODE-solvers, their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step. This suggests the use of a posteriors estimates of the convergence radii of the corrector. It is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about F. However, it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors. This is used for the design of various step-algorithms for continuation processes; two of them are based on a Newton-corrector while the third one is applicable to any corrector. Some numerical results show the effectiveness of the three algorithms. Finally some asymptotic analysis of continuation steps is given.

On the computation of multi-dimensional solution manifolds of parametrized equations

1988-01-01
Werner C. Rheinboldt

Numerical Methods for a Class of Finite Dimensional Bifurcation Problems

1978-02-01
Werner C. Rheinboldt
For an equation $H(y,t) = 0$, where $H:D \subset R^{n + 1} \to R^n $, let $p:J \subset R^1 \to R^n $ be a primary solution on which a simple … For an equation $H(y,t) = 0$, where $H:D \subset R^{n + 1} \to R^n $, let $p:J \subset R^1 \to R^n $ be a primary solution on which a simple bifurcation point $p^ * = p(t^ * )$ with rank $H_y = (p^ * ,t^ * ) = n - 1$ has been detected and a secondary solution is branching off. An iterative process is presented which starts at a point $p^0 = p(t_0 )$ near $p^ * $ and converges to a point on the secondary curve. It is similar in form to methods proposed by H. B. Kelley and others but has considerably lower computational complexity. The process represents a chord iteration with singular iteration matrix and its convergence is derived from a general result for such singular chord iterations. Computational details for the implementation of the method and an informal program are given. Finally, some comments about extensions to the case rank $H_y (p^ * ,t^ * ) < n - 1$ are made.

On a generalization of an inequality of L. V. Kantorovich

1959-06-01
Werner Greub , Werner C. Rheinboldt
G. E. Forsythe, who edited the translation of Kantorovich's paper, included the following remark about this footnote: It is not clear to me that Kantorovich's inequality really is a special … G. E. Forsythe, who edited the translation of Kantorovich's paper, included the following remark about this footnote: It is not clear to me that Kantorovich's inequality really is a special case of that of Polya and Szego. Examining the relation between the two inequalities more closely we found that this remark is well justified and can be made even more specific in that the inequality of Polya and Szeg6 in the form (4) is a special case of the Kantorovich inequality
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On Impasse Points of Quasilinear Differential-Algebraic Equations

1994-01-01
Patrick J. Rabier , Werner C. Rheinboldt

Local mapping relations and global implicit function theorems

1969-01-01
Werner C. Rheinboldt
Introduction.Consider the problem of solving a nonlinear equation (1) F(x,y) = zwith respect to y for given x and z.If, for example, x, y, z are elements of some Banach … Introduction.Consider the problem of solving a nonlinear equation (1) F(x,y) = zwith respect to y for given x and z.If, for example, x, y, z are elements of some Banach space, then under appropriate conditions about F, the well-known implicit function theorem ensures the "local" solvability of (1).The question arises when such a local result leads to "global" existence theorems.Several authors, for example, Cesari [2], Ehrmann [5], Hildebrandt and Graves [8], and Levy [9], have obtained results along this line.But these results are closely tied to the classical implicit function theorem, and no general theory appears to exist which permits the deduction of global solvability results once a local one is known.The problem has considerable interest in numerical analysis.In fact, when an operator equation(2) Fy = x is to be solved iteratively for y, the process converges usually only in a neighborhood of a solution.If for some other operator F0 the equation F0y = x has been solved, then one may try to "connect" F and F0 by an operator homotopy H(t,y) such that 77(0, y) = F0y and H(l,y) = Fy, and to "move" along the solution "curve" y(t) of H(t,y)=x, O^i^l, from the known solution y(0) to the unknown y(l), for instance, by using a locally convergent iterative process.This is the so-called "continuation method".For a review of earlier work about this method see, for example, the introduction of Ficken [6] who then proceeds to develop certain results about the global solvability of H(t, y) = x by using the local solvability provided by the implicit function theorem.Other results are due to Davidenko [3]; see also Yakovlev [14] and, more recently, Davis [4], and Meyer [10].In this paper we shall consider the problem of finding global existence theorems for (1) in the following setting.For fixed z the problem depends only on the (multivalued) relation between x and y given by (1).We therefore consider abstract
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A comparison of methods for determining turning points of nonlinear equations

1982-09-01
Rami Melhem , Werner C. Rheinboldt

Classical and generalized solutions of time-dependent linear differential-algebraic equations

1996-09-01
Patrick J. Rabier , Werner C. Rheinboldt

On a Theory of Mesh-Refinement Processes

1980-12-01
Werner C. Rheinboldt
A general theory of mesh-refinement processes is developed. The fundamental structure is a locally finite, rooted tree with nodes representing the subdivision cells. The possible meshes then constitute a distributive … A general theory of mesh-refinement processes is developed. The fundamental structure is a locally finite, rooted tree with nodes representing the subdivision cells. The possible meshes then constitute a distributive lattice. Under mild conditions on the given cell-size and error-indicator functions a local Pareto-type optimality property is introduced for the meshes. This in turn is used to prove some general rate-of-convergence and global optimality properties which contain various known results of this type for specific problems.

GENERAL ITERATIVE METHODS

1970-01-01
James M. Ortega , Werner C. Rheinboldt

On the existence and uniqueness of solutions of nonlinear semi-implicit differential-algebraic equations

1991-01-01
Werner C. Rheinboldt

On the Computation of Impasse Points of Quasi-Linear Differential-Algebraic Equations

1994-01-01
Patrick J. Rabier , Werner C. Rheinboldt
Abstract : We present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential-algebraic equations. Our method combines a reduction step transforming the DAE into … Abstract : We present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential-algebraic equations. Our method combines a reduction step transforming the DAE into a singular ODE with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.

Applications of Functional Analysis in Mathematical Physics

1964-07-01
Werner C. Rheinboldt , С. Л. Соболев

Computational Methods for Determining Lower Bounds for Eigenvalues of Operators in Hilbert Spaces

1966-10-01
David W. Fox , Werner C. Rheinboldt
Next article Computational Methods for Determining Lower Bounds for Eigenvalues of Operators in Hilbert SpacesDavid W. Fox and Werner C. RheinboldtDavid W. Fox and Werner C. Rheinboldthttps://doi.org/10.1137/1008101PDFBibTexSections ToolsAdd to favoritesExport … Next article Computational Methods for Determining Lower Bounds for Eigenvalues of Operators in Hilbert SpacesDavid W. Fox and Werner C. RheinboldtDavid W. Fox and Werner C. Rheinboldthttps://doi.org/10.1137/1008101PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1A] N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1961xi+147 MR0264420 0098.30702 Google Scholar[1B] N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. II, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1963v+218 MR0264421 Google Scholar[2A] N. Aronszajn, Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I. Operators in a Hilbert space, Proc. Nat. Acad. Sci. U. S. A., 34 (1948), 474–480 MR0027955 0031.40601 CrossrefISIGoogle Scholar[2B] N. Aronszajn, The Rayleigh-Ritz and the Weinstein methods for approximation of eigenvalues. II. Differential operators, Proc. Nat. Acad. Sci. U. S. A., 34 (1948), 594–601 MR0027956 0038.24803 CrossrefISIGoogle Scholar[3] N. Aronszajn, Approximation methods for eigenvalues of completely continuous symmetric operators, Proceedings of the Symposium on Spectral Theory and Differential Problems, Oklahoma Agricultural and Mechanical College, Stillwater, Okla., 1951, 179–202 MR0044736 0067.09101 Google Scholar[4] N. Aronszajn and , W. F. 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Bazley, , W. Borsch-Supan and , D. W. Fox, Lower bounds for eigenvalues of a quadratic form relative to a positive quadratic form, Tech. Memo, TG-657, Applied Physics Laboratory, The Johns Hopkins University, 1965 Google Scholar[10] Norman W. Bazley and , David W. Fox, Truncations in the method of intermediate problems for lower bounds to eigenvalues, J. Res. Nat. Bur. Standards Sect. B, 65B (1961), 105–111 MR0142897 0108.11303 CrossrefISIGoogle Scholar[11] Norman W. Bazley and , David W. Fox, Lower bounds for eigenvalues of Schrödinger's equation, Phys. Rev. (2), 124 (1961), 483–492 10.1103/PhysRev.124.483 MR0142898 0121.08501 CrossrefISIGoogle Scholar[12] Norman W. Bazley and , David W. Fox, Lower bounds to eigenvalues using operator decompositions of the form $B\sp{\ast} B$, Arch. Rational Mech. Anal., 10 (1962), 352–360 10.1007/BF00281201 MR0148216 0104.33702 CrossrefISIGoogle Scholar[13] Norman W. Bazley and , David W. Fox, Error bounds for eigenvectors of self-adjoint operators, J. Res. Nat. Bur. Standards Sect. B, 66B (1962), 1–4 MR0139265 0106.09101 CrossrefGoogle Scholar[14] Norman W. Bazley and , David W. Fox, A procedure for estimating eigenvalues, J. Mathematical Phys., 3 (1962), 469–471 10.1063/1.1724246 MR0144454 0108.11304 CrossrefISIGoogle Scholar[15] Norman W. Bazley and , David W. Fox, Error bounds for expectation values, Rev. Modern Phys., 35 (1963), 712–716 10.1103/RevModPhys.35.712 MR0160452 0132.43402 CrossrefGoogle Scholar[16] Norman W. Bazley and , David W. Fox, Lower bounds for energy levels of molecular systems, J. Mathematical Phys., 4 (1963), 1147–1153 10.1063/1.1704045 MR0165238 CrossrefISIGoogle Scholar[17] Normann W. Bazley and , David W. Fox, Improvement of bounds to eigenvalues of operators of the form $T\sp{\ast} T$, J. Res. Nat. Bur. Standards Sect. B, 68B (1964), 173–183 MR0179923 0163.38902 CrossrefGoogle Scholar[18] Norman W. Bazley and , David W. Fox, Error bounds for approximations to expectation values of unbounded operators, J. Mathematical Phys., 7 (1966), 413–416 10.1063/1.1704947 MR0202006 0158.45504 CrossrefISIGoogle Scholar[19] Norman W. Bazley and , David W. Fox, Comparison operators for lower bounds to eigenvalues, J. Reine Angew. Math., 223 (1966), 142–149 MR0203925 0144.17603 ISIGoogle Scholar[20] N. W. Bazley and , D. W. Fox, Methods for lower bounds to frequencies of continuous elastic systems, Z. Angew. Math. Phys., to appear Google Scholar[21] N. W. Bazley and , D. W. Fox, On a recent paper of Gay, Tech. Memo., TG-743, Applied Physics Laboratory, The Johns Hopkins University, 1965 Google Scholar[22] N. W. Bazley, , D. W. Fox and , J. T. Stadter, Upper and lower bounds for the frequencies of rectangular clamped plates, Tech. Memo., TG-626, Applied Physics Laboratory, The Johns Hopkins University, 1965 Google Scholar[23] N. W. Bazley, , D. W. Fox and , J. T. Stadter, Upper and lower bounds for the frequencies of rectangular cantilever plates, Tech. Memo., TG-705, Applied Physics Laboratory, The Johns Hopkins University, 1965 Google Scholar[24] N. W. Bazley, , D. W. Fox and , J. T. Stadter, Upper and lower bounds for the frequencies of rectangular free plates, Tech. Memo., TG-707, Applied Physics Laboratory, The Johns Hopkins University, 1965 Google Scholar[25] Earl A. Coddington and , Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955xii+429 MR0069338 0064.33002 Google Scholar[26] R. Courant and , D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953xv+561 MR0065391 0053.02805 Google Scholar[27] Nelson Dunford and , Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. 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Part II: Eigenvalue problemsESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, No. 5 | 23 October 2017 Cross Ref Guaranteed and Robust a Posteriori Bounds for Laplace Eigenvalues and Eigenvectors: Conforming ApproximationsEric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, and Martin VohralíkSIAM Journal on Numerical Analysis, Vol. 55, No. 5 | 19 September 2017AbstractPDF (2524 KB)Two-Sided Bounds for Eigenvalues of Differential Operators with Applications to Friedrichs, Poincaré, Trace, and Similar ConstantsIvana Šebestová and Tomáš VejchodskýSIAM Journal on Numerical Analysis, Vol. 52, No. 1 | 6 February 2014AbstractPDF (930 KB)Analysis of dominant and higher order modes for transmission lines using parallel cylindersIEEE Transactions on Microwave Theory and Techniques, Vol. 42, No. 4 | 1 Apr 1994 Cross Ref A diffusion mechanism for obstacle detection from size-change informationIEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 16, No. 1 | 1 Jan 1994 Cross Ref Related upper and lower bounds to atomic binding energiesInternational Journal of Quantum Chemistry, Vol. 46, No. 6 | 1 Jan 1993 Cross Ref Some Remarks Concerning Closure Rates for Aronszajn’s MethodNumerical Treatment of Eigenvalue Problems Vol. 5 / Numerische Behandlung von Eigenwertaufgaben Band 5 | 1 Jan 1991 Cross Ref Systematic lower bounds for lattice HamiltoniansJournal of Physics A: Mathematical and General, Vol. 22, No. 10 | 1 January 1999 Cross Ref An Extension of Aronszajn’S Rule: Slicing the Spectrum for Intermediate ProblemsChristopher BeattieSIAM Journal on Numerical Analysis, Vol. 24, No. 4 | 14 July 2006AbstractPDF (1749 KB)The eigenvalue problem for −Δu=λu with dirichlet boundary conditions for a certain class of two-dimensional regionsJournal of Sound and Vibration, Vol. 112, No. 3 | 1 Feb 1987 Cross Ref Lower Bounds for the Resonant Frequencies of Nonuniform Frame StructuresZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 67, No. 11 | 1 Jan 1987 Cross Ref Ein Stufenverfahren zur Berechnung von EigenwertschrankenNumerical Treatment of Eigenvalue Problems Vol.4 / Numerische Behandlung von Eigenwertaufgaben Band 4 | 1 Jan 1987 Cross Ref Convergence theorems for intermediate problemsProceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 100, No. 1-2 | 14 November 2011 Cross Ref Eigenvalues of the Laplacian in Two DimensionsJ. R. Kuttler and V. G. SigillitoSIAM Review, Vol. 26, No. 2 | 10 July 2006AbstractPDF (2896 KB)A New Method for Calculating TE and TM Cutoff Frequencies of Uniform Waveguides with Lunar or Eccentric Annular Cross SectionIEEE Transactions on Microwave Theory and Techniques, Vol. 32, No. 4 | 1 Apr 1984 Cross Ref Sloshing frequenciesZAMP Zeitschrift f�r angewandte Mathematik und Physik, Vol. 34, No. 5 | 1 Sep 1983 Cross Ref The Computation of Convergent Lower Bounds in Quantum Mechanical Eigenvalue ProblemsNumerical Treatment of Eigenvalue Problems Vol. 3 / Numerische Behandlung von Eigenwertaufgaben Band 3 | 1 Jan 1983 Cross Ref Zur Einschliessung von EigenwertenNumerische Behandlung von Differentialgleichungen Band 3 | 1 Jan 1981 Cross Ref Useful Technical Devices in Intermediate ProblemsNumerische Behandlung von Differentialgleichungen Band 3 | 1 Jan 1981 Cross Ref Upper and lower bounds for sloshing frequencies by intermediate problemsZAMP Zeitschrift f�r angewandte Mathematik und Physik, Vol. 32, No. 6 | 1 Jan 1981 Cross Ref Bounding Eigenvalues of Elliptic OperatorsJ. R. Kuttler and V. G. SigillitoSIAM Journal on Mathematical Analysis, Vol. 9, No. 4 | 17 February 2012AbstractPDF (473 KB)An Eigenvalue Estimation Method of Weinberger and Weinstein’s Intermediate ProblemsDavid W. Fox and James T. StadterSIAM Journal on Mathematical Analysis, Vol. 8, No. 3 | 17 February 2012AbstractPDF (1116 KB)Computation of upper and lower bounds to the frequencies of clamped cylindrical shellsEarthquake Engineering & Structural Dynamics, Vol. 4, No. 6 | 1 Oct 1976 Cross Ref The Rayleigh–Ritz and Weinstein–Bazley Methods Applied to a Class of Ordinary Differential Equations of the Second Order. IITetsuro YamamotoSIAM Journal on Numerical Analysis, Vol. 12, No. 3 | 14 July 2006AbstractPDF (682 KB)Direct Methods for Computing Eigenvalues of the Finite-Difference LaplacianJ. R. KuttlerSIAM Journal on Numerical Analysis, Vol. 11, No. 4 | 14 July 2006AbstractPDF (720 KB)Computation of upper and lower bounds to the frequencies of elastic systems by the method of Lehmann and MaehlyInternational Journal for Numerical Methods in Engineering, Vol. 6, No. 3 | 1 Jan 1973 Cross Ref Über stabile Eigenwerte und die Konvergenz des Weinstein-Aronszajn-Bazley-Fox-Verfahrens bei nicht notwendig rein diskretem SpektrumComputing, Vol. 9, No. 3 | 1 Sep 1972 Cross Ref New lower bounds for energies of radial lithiumChemical Physics Letters, Vol. 14, No. 5 | 1 Jul 1972 Cross Ref Chapter 7 Introduction to Variational PrinciplesThe Method of Weighted Residuals and Variational Principles - With Application in Fluid Mechanics, Heat and Mass Transfer | 1 Jan 1972 Cross Ref BibliographyMethods of Intermediate Problems for Eigenvalues: Theory and Ramifications | 1 Jan 1972 Cross Ref Natural frequencies and elastic stability of a simply-supported rectangular plate under linearly varying compressive loadsInternational Journal of Solids and Structures, Vol. 7, No. 5 | 1 May 1971 Cross Ref Untere Schranken f�r die Eigenwerte selbstadjungierter positiv-definiter OperatorenNumerische Mathematik, Vol. 17, No. 2 | 1 Apr 1971 Cross Ref Effect of a thermal gradient on the natural frequencies of a rectangular plateInternational Journal of Mechanical Sciences, Vol. 12, No. 2 | 1 Feb 1970 Cross Ref Volume 8, Issue 4| 1966SIAM Review427-591 History Submitted:22 March 1966Published online:18 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1008101Article page range:pp. 427-462ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

On Discretization and Differentiation of Operators with Application to Newton’s Method

1966-03-01
James M. Ortega , Werner C. Rheinboldt
In the numerical solution of operator equations $Fx = 0$, discretization of the equation and then application of Newton's method results in the same linear algebraic system of equations as … In the numerical solution of operator equations $Fx = 0$, discretization of the equation and then application of Newton's method results in the same linear algebraic system of equations as application of Newton's method followed by discretization. This leads to the general problem of determining when the two frequently used operations of discretization and (Frechet) differentiation applied to a nonlinear operator are commutative. A theory of discretization processes is developed here which proves that for a wide class of operators of interest in applications, discretization and differentiation indeed "commute". The fundamental concept of the theory is a distinction between the discretization of the linear spaces involved and the replacement of the infinitesimal parts of the operator F, i.e., those parts involving, e.g., differentiation and integration, by a discrete analogue. Using this distinction in an abstract way, a "complete" discretization process is defined precisely and the cited commutativity results are proven. The results are then applied to Newton's method.
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Numerical continuation methods: a perspective

2000-12-01
Werner C. Rheinboldt

MANPAK: A set of algorithms for computations on implicitly defined manifolds

1996-12-01
Werner C. Rheinboldt

On the Computation of Manifolds of Foldpoints for Parameter-Dependent Problems

1990-04-01
R.X. Dai , Werner C. Rheinboldt
For parameter-dependent nonlinear equations in $R^n $, a new augmentation method for computing submanifolds of foldpoints is presented. The underlying theory is based on a characterization, developed earlier, of foldpoints … For parameter-dependent nonlinear equations in $R^n $, a new augmentation method for computing submanifolds of foldpoints is presented. The underlying theory is based on a characterization, developed earlier, of foldpoints on solution manifolds of parameterized equations. The method allows for a locally convergent iterative computation of foldpoints of a certain class, for a continuation along paths of such foldpoints, and for a computation of simplicial approximations of higher-dimensional submanifolds of foldpoints of the same type.

On a Generalization of an Inequality of L. V. Kantorovich

1959-06-01
Werner Greub , Werner C. Rheinboldt
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A Geometric Framework for the Numerical Study of Singular Points

1987-06-01
James P. Fink , Werner C. Rheinboldt
While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such … While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter-dependent equation $F(z,\lambda ) = 0$. The presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points.

On the Numerical Solution of the Euler–Lagrange Equations

1995-02-01
Patrick J. Rabier , Werner C. Rheinboldt
The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the … The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the constraint manifold. The algorithm guarantees that the constraints are automatically satisfied and requires a minimal number of evaluations of second-order derivative terms. In fact, second-order derivatives are involved only through the second fundamental tensor of the constraint manifold. This tensor may be computed either explicitly when second derivatives are available or via an approximation procedure with excellent accuracy. Examples are given along with comparisons with state-of-the-art software.

Solution manifolds and submanifolds of parametrized equations and their discretization errors

1984-10-01
James P. Fink , Werner C. Rheinboldt

On a theorem of S. Smale about Newton's method for analytic mappings

1988-01-01
Werner C. Rheinboldt

A general existence and uniqueness theory for implicit differential-algebraic equations

1991-01-01
Patrick J. Rabier , Werner C. Rheinboldt
This paper presents a general existence and uniqueness theory for differentialalgebraic equations extending the well-known ODE theory.Both local and global aspects are considered, and the definition of the index for … This paper presents a general existence and uniqueness theory for differentialalgebraic equations extending the well-known ODE theory.Both local and global aspects are considered, and the definition of the index for nonlinear problems is elucidated.For the case of linear problems with constant coefficients the results are shown to provide an alternate treatment equivalent to the standard approach in terms of matrix pencils.Also, it is proved that general differential-algebraic equations carry a geometric content, in that they are locally equivalent to ODEs on a "constraint" manifold.A simple example from particle dynamics is given to illustrate our approach.
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ON CLASSES OF n-DIMENSIONAL NONLINEAR MAPPINGS GENERALIZING SEVERAL TYPES OF MATRICES**This work was supported in part by the National Science Foundation under Grant GJ-231 and the National Aeronautics and Space Administration under Grant NGL-21-002-008.

1971-01-01
Werner C. Rheinboldt

Time-dependent linear DAEs with discontinuous inputs

1996-11-01
Patrick J. Rabier , Werner C. Rheinboldt

Differential-Algebraic Systems as Differential Equations on Manifolds

1984-10-01
Werner C. Rheinboldt
Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in … Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.
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On a class of approximate iterative processes

1967-01-01
James M. Ortega , Werner C. Rheinboldt
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On Piecewise Affine Mappings in $R^n $

1975-12-01
Werner C. Rheinboldt , James S. Vandergraft
Let $H_1 , \cdots ,H_p $, be given hyperplanes which divide $R^n $ into finitely many closed, convex polytopes $\bar C_1 , \cdots \bar C_q $, and consider a continuous, … Let $H_1 , \cdots ,H_p $, be given hyperplanes which divide $R^n $ into finitely many closed, convex polytopes $\bar C_1 , \cdots \bar C_q $, and consider a continuous, piecewise affine mapping $F:R^n \to R^n ,Fx = A_j x + a^J ,\forall x \in \bar C_J ,j = 1, \cdots ,q$, with nonsingular $A_J \in L( {R^n } )$. Such functions arise, for instance, in the piecewise linear analysis of nonlinear resistive electric networks. We prove here that F is surjective if the signs of the determinants of all matrices $A_j $ are the same, and that then the Katzenelson algorithm for solving $Fx = b$ will always reach a solution. This extends recent results of Fujisawa and Kuh who proved a homeomorphism theorem for piecewise affine mappings and considered the Katzenelson algorithm in that case. We also augment their basic theorem in another direction by showing that when all $A_j $ are P- or M-matrices then F is a (surjective) P- or M-function, respectively. In the M-function case this implies, for example, the global convergence of the nonlinear Gauss–Seidel process.

On a computational method for the second fundamental tensor and its application to bifurcation problems

1990-12-01
Patrick J. Rabier , Werner C. Rheinboldt

On measures of ill-conditioning for nonlinear equations

1976-01-01
Werner C. Rheinboldt
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{x^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a solution of the nonlinear … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{x^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a solution of the nonlinear equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F x equals b"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Fx = b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a normed linear space and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{y^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that of a perturbed equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G x equals c"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Gx = c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Estimates for the relativized error between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{x^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{y^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are derived which extend a known estimate for the corresponding matrix case. The condition number of <italic>F</italic> depends now also on the domain, and special considerations are needed to determine the existence of the solution of the perturbed equation. For differentiable <italic>F</italic>, when the domain shrinks to a point, the condition number of <italic>F</italic> is shown to reduce to that of the derivative at that point.
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A posteriori finite element error estimators for parametrized nonlinear boundary value problems

1996-01-01
Jinn‐Liang Liu , Werner C. Rheinboldt
A posteriori error estimators for finite element solutions of multi—parameter nonlinear partial differential equations are based on an element—by—element solution of local linearizations of the nonlinear equation. In general, the … A posteriori error estimators for finite element solutions of multi—parameter nonlinear partial differential equations are based on an element—by—element solution of local linearizations of the nonlinear equation. In general, the associated bilinear form of the linearized Problems satisfies a Gårding—type inequality. Under appropriate assumption it is shown that the error estimators are bounded by constant multiples of the true error in a suitable norm. Computational experiments indicate that the estimators are effective, inexpensive, and insensitive to the choice of the local coordinate system on the solution manifold.

On the monotone convergence of Newton's method

1986-03-01
Florian A. Potra , Werner C. Rheinboldt

A General Convergence Result for Unconstrained Minimization Methods

1972-03-01
James M. Ortega , Werner C. Rheinboldt
This note extends an observation of J. Daniel and presents a general convergence theorem for iterative methods for unconstrained minimization problems. The key point is the concept of an essentially … This note extends an observation of J. Daniel and presents a general convergence theorem for iterative methods for unconstrained minimization problems. The key point is the concept of an essentially gradient-related sequence which includes the previously studied gradient-related sequences, as well as sequences which arise from univariate relaxation methods.

Local Error Estimates for Parametrized Nonlinear Equations

1985-08-01
P. James Fink , Werner C. Rheinboldt
This paper is concerned with the development of error estimates for parametrized nonlinear equations $F(z,\lambda ) = 0$ and their discretizations $F_h (z,\lambda ) = 0$. The estimates obtained are … This paper is concerned with the development of error estimates for parametrized nonlinear equations $F(z,\lambda ) = 0$ and their discretizations $F_h (z,\lambda ) = 0$. The estimates obtained are local error estimates in the sense of the local error in the numerical solution of ordinary differential equations. They represent a different approach to the general problem of error estimation, an approach which involves only a single discretized equation instead of a converging family of such equations.

Computing smooth solutions of DAEs for elastic multibody systems

1999-03-01
Werner C. Rheinboldt , Bernd Simeon

On the Local Convergence of Update Methods

1974-10-01
Werner C. Rheinboldt , James S. Vandergraft
This paper concerns a class of update methods, or, as they are also called, quasi-Newton methods, variable metric methods, or modification methods. A general theory of rank-one and symmetric rank-two … This paper concerns a class of update methods, or, as they are also called, quasi-Newton methods, variable metric methods, or modification methods. A general theory of rank-one and symmetric rank-two update formulas is presented which covers many of the special methods proposed in the literature. Recently, Broyden, Dennis and Moré found a local convergence theorem for a class of these methods. A new and unified proof of this theorem is given here which uses a geometrically more intuitive and also more general convergence condition than the original theorem. The proof utilizes elliptic-norm estimates to derive a majorizing system of difference inequalities. Then more careful estimates involving a generalized Frobenius norm show that under rather general conditions the methods under consideration are superlinearly convergent.

Theoretical and numerical analysis of differential-algebraic equations

2002-01-01
Patrick J. Rabier , Werner C. Rheinboldt

Numerical continuation methods: a perspective

2000-12-01
Werner C. Rheinboldt

Computing smooth solutions of DAEs for elastic multibody systems

1999-03-01
Werner C. Rheinboldt , Bernd Simeon

8. Unconstrained Minimization Methods

1998-01-01
Werner C. Rheinboldt
As we observed, the problem of solving a system of nonlinear equations may be replaced by a problem of minimizing a nonlinear functional on Rn . The literature in this … As we observed, the problem of solving a system of nonlinear equations may be replaced by a problem of minimizing a nonlinear functional on Rn . The literature in this area is extensive; see, e.g., the books by Fletcher [96], Gill, Murray, and Wright [109], and Dennis and Schnabel [66], as well as the bibliography by Berman [23]. We present here only a very modest introduction to some convergence theory for a few classes of methods for solving unconstrained minimization problems.

2. Model Problems

1998-01-01
Werner C. Rheinboldt
Systems of finitely many nonlinear equations in several real variables arise in connection with numerous scientific and technical problems and, correspondingly, they differ widely in form and properties. This chapter … Systems of finitely many nonlinear equations in several real variables arise in connection with numerous scientific and technical problems and, correspondingly, they differ widely in form and properties. This chapter introduces some typical examples of such systems without attempting to be exhaustive or to enter into details of the underlying problem areas. For further nonlinear model problems see [OR] and the collection by Moré [179].

7. Parametrized Systems of Equations

1998-01-01
Werner C. Rheinboldt
7.1 Submanifolds of Rn 7.1 Submanifolds of Rn

4. Methods of Newton Type

1998-01-01
Werner C. Rheinboldt
This chapter introduces several basic types of iterative methods derived by linearizations, including, in particular, the classical Newton method and a few of its modifications. Throughout the chapter, F:E⊂Rn→Rn denotes … This chapter introduces several basic types of iterative methods derived by linearizations, including, in particular, the classical Newton method and a few of its modifications. Throughout the chapter, F:E⊂Rn→Rn denotes a mapping defined on some open subset E of Rn .

Solving algebraically explicit DAEs with the MANPAK-manifold-algorithms

1997-02-01
Werner C. Rheinboldt

Differential-geometric computational methods for nonlinear problems

1996-12-31
Werner C. Rheinboldt

MANPAK: A set of algorithms for computations on implicitly defined manifolds

1996-12-01
Werner C. Rheinboldt

Discontinuous solutions of semilinear differential-algebraic equations—I. Distribution solutions

1996-12-01
Patrick J. Rabier , Werner C. Rheinboldt

Discontinuous solutions of semilinear differential-algebraic equations—II. P-consistency

1996-12-01
Patrick J. Rabier , Werner C. Rheinboldt

Time-dependent linear DAEs with discontinuous inputs

1996-11-01
Patrick J. Rabier , Werner C. Rheinboldt

Classical and generalized solutions of time-dependent linear differential-algebraic equations

1996-09-01
Patrick J. Rabier , Werner C. Rheinboldt

Geometric notes on optimization with equality constraints

1996-05-01
Werner C. Rheinboldt

A posteriori finite element error estimators for parametrized nonlinear boundary value problems

1996-01-01
Jinn‐Liang Liu , Werner C. Rheinboldt
A posteriori error estimators for finite element solutions of multi—parameter nonlinear partial differential equations are based on an element—by—element solution of local linearizations of the nonlinear equation. In general, the … A posteriori error estimators for finite element solutions of multi—parameter nonlinear partial differential equations are based on an element—by—element solution of local linearizations of the nonlinear equation. In general, the associated bilinear form of the linearized Problems satisfies a Gårding—type inequality. Under appropriate assumption it is shown that the error estimators are bounded by constant multiples of the true error in a suitable norm. Computational experiments indicate that the estimators are effective, inexpensive, and insensitive to the choice of the local coordinate system on the solution manifold.

On the Numerical Solution of the Euler–Lagrange Equations

1995-02-01
Patrick J. Rabier , Werner C. Rheinboldt
The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the … The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the constraint manifold. The algorithm guarantees that the constraints are automatically satisfied and requires a minimal number of evaluations of second-order derivative terms. In fact, second-order derivatives are involved only through the second fundamental tensor of the constraint manifold. This tensor may be computed either explicitly when second derivatives are available or via an approximation procedure with excellent accuracy. Examples are given along with comparisons with state-of-the-art software.

Performance analysis of some methods for solving Euler-Lagrange equations

1995-01-01
Werner C. Rheinboldt , Bernd Simeon

Finite difference methods for time dependent, linear differential algebraic equations

1994-03-01
Patrick J. Rabier , Werner C. Rheinboldt

On Impasse Points of Quasilinear Differential-Algebraic Equations

1994-01-01
Patrick J. Rabier , Werner C. Rheinboldt

On the Computation of Impasse Points of Quasi-Linear Differential-Algebraic Equations

1994-01-01
Patrick J. Rabier , Werner C. Rheinboldt
Abstract : We present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential-algebraic equations. Our method combines a reduction step transforming the DAE into … Abstract : We present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential-algebraic equations. Our method combines a reduction step transforming the DAE into a singular ODE with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.

On the computation of impasse points of quasilinear differential-algebraic equations

1994-01-01
Patrick J. Rabier , Werner C. Rheinboldt
We present computational algorithms for the calculation of impasse points and higher-order singularities in quasi-linear differential-algebraic equations. Our method combines a reduction step, transforming the DAE into a singular ODE, … We present computational algorithms for the calculation of impasse points and higher-order singularities in quasi-linear differential-algebraic equations. Our method combines a reduction step, transforming the DAE into a singular ODE, with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.
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On the Sensitivity of Solutions of Parameterized Equations

1993-04-01
Werner C. Rheinboldt
The sensitivity of a solution of a parameterized equation $F(z,\lambda ) = 0$ with respect to the parameter vector $\lambda $ is usually defined as the change of the state … The sensitivity of a solution of a parameterized equation $F(z,\lambda ) = 0$ with respect to the parameter vector $\lambda $ is usually defined as the change of the state z in dependence of $\lambda $. In other words, for any solution expressible in the form $(z(\lambda ),\lambda )$ with some smooth function $z = z(\lambda )$, the sensitivity is the derivative $Dz(\lambda )$. Typically the solutions form a manifold M in the product of 1the state space and the parameter space and this sensitivity is available only at those points of M where the parameters can be used to define a local coordinate system. This paper introduces a general sensitivity concept which applies at all solutions on M and which includes the earlier definition. Some general geometric interpretations of the new measure are presented and it is shown that the sensitivity analysis can be easily integrated into the solution process. The theory also suggests the introduction of a readily computable second-order sensitivity measure reflecting the curvature behavior of M. Two numerical examples illustrate the discussion.

On Impasse Points of Quasilinear Differential Algebraic Equations

1992-06-01
Patrick J. Rabier , Werner C. Rheinboldt
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On the existence and uniqueness of solutions of nonlinear semi-implicit differential-algebraic equations

1991-01-01
Werner C. Rheinboldt

Continuation Techniques and Bifurcation Problems.

1991-01-01
Werner C. Rheinboldt , Hans D. Mittelmann , Dirk Roose
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A general existence and uniqueness theory for implicit differential-algebraic equations

1991-01-01
Patrick J. Rabier , Werner C. Rheinboldt
This paper presents a general existence and uniqueness theory for differentialalgebraic equations extending the well-known ODE theory.Both local and global aspects are considered, and the definition of the index for … This paper presents a general existence and uniqueness theory for differentialalgebraic equations extending the well-known ODE theory.Both local and global aspects are considered, and the definition of the index for nonlinear problems is elucidated.For the case of linear problems with constant coefficients the results are shown to provide an alternate treatment equivalent to the standard approach in terms of matrix pencils.Also, it is proved that general differential-algebraic equations carry a geometric content, in that they are locally equivalent to ODEs on a "constraint" manifold.A simple example from particle dynamics is given to illustrate our approach.
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On a computational method for the second fundamental tensor and its application to bifurcation problems

1990-12-01
Patrick J. Rabier , Werner C. Rheinboldt

On the Computation of Manifolds of Foldpoints for Parameter-Dependent Problems

1990-04-01
R.X. Dai , Werner C. Rheinboldt
For parameter-dependent nonlinear equations in $R^n $, a new augmentation method for computing submanifolds of foldpoints is presented. The underlying theory is based on a characterization, developed earlier, of foldpoints … For parameter-dependent nonlinear equations in $R^n $, a new augmentation method for computing submanifolds of foldpoints is presented. The underlying theory is based on a characterization, developed earlier, of foldpoints on solution manifolds of parameterized equations. The method allows for a locally convergent iterative computation of foldpoints of a certain class, for a continuation along paths of such foldpoints, and for a computation of simplicial approximations of higher-dimensional submanifolds of foldpoints of the same type.

On the computation of finite invariant sets of mappings

1989-01-01
Alex Gelman , Werner C. Rheinboldt
This paper suggests a new computational method for determining closed curves that are invariant under a given mapping. Unlike other authors, we discretize not only the curve but also the … This paper suggests a new computational method for determining closed curves that are invariant under a given mapping. Unlike other authors, we discretize not only the curve but also the mapping itself. This allows us to avoid completely the computational difficulties connected with the numerical solution of large linear systems. The method uses simple recurrence formulas, which greatly reduce the execution times.
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Lectures on Numerical Methods in Bifurcation Problems (Herbert B. Kelley)

1988-12-01
Werner C. Rheinboldt

Error Questions in the Computation of Solution Manifolds of Parametrized Equations11This work was in part supported by the Office of Naval Research under contract N-00014–80-C-9455 and the National Science Foundation under grant DCR-8309926.

1988-01-01
Werner C. Rheinboldt

On the computation of multi-dimensional solution manifolds of parametrized equations

1988-01-01
Werner C. Rheinboldt

On a theorem of S. Smale about Newton's method for analytic mappings

1988-01-01
Werner C. Rheinboldt

A Geometric Framework for the Numerical Study of Singular Points

1987-06-01
James P. Fink , Werner C. Rheinboldt
While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such … While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter-dependent equation $F(z,\lambda ) = 0$. The presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points.

Folds on the Solution Manifold of a Parametrized Equation

1986-08-01
James P. Fink , Werner C. Rheinboldt
This paper presents a framework for studying singular points on the solution manifold of a parameter-dependent nonlinear equation $F(z,\lambda ) = 0$. The approach is based on a systematic combination … This paper presents a framework for studying singular points on the solution manifold of a parameter-dependent nonlinear equation $F(z,\lambda ) = 0$. The approach is based on a systematic combination of general constrained mappings with the tangent map of differential geometry. This framework is then used to develop a geometrically instructive and coordinate-free treatment of fold points on the solution manifold. The treatment includes a detailed analysis of the types of points that may occur on fold lines.

Numerical analysis of parametrized nonlinear equations

1986-04-01
Werner C. Rheinboldt

On the monotone convergence of Newton's method

1986-03-01
Florian A. Potra , Werner C. Rheinboldt

A Mesh-Independence Principle for Operator Equations and Their Discretizations

1986-02-01
Eugene L. Allgower , Klaus Böhmer , F. A. Potra +1 more
The mesh-independence principle asserts that, when Newton’s method is applied to a nonlinear equation between some Banach spaces as well as to some finite-dimensional discretization of that equation, then the … The mesh-independence principle asserts that, when Newton’s method is applied to a nonlinear equation between some Banach spaces as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration and, as a consequence, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved only for certain classes of boundary value problems. In this paper a proof is presented for a general class of operator equations and discretizations. It covers the earlier results and extends them well beyond the cases that have been considered before.

Local Error Estimates for Parametrized Nonlinear Equations

1985-08-01
P. James Fink , Werner C. Rheinboldt
This paper is concerned with the development of error estimates for parametrized nonlinear equations $F(z,\lambda ) = 0$ and their discretizations $F_h (z,\lambda ) = 0$. The estimates obtained are … This paper is concerned with the development of error estimates for parametrized nonlinear equations $F(z,\lambda ) = 0$ and their discretizations $F_h (z,\lambda ) = 0$. The estimates obtained are local error estimates in the sense of the local error in the numerical solution of ordinary differential equations. They represent a different approach to the general problem of error estimation, an approach which involves only a single discretized equation instead of a converging family of such equations.

A program design for an adaptive, nonlinear finite element solver

1985-01-01
Frank R. Sledge , Werner C. Rheinboldt

Solution manifolds and submanifolds of parametrized equations and their discretization errors

1984-10-01
James P. Fink , Werner C. Rheinboldt

Differential-Algebraic Systems as Differential Equations on Manifolds

1984-10-01
Werner C. Rheinboldt
Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in … Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.
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Differential-algebraic systems as differential equations on manifolds

1984-01-01
Werner C. Rheinboldt
Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in … Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.
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The Role of the Tangent Mapping in Analyzing Bifurcation Behaviour

1984-01-01
James P. Fink , Werner C. Rheinboldt
Abstract In the study of solution manifolds of parameter‐dependent nonlinear equations extended systems of equations play an important role, especially for the computation of singular points, such as turning points, … Abstract In the study of solution manifolds of parameter‐dependent nonlinear equations extended systems of equations play an important role, especially for the computation of singular points, such as turning points, bifurcation points, etc. Various extended systems have been proposed in the literature. Here it is shown that a central feature in the construction of extended systems is the tangent map of differential geometry. A theory of extended equations based on the tangent map is presented which also exhibits the close connection with the choice of local coordinate systems. The ideas and results are illustrated with an example of a continuously stirred chemical reactor.

A locally parameterized continuation process

1983-06-01
Werner C. Rheinboldt , John Burkardt
article Free AccessA locally parameterized continuation process Authors: Werner C. Rheinboldt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational … article Free AccessA locally parameterized continuation process Authors: Werner C. Rheinboldt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile , John V. Burkardt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 9Issue 2pp 215–235https://doi.org/10.1145/357456.357460Published:01 June 1983Publication History 195citation923DownloadsMetricsTotal Citations195Total Downloads923Last 12 Months29Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
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A comparison of methods for determining turning points of nonlinear equations

1982-09-01
Rami Melhem , Werner C. Rheinboldt

On Steplength Algorithms for a Class of Continuation Methods

1981-10-01
C. den Heijer , Werner C. Rheinboldt
The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + … The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + 1} \to R^n $, for given $b \in R^n $. While these methods are similar in structure to those used for ODE-solvers, their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step. This suggests the use of a posteriors estimates of the convergence radii of the corrector. It is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about F. However, it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors. This is used for the design of various step-algorithms for continuation processes; two of them are based on a Newton-corrector while the third one is applicable to any corrector. Some numerical results show the effectiveness of the three algorithms. Finally some asymptotic analysis of continuation steps is given.

Numerical analysis of continuation methods for nonlinear structural problems

1981-06-01
Werner C. Rheinboldt

NUMERICAL ANALYSIS OF CONTINUATION METHODS FOR NONLINEAR STRUCTURAL PROBLEMS††This work was in part supported by the National Science Foundation under grant MCS-78-05299.

1981-01-01
Werner C. Rheinboldt
Coauthor Papers Together
James M. Ortega 16
Patrick J. Rabier 13
James P. Fink 4
Werner Greub 3
James S. Vandergraft 3
Bernd Simeon 2
John Burkardt 1
Rami Melhem 1
Eugene L. Allgower 1
L. J. Slater 1
Hans D. Mittelmann 1
С. Л. Соболев 1
E. I. 1
Florian A. Potra 1
C. K. Mesztenyi 1
P. James Fink 1
Dirk Roose 1
F. A. Potra 1
Alex Gelman 1
David W. Fox 1
R.X. Dai 1
Jorge J. Morè 1
A. S. Householder 1
Frank R. Sledge 1
Jinn‐Liang Liu 1
J. M. Ortega 1
C. den Heijer 1
Klaus Böhmer 1

On the computation of multi-dimensional solution manifolds of parametrized equations

1988-01-01
Werner C. Rheinboldt

Numerical analysis of parametrized nonlinear equations

1986-04-01
Werner C. Rheinboldt

Iterative Solution of Nonlinear Equations in Several Variables

1971-04-01
E. I. , J. M. Ortega , Werner C. Rheinboldt

Solution Fields of Nonlinear Equations and Continuation Methods

1980-04-01
Werner C. Rheinboldt
Contihuation methods are considered here in the broad sense as algorithms for the computational analysis of specified parts of the solution field of equations of the form $Fx = b$ … Contihuation methods are considered here in the broad sense as algorithms for the computational analysis of specified parts of the solution field of equations of the form $Fx = b$ , where $F:R^m \to R^n $ is a given mapping and $m > n$. Such problems arise, for instance, in structural mechanics and then usually $m - n$ of the variables $x_i $ are designated as parameters. For the case $m = n + 1$ an existence theory for the regular curves of the solution field is developed here. Then approximate solutions are considered and shown to be solutions of certain perturbed problems. These results are used to prove that for the continuation methods with Eider-predictor and Newton-corrector a particular steplength algorithm is guaranteed to trace any regular solution of the field. Some numerical aspects of the procedure are discussed and a numerical example is included to illustrate the effectiveness of the approach.

Solution manifolds and submanifolds of parametrized equations and their discretization errors

1984-10-01
James P. Fink , Werner C. Rheinboldt

On a computational method for the second fundamental tensor and its application to bifurcation problems

1990-12-01
Patrick J. Rabier , Werner C. Rheinboldt

A locally parameterized continuation process

1983-06-01
Werner C. Rheinboldt , John Burkardt
article Free AccessA locally parameterized continuation process Authors: Werner C. Rheinboldt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational … article Free AccessA locally parameterized continuation process Authors: Werner C. Rheinboldt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile , John V. Burkardt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 9Issue 2pp 215–235https://doi.org/10.1145/357456.357460Published:01 June 1983Publication History 195citation923DownloadsMetricsTotal Citations195Total Downloads923Last 12 Months29Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
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A Geometric Framework for the Numerical Study of Singular Points

1987-06-01
James P. Fink , Werner C. Rheinboldt
While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such … While bifurcation theory has developed rapidly in recent years, there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems. This paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter-dependent equation $F(z,\lambda ) = 0$. The presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points.

Perturbed bifurcation theory

1973-01-01
James P. Keener , Herbert B. Keller

Global Homotopies and Newton Methods

1978-01-01
Herbert B. Keller

Implicit differential equations near a singular point

1989-12-01
Patrick J. Rabier

An adaptive continuation process for solving systems of nonlinear equations

1978-01-01
Werner C. Rheinboldt
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The continuation method for functional equations

1951-11-01
F. A. Ficken

Differential-algebraic equations and their numerical treatment

1986-01-01
E. Griepentrog , Roswitha März

Characterization and Computation of Generalized Turning Points

1984-02-01
Andreas Griewank , G. W. Reddien
A minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency. Applications to determining … A minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency. Applications to determining turning points, bifurcation points and constrained optima are presented. The numerical solution of the equations using variants of Newton’s method is also discussed.

On Steplength Algorithms for a Class of Continuation Methods

1981-10-01
C. den Heijer , Werner C. Rheinboldt
The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + … The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + 1} \to R^n $, for given $b \in R^n $. While these methods are similar in structure to those used for ODE-solvers, their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step. This suggests the use of a posteriors estimates of the convergence radii of the corrector. It is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about F. However, it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors. This is used for the design of various step-algorithms for continuation processes; two of them are based on a Newton-corrector while the third one is applicable to any corrector. Some numerical results show the effectiveness of the three algorithms. Finally some asymptotic analysis of continuation steps is given.

On the Computation of Impasse Points of Quasi-Linear Differential-Algebraic Equations

1994-01-01
Patrick J. Rabier , Werner C. Rheinboldt
Abstract : We present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential-algebraic equations. Our method combines a reduction step transforming the DAE into … Abstract : We present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential-algebraic equations. Our method combines a reduction step transforming the DAE into a singular ODE with an augmentation procedure inspired by numerical bifurcation theory. Singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory. Two numerical examples with physical relevance are given.

Solving Ordinary Differential Equations II

1996-01-01
Ernst Hairer , Gerhard Wanner

An efficient algorithm for the determination of certain bifurcation points

1978-01-01
James P. Abbott

On the existence and uniqueness of solutions of nonlinear semi-implicit differential-algebraic equations

1991-01-01
Werner C. Rheinboldt

A comparison of methods for determining turning points of nonlinear equations

1982-09-01
Rami Melhem , Werner C. Rheinboldt

On the Numerical Solution of the Euler–Lagrange Equations

1995-02-01
Patrick J. Rabier , Werner C. Rheinboldt
The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the … The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the constraint manifold. The algorithm guarantees that the constraints are automatically satisfied and requires a minimal number of evaluations of second-order derivative terms. In fact, second-order derivatives are involved only through the second fundamental tensor of the constraint manifold. This tensor may be computed either explicitly when second derivatives are available or via an approximation procedure with excellent accuracy. Examples are given along with comparisons with state-of-the-art software.

Applications of the blowing-up construction and algebraic geometry to bifurcation problems

1983-06-01
Michael Büchner , Jerrold E. Marsden , Stephen Schecter

Monotone Iterations for Nonlinear Equations with Application to Gauss-Seidel Methods

1967-06-01
James M. Ortega , Werner C. Rheinboldt
Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods
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Some Efficient Algorithms for Solving Systems of Nonlinear Equations

1973-04-01
Richard P. Brent
We compare the Ostrowski efficiency of some methods for solving systems of nonlinear equations without explicitly using derivatives. The methods considered include the discrete Newton method, Shamanskii’s method, the two-point … We compare the Ostrowski efficiency of some methods for solving systems of nonlinear equations without explicitly using derivatives. The methods considered include the discrete Newton method, Shamanskii’s method, the two-point secant method, and Brown’s methods. We introduce a class of secant methods and a class of methods related to Brown’s methods, but using orthogonal rather than stabilized elementary transformations. The idea of these methods is to avoid finding a new approximation to the Jacobian matrix of the system at each step, and thus increase the efficiency. Local convergence theorems are proved, and the efficiencies of the methods are calculated. Numerical results are given, and some possible extensions are mentioned.

Constrained equations; a study of implicit differential equations and their discontinuous solutions

1976-01-01
Floris Takens

Numerical Methods for a Class of Finite Dimensional Bifurcation Problems

1978-02-01
Werner C. Rheinboldt
For an equation $H(y,t) = 0$, where $H:D \subset R^{n + 1} \to R^n $, let $p:J \subset R^1 \to R^n $ be a primary solution on which a simple … For an equation $H(y,t) = 0$, where $H:D \subset R^{n + 1} \to R^n $, let $p:J \subset R^1 \to R^n $ be a primary solution on which a simple bifurcation point $p^ * = p(t^ * )$ with rank $H_y = (p^ * ,t^ * ) = n - 1$ has been detected and a secondary solution is branching off. An iterative process is presented which starts at a point $p^0 = p(t_0 )$ near $p^ * $ and converges to a point on the secondary curve. It is similar in form to methods proposed by H. B. Kelley and others but has considerably lower computational complexity. The process represents a chord iteration with singular iteration matrix and its convergence is derived from a general result for such singular chord iterations. Computational details for the implementation of the method and an informal program are given. Finally, some comments about extensions to the case rank $H_y (p^ * ,t^ * ) < n - 1$ are made.

On mildly nonlinear partial difference equations of elliptic type

1953-11-01
Lipman Bers
The use of t he fini te d ifferences met hod is in solving t he boundar y value problem of t he first kind for t he nonlinear elli … The use of t he fini te d ifferences met hod is in solving t he boundar y value problem of t he first kind for t he nonlinear elli ptic equation A</ > = F (X,y,</>, </>., cf>u) is justified by first showing t hat t h e problem of the corresp onding difference equat ion has a uniqu e solution, and t h en t hat t he solution of t he differe nce e quat ion tends t o that of t he different ial e quation when t he net uni t te nds t o zero.Also a numerical m cthod of t he Liebmann type for t he computa tion of t he solution of t he diffe re nce equat ion is deyeloped , and t hese result s are extended to more general n onlinear ellipt ic equat ions.

Numerical Solution of Systems of Nonlinear Equations

1963-10-01
Ferdinand Freudenstein , Bernhard Roth
article Free AccessNumerical Solution of Systems of Nonlinear Equations Authors: Ferdinand Freudenstein Department of Mechanical Engineering, Columbia University and Stanford University Department of Mechanical Engineering, Columbia University and Stanford UniversityView … article Free AccessNumerical Solution of Systems of Nonlinear Equations Authors: Ferdinand Freudenstein Department of Mechanical Engineering, Columbia University and Stanford University Department of Mechanical Engineering, Columbia University and Stanford UniversityView Profile , Bernhard Roth Department of Mechanical Engineering, Columbia University and Stanford University Department of Mechanical Engineering, Columbia University and Stanford UniversityView Profile Authors Info & Claims Journal of the ACMVolume 10Issue 4pp 550–556https://doi.org/10.1145/321186.321200Published:01 October 1963Publication History 92citation2,067DownloadsMetricsTotal Citations92Total Downloads2,067Last 12 Months395Last 6 weeks105 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
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On Impasse Points of Quasilinear Differential-Algebraic Equations

1994-01-01
Patrick J. Rabier , Werner C. Rheinboldt

MANPAK: A set of algorithms for computations on implicitly defined manifolds

1996-12-01
Werner C. Rheinboldt

Folds in Solutions of Two Parameter Systems and Their Calculation. Part I

1985-04-01
Allan Jepson , A. Spence
This paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters. It is well known that these paths are solutions of a particular extended … This paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters. It is well known that these paths are solutions of a particular extended system of nonlinear equations. In this paper both regular points and simple turning points in the extended system are related to the local geometry of the solution surface of the original nonlinear system. A description is given of numerical methods both for solving the extended system and for calculating certain quantities which determine the local geometry of the solution surface. Applications to perturbed bifurcation, to the formation of isolas, and to the calculation of the multiplicity of solutions are also discussed. Numerical examples are given.

Iteration methods for nonlinear problems

1962-01-01
Samuel Schechter
Analogous methods have been used in practice, with apparent success, on nonlinear problems as well. For the most part, these have not been justified mathematically and this work is an … Analogous methods have been used in practice, with apparent success, on nonlinear problems as well. For the most part, these have not been justified mathematically and this work is an attempt to fill this gap. In particular it is shown that the relaxation methods yield solutions to problems arising from the minimization of certain convex functions. In practice, these functions are obtained by approximating multiple integrals in a calculus of variations problem. It is shown that an approximate Plateau problem may be solved by a successive displacements method, or a method analogous to Liebmann's method. We at the same time obtain an extension of a free steering theorem for positive definite symmetric matrices given as Theorem 4 of [3], and results of Ostrowski [2].
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An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold

1985-04-01
Eugene L. Allgower , Phillip H. Schmidt
A simplicial method is used to approximate the solution manifold to a system of nonlinear equations, $H(x) = \theta $, where $H:\mathbb{R}^{N + K} \to \mathbb{R}^N $ The method begins … A simplicial method is used to approximate the solution manifold to a system of nonlinear equations, $H(x) = \theta $, where $H:\mathbb{R}^{N + K} \to \mathbb{R}^N $ The method begins at a point $x_0 $ in the solution set where the derivative $DH(x_0 )$ is of full rank. Given any $\varepsilon > 0$, a piecewise linear manifold is constructed along which $\| {H(x)} \|_\infty < \varepsilon $. An algorithm is presented to carry out this construction in an efficient fashion.

On Discretization and Differentiation of Operators with Application to Newton’s Method

1966-03-01
James M. Ortega , Werner C. Rheinboldt
In the numerical solution of operator equations $Fx = 0$, discretization of the equation and then application of Newton's method results in the same linear algebraic system of equations as … In the numerical solution of operator equations $Fx = 0$, discretization of the equation and then application of Newton's method results in the same linear algebraic system of equations as application of Newton's method followed by discretization. This leads to the general problem of determining when the two frequently used operations of discretization and (Frechet) differentiation applied to a nonlinear operator are commutative. A theory of discretization processes is developed here which proves that for a wide class of operators of interest in applications, discretization and differentiation indeed "commute". The fundamental concept of the theory is a distinction between the discretization of the linear spaces involved and the replacement of the infinitesimal parts of the operator F, i.e., those parts involving, e.g., differentiation and integration, by a discrete analogue. Using this distinction in an abstract way, a "complete" discretization process is defined precisely and the cited commutativity results are proven. The results are then applied to Newton's method.
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The solvability of non-linear functional equations

1963-12-01
Felix E. Browder

Finite difference methods for time dependent, linear differential algebraic equations

1994-03-01
Patrick J. Rabier , Werner C. Rheinboldt

The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods

1989-01-01
Ernst Hairer , Michel Roche , Christian Lubich
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Principles of functional analysis

2018-08-08
Martin Schechter
Basic notions Duality Linear operators The Riesz theory for compact operators Fredholm operators Spectral theory Unbounded operators Reflexive Banach spaces Banach algebras Semigroups Hilbert space Bilinear forms Selfadjoint operators Measures … Basic notions Duality Linear operators The Riesz theory for compact operators Fredholm operators Spectral theory Unbounded operators Reflexive Banach spaces Banach algebras Semigroups Hilbert space Bilinear forms Selfadjoint operators Measures of operators Examples and applications Glossary Major Theorems Bibliography Index.
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Zur L�sung parameterabh�ngiger nichtlinearer Gleichungen mit singul�ren Jacobi-Matrizen

1978-03-01
R. Menzel , H. Schwetlick

Solving algebraically explicit DAEs with the MANPAK-manifold-algorithms

1997-02-01
Werner C. Rheinboldt

Characterization and Computation of Singular Points with Maximum Rank Deficiency

1986-10-01
Patrick J. Rabier , G. W. Reddien
Characterizations are given of points at which a mapping has a rank drop possibly greater than one. Multiple bifurcation points can then be computed without needing a full unfolding of … Characterizations are given of points at which a mapping has a rank drop possibly greater than one. Multiple bifurcation points can then be computed without needing a full unfolding of the singularity. Nondegeneracy conditions are studied in detail.

A General Form for Solvable Linear Time Varying Singular Systems of Differential Equations

1987-07-01
Stephen L. Campbell
A canonical form is derived for all linear solvable systems $E(t)x'(t) + F(t)x(t) = f(t)$ with sufficiently smooth coefficients E, F Using this form it is shown that for all … A canonical form is derived for all linear solvable systems $E(t)x'(t) + F(t)x(t) = f(t)$ with sufficiently smooth coefficients E, F Using this form it is shown that for all smooth enough solvable systems a class of recently defined numerical imbedding methods and an algorithm to compute the manifold of consistent initial conditions always work. In addition, necessary and sufficient conditions are given on $E(t)$, $F(t)$ to insure solvability in the case when $E(t)$, $F(t)$ are infinitely differentiable.

Perturbation Theory for Linear Operators

1995-01-01
Tosio Kato
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The Calculation of Turning Points of Nonlinear Equations

1980-08-01
Gerald Moore , A. Spence
This paper is concerned with the determination of a type of singular point, called a "turning" or "limit" point, of nonlinear equations depending on a parameter. An enlarged system is … This paper is concerned with the determination of a type of singular point, called a "turning" or "limit" point, of nonlinear equations depending on a parameter. An enlarged system is introduced for which the turning point is a nonsingular solution and thus standard methods can be used to compute it. An efficient implementation of Newton's method in the finite-dimensional case is presented and numerical results for discretizations of a differential equation and an integral equation are given.

Canonical forms for linear differential-algebraic equations with variable coefficients

1994-12-01
Peter Kunkel , Volker Mehrmann

On the solutions of linear differential equations with singular coefficients

1982-12-01
Daniel Cobb

Extrapolation integrators for constrained multibody systems

1991-09-01
Ch. Lubich

Existence of solutions of certain systems of non-linear inequalities

1968-11-01
S. Karamardian

On the Computation of Manifolds of Foldpoints for Parameter-Dependent Problems

1990-04-01
R.X. Dai , Werner C. Rheinboldt
For parameter-dependent nonlinear equations in $R^n $, a new augmentation method for computing submanifolds of foldpoints is presented. The underlying theory is based on a characterization, developed earlier, of foldpoints … For parameter-dependent nonlinear equations in $R^n $, a new augmentation method for computing submanifolds of foldpoints is presented. The underlying theory is based on a characterization, developed earlier, of foldpoints on solution manifolds of parameterized equations. The method allows for a locally convergent iterative computation of foldpoints of a certain class, for a continuation along paths of such foldpoints, and for a computation of simplicial approximations of higher-dimensional submanifolds of foldpoints of the same type.