Efficient Numerical Pathfollowing Beyond Critical Points The paper discusses methods for the computation of solution paths of a nonlinear system F(y)=0, F:D⊂R n+1 →R n with rank F’(y)=n for y∈D. In particular, for the corrector iteration the Gauss-Newton method is used where the QR-factorization of the Jacobian is computed only once and thereafter a rank-one update is applied. The choice of the steplength of the predicted step along the tangent line is theoretically analyzed in terms of the convergence properties of the corrector. Then the detection and determination of critical points is discussed including, in particular, the computation of simple bifurcation points and branch switching at such points. Some numerical comparisons illustrate the efficiency of the implementation of these techniques in the authors’ continuation code ALCON. ALCON1: (Al)gebraic system of equations (Con)tinuation method. Pathfollowing method for parameter-dependent nonlinear systems of equations, without using the analytical Jacobian of the system, and without computation of bifurcations: ALCON2: (Al)gebraic system of equations (Con)tinuation method. Pathfollowing method for parameter-dependent nonlinear systems of equations, using the analytical Jacobian of the system, and with computation of simple bifurcations. (Source:

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  1. Andò, Alessia; Breda, Dimitri; Scarabel, Francesca: Numerical continuation and delay equations: a novel approach for complex models of structured populations (2020)
  2. Deuflhard, Peter: The grand four: affine invariant globalizations of Newton’s method (2018)
  3. Wulff, Claudia; Schebesch, Andreas: Numerical continuation of Hamiltonian relative periodic orbits (2008)
  4. Bykov, Viatcheslav; Goldfarb, Igor; Gol’dshtein, Vladimir; Maas, Ulrich: On a modified version of ILDM approach: asymptotic analysis based on integral manifolds (2006)
  5. Leykin, Anton; Verschelde, Jan; Zhao, Ailing: Newton’s method with deflation for isolated singularities of polynomial systems (2006)
  6. Jackiewicz, Z.: Construction and implementation of general linear methods for ordinary differential equations: a review (2005)
  7. Deuflhard, Peter; Hohmann, Andreas: Numerical analysis in modern scientific computing. An introduction. (2003)
  8. Beyn, Wolf-Jürgen; Champneys, Alan; Doedel, Eusebius; Govaerts, Willy; Kuznetsov, Yuri A.; Sandstede, Björn: Numerical continuation, and computation of normal forms. (2002)
  9. De Almeida, Valmor F.; Derby, Jeffrey J.: Construction of solution curves for large two-dimensional problems of steady-state flows of incompressible fluids (2000)
  10. Govaerts, Willy J. F.: Numerical methods for bifurcations of dynamical equilibria (2000)
  11. Rabier, Patrick J.: The Hopf bifurcation theorem for quasilinear differential-algebraic equations. (1999)
  12. Maas, Ulrich: Efficient calculation of intrinsic low-dimensional manifolds for the simplification of chemical kinetics (1998)
  13. Hinze, Michael: On the numerical approximation and computation of minimal surface continua bounded by one-parameter families of polygonal contours (1997)
  14. Reif, Konrad; Weinzierl, Klaus; Zell, Andreas; Unbehauen, Rolf: Nonlinear feedback stabilization by tangential linearization (1997)
  15. Butcher, J. C.; Jackiewicz, Z.: Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations (1996)
  16. Hohmann, Andreas: Inexact Gauss Newton methods for parameter dependent nonlinear problems (1994)
  17. Beyn, W.-J.: On smoothness and invariance properties of the Gauss-Newton method (1993)
  18. Van de Velde, Eric F.; Lorenz, Jens: Adaptive data distribution for concurrent continuation (1992)
  19. Gatermann, Karin: Mixed symbolic-numeric solution of symmetrical nonlinear systems (1991)
  20. Gatermann, Karin; Hohmann, Andreas: Symbolic exploitation of symmetry in numerical pathfollowing (1991)

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