Advertisement

Mathematical Programming

, Volume 62, Issue 1–3, pp 427–459 | Cite as

Local convergence analysis of tensor methods for nonlinear equations

  • Dan Feng
  • Paul D. Frank
  • Robert B. Schnabel
Article

Abstract

Tensor methods for nonlinear equations base each iteration upon a standard linear model, augmented by a low rank quadratic term that is selected in such a way that the mode is efficient to form, store, and solve. These methods have been shown to be very efficient and robust computationally, especially on problems where the Jacobian matrix at the root has a small rank deficiency. This paper analyzes the local convergence properties of two versions of tensor methods, on problems where the Jacobian matrix at the root has a null space of rank one. Both methods augment the standard linear model by a rank one quadratic term. We show under mild conditions that the sequence of iterates generated by the tensor method based upon an “ideal” tensor model converges locally and two-step Q-superlinearly to the solution with Q-order 3/2, and that the sequence of iterates generated by the tensor method based upon a practial tensor model converges locally and three-step Q-superlinearly to the solution with Q-order 3/2. In the same situation, it is known that standard methods converge linearly with constant converging to 1/2. Hence, tensor methods have theoretical advantages over standard methods. Our analysis also confirms that tensor methods converge at least quadratically on problems where the Jacobian matrix at the root is nonsingular.

Key words

Nonlinear equations singular Jacobian tensor methods local convergence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Bouaricha, “Solving large sparse systems of nonlinear equations and nonlinear least squares problems using tensor methods on sequential and parallel computers,” Ph.D. dissertation, Department of Computer Science, University of Colorado (Boulder, CO, 1992).Google Scholar
  2. [2]
    A. Bouaricha and R.B. Schnabel, “A software package for tensor methods for nonlinear equations and nonlinear least squares,” in preparation.Google Scholar
  3. [3]
    D.W. Decker, H.B. Keller and C.T. Kelley, “Convergence rate for Newton's method at singular points,”SIAM Journal on Numerical Analysis 20 (1983) 293–314.Google Scholar
  4. [4]
    D.W. Decker and C.T. Kelley, “Newton's method at singular points I,”SIAM Journal on Numerical Analysis 17 (1980) 66–70.Google Scholar
  5. [5]
    D.W. Decker and C.T. Kelley, “Newton's method at singular points II,”SIAM Journal on Numerical Analysis 17 (1980) 465–471.Google Scholar
  6. [6]
    D.W. Decker and C.T. Kelley, “Convergence acceleration for Newton's method at singular points,”SIAM Journal on Numerical Analysis 19 (1981) 219–229.Google Scholar
  7. [7]
    J.E. Dennis Jr. and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).Google Scholar
  8. [8]
    J.J. Dongarra, J.R. Bunch, C.B. Moler and G.W. Stewart,LINPACK Users Guide (SIAM, Philadelphia, PA, 1979).Google Scholar
  9. [9]
    P.D. Frank, “Tensor methods for solving systems of nonlinear equations,” Ph.D. dissertation, Department of Computer Science, University of Colorado (Boulder, CO, 1982).Google Scholar
  10. [10]
    A. Griewank, “Analysis and modification of Newton's method at singularities,” Ph.D. dissertation, Australian National University (Canberra, ACT, 1980).Google Scholar
  11. [11]
    A. Griewank, “On solving nonlinear equations with simple singularities or nearly singular solutions,”SIAM Review (1985) 537–563.Google Scholar
  12. [12]
    A. Griewank and M.R. Osborne, “Analysis of Newton's method at irregular singularities,”SIAM Journal on Numerical Analysis 20 (1983) 747–773.Google Scholar
  13. [13]
    C.T Kelley and R. Suresh, “A new acceleration method for Newton's method at singular points,”SIAM Journal on Numerical Analysis 20 (1983) 1001–1009.Google Scholar
  14. [14]
    J.M. Ortega,Numerical Analysis (Academic Press, New York, 1972).Google Scholar
  15. [15]
    G.W. Reddien, “On Newton's method for singular problems,”SIAM Journal on Numerical Analysis 15 (1978) 993–996.Google Scholar
  16. [16]
    R.B. Schnabel and T. Chow, “Tensor methods for unconstrained optimization using second derivatives,”SIAM Journal on Optimization 1 (1991) 293–315.Google Scholar
  17. [17]
    R.B. Schnabel and P.D. Frank, Tensor methods for nonlinear equations,”SIAM Journal on Numerical Analysis 21 (1984) 815–843.Google Scholar
  18. [18]
    G.W. Stewart, “Error and perturbation bounds for subspaces associated with certain eigenvalue problems,”SIAM Review 15(4) (1973) 727–764.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Dan Feng
    • 1
  • Paul D. Frank
    • 2
  • Robert B. Schnabel
    • 1
  1. 1.Department of Computer Science, ECOT 7-7 Engineering CenterUniversity of Colorado at BoulderBoulderUSA
  2. 2.Boeing Computer ServicesSeattleUSA

Personalised recommendations