Kantorovich theorem

theorem about initial conditions that insure the convergence of Newton's method

In mathematics (especially numerical analysis), the Kantorovich theorem (or the Newton-Kantorovich theorem) is a mathematical statement about the convergence of Newton's method.[1][2][3] This theorem was named after Leonid Kantorovich,[1] and it is frequently used in the field of validated numerics.[4]


  1. 1.0 1.1 Deuflhard, P. (2004). Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics. Vol. 35. Berlin: Springer. ISBN 3-540-21099-7.
  2. Zeidler, E. (1985). Nonlinear Functional Analysis and its Applications: Part 1: Fixed-Point Theorems. New York: Springer. ISBN 0-387-96499-1.
  3. Ortega, J. M.; Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial and Applied Mathematics. OCLC 95021.
  4. M. Nakao, M. Plum, Y. Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics).

Further reading

  • Yamamoto, T. (2001). "Historical Developments in Convergence Analysis for Newton's and Newton-like Methods". In Brezinski, C.; Wuytack, L. (eds.). Numerical Analysis : Historical Developments in the 20th Century. North-Holland. pp. 241–263.