Math 2.0

Numerical analysis Theorem

Newton–Kantorovich theorem

Builds on Banach spaceFréchet derivativeNewton's methodLipschitz condition

Statement

Let XX and YY be Banach spaces, let D⊆XD \subseteq X be open, and let F ⁣:D→YF \colon D \to Y be Fréchet differentiable with derivative F′(x)F'(x) at xx. Let x0∈Dx_0 \in D be a point where F′(x0)F'(x_0) is invertible with bounded inverse. Assume that there are constants η>0\eta > 0 and L>0L > 0 such that:

  1. ∥F′(x0)−1F(x0)∥≤η\norm{F'(x_0)^{-1} F(x_0)} \le \eta;
  2. ∥F′(x0)−1(F′(x)−F′(y))∥≤L∥x−y∥\norm{F'(x_0)^{-1} (F'(x) - F'(y))} \le L \norm{x - y} for all x,y∈Dx, y \in D, a Lipschitz condition;
  3. h=Lη≤12h = L \eta \le \tfrac12;
  4. the closed ball Bˉ(x0,t∗)\bar B(x_0, t^*) lies in DD, where t∗=1−1−2hLt^* = \dfrac{1 - \sqrt{1 - 2h}}{L}.

Then the Newton iterates xk+1=xk−F′(xk)−1F(xk)x_{k+1} = x_k - F'(x_k)^{-1} F(x_k) are well defined, they lie in Bˉ(x0,t∗)\bar B(x_0, t^*), and they converge to a point x∗∈Bˉ(x0,t∗)x^* \in \bar B(x_0, t^*) with F(x∗)=0F(x^*) = 0.

Remarks

The proof compares the iterates with Newton's method for the real quadratic p(t)=L2t2−t+ηp(t) = \tfrac{L}{2} t^2 - t + \eta, started at t0=0t_0 = 0. Those scalar iterates increase to the smallest root t∗t^* of pp, and an induction shows ∥xk+1−xk∥≤tk+1−tk\norm{x_{k+1} - x_k} \le t_{k+1} - t_k, so (xk)(x_k) is a Cauchy sequence. The quadratic pp has a real root exactly when h≤12h \le \tfrac12, which explains the condition on hh. If h<12h < \tfrac12, the convergence is quadratic; if h=12h = \tfrac12, it may be only linear. Example: for F(x)=x2−2F(x) = x^2 - 2 on ℝ\R and x0=3/2x_0 = 3/2, one has η=1/12\eta = 1/12 and L=2/3L = 2/3 on D=ℝD = \R, so h=1/18≤1/2h = 1/18 \le 1/2, and Newton's method converges to 2\sqrt 2.

Going deeper

For a student

The theorem works in any Banach space, so it applies to nonlinear integral and differential equations, not only to systems in ℝn\R^n. It is also an existence theorem: it proves that a solution exists near x0x_0 without knowing it in advance. This is the basis of computer-assisted proofs, where η\eta and LL are bounded rigorously by interval arithmetic. Compared with the Banach fixed point theorem, it needs a derivative but gives quadratic convergence when h<12h < \tfrac12.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Banach space, concept, no evidence levelBanach spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLipschitz condition, concept, no evidence levelLipschitz conditionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionFréchet derivative, concept, no evidence levelFréchet derivativeconceptDerivative, definition, no evidence levelDerivativedefinitionTangent line, concept, no evidence levelTangent lineconceptNewton's method, concept, no evidence levelNewton's methodconceptNewton–Kantorovich theorem, statement, no evidence levelNewton–Kantorovich th…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Banach space concept
  2. Euclidean space object
  3. Limit point of a set definition
  4. Lipschitz condition concept
  5. Total derivative concept
  6. Limit of a function definition
  7. Fréchet derivative concept
  8. Derivative definition
  9. Tangent line concept
  10. Newton's method concept
  11. Newton–Kantorovich theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Kantorovich_theorem
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Kindstatement, theorem
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AboutNewton's method
UsesBanach space, Fréchet derivative, Newton's method, Lipschitz condition
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Changed2 November 2026

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