1.3 Newton’s Method

Newton’s method transforms the problem of finding zeros into the problem of finding fixed points. This section establishes the fundamental equivalence and introduces Newton-like operators.

Definition 1.3.1 Newton-like map

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Let \(E, F\) be Banach spaces, \(f: E \rightarrow F\) a function, and \(A: F \rightarrow E\) a continuous linear map. The Newton-like map is defined by

\begin{equation*} T(x) = x - A(f(x)). \end{equation*}

Proposition 1.3.2 Fixed points \(\Leftrightarrow \) Zeros (Proposition 2.3.1)

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Let \(E, F\) be vector spaces, \(f: E \rightarrow F\), and \(A: F \rightarrow E\) an injective linear map. Let \(T(x) = x - A(f(x))\) be the Newton-like operator. Then:

\begin{equation*} T(x) = x \quad \Leftrightarrow \quad f(x) = 0. \end{equation*}

Proof ▶

(\(\Rightarrow \)) If \(T(x) = x\), then \(x - A(f(x)) = x\), so \(A(f(x)) = 0\). Since \(A\) is linear, \(A(0) = 0\). By injectivity of \(A\), \(A(f(x)) = A(0)\) implies \(f(x) = 0\).

(\(\Leftarrow \)) If \(f(x) = 0\), then \(T(x) = x - A(0) = x - 0 = x\).