Notations

General Notations

\(E, F, X, Y\) — Banach spaces over \(\mathbb {R}\)

\(\| \cdot \| \) — norm on a normed space

\(\| \cdot \| _{\mathcal{B}(X,Y)}\) — operator norm for continuous linear maps

\(B_r(x)\) — open ball of radius \(r\) centered at \(x\)

\(\overline{B}_r(x)\) — closed ball of radius \(r\) centered at \(x\)

\(I_E\) — identity operator on space \(E\)

\(\text{Df}(x)\) — Fréchet derivative of \(f\) at \(x\)

\(\bar{x}\) — approximate zero or initial guess

\(\tilde{x}\) — exact zero of function \(f\)

\(A\) — approximate inverse operator

\(A^\dagger \) — approximate derivative operator

\(Y_0\) — bound on \(\| A(f(\bar{x}))\| \) (initial defect)

\(Z_0\) — bound on \(\| I_E - AA^\dagger \| \) (composition error)

\(Z_1\) — bound on \(\| A[\text{Df}(\bar{x}) - A^\dagger ]\| \) (derivative approximation error)

\(Z_2(r)\) — bound on \(\| A[\text{Df}(c) - \text{Df}(\bar{x})]\| \) for \(c \in \overline{B}_r(\bar{x})\)

\(Z(r)\) — combined bound \(Z_0 + Z_1 + Z_2(r) \cdot r\)

\(p(r)\) — radii polynomial

When \(E = F\) and \(A^\dagger = \text{Df}(\bar{x})\), we have \(Z_1 = 0\).

Simple radii polynomial: \(p(r) = Z_2(r)r^2 - (1-Z_0)r + Y_0\)

General radii polynomial: \(p(r) = Z_2(r)r^2 - (1-Z_0-Z_1)r + Y_0\)

\(\nu \) — weight parameter (positive real)

\(\ell ^1_\nu \) — weighted \(\ell ^1\) space with norm \(\| a\| _{\ell ^1_\nu } = \sum _{n=0}^\infty |a_n| \nu ^n\)

\(a \star b\) — Cauchy product: \((a \star b)_n = \sum _{k=0}^n a_k b_{n-k}\)

\(\bar{a}\) — approximate solution (finite coefficient sequence)

\(\tilde{a}\) — exact solution in \(\ell ^1_\nu \)

\(\lambda _0\) — base point for power series expansion

\(c\) — parameter sequence: \(c_0 = \lambda _0\), \(c_1 = 1\), \(c_n = 0\) for \(n \geq 2\)

\(F(a)\) — fixed-point map: \(F(a)_n = (a \star a)_n - c_n\)

toPowerSeries\((a)\) — formal power series embedding \(\ell ^1_\nu \hookrightarrow \mathbb {R}[\! [ X ]\! ]\)

eval\((a, z)\) — analytic evaluation: \(\sum _{n=0}^\infty a_n z^n\) for \(|z| \leq \nu \)