Notation
\[ \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\E}{\mathbb{E}} \newcommand{\eps}{\varepsilon} \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} \newcommand{\ninf}[1]{\left\lVert #1 \right\rVert_{\infty}} \newcommand{\ntwo}[1]{\left\lVert #1 \right\rVert_{2}} \newcommand{\none}[1]{\left\lVert #1 \right\rVert_{1}} \newcommand{\inner}[2]{\left\langle #1,\, #2 \right\rangle} \newcommand{\bigO}{\mathcal{O}} \newcommand{\dd}{\mathrm{d}} \newcommand{\diff}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}} \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\spec}{\rho} \newcommand{\cond}{\kappa} \newcommand{\Span}{\operatorname{span}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\argmax}{\operatorname*{arg\,max}} \newcommand{\argmin}{\operatorname*{arg\,min}} \newcommand{\defeq}{:=} \]
This chapter is the master notation table for the whole book. Because Part I (128A) and Part II (128B) come from different sources with different habits, the conventions here are chosen once and applied across both parts; the canonical word for each concept lives in CONTEXT.md. The traps worth watching are flagged with callouts.
General symbols
| Symbol | Meaning |
|---|---|
| \(\R,\ \R^n,\ \C\) | reals, \(n\)-space, complex numbers |
| \(\abs{x}\) | absolute value / modulus |
| \(\norm{x},\ \none{x},\ \ntwo{x},\ \ninf{x}\) | vector norm; \(1\)-, \(2\)-, \(\infty\)-norm |
| \(\inner{x}{y}=x^\top y\) | inner product |
| \(\bigO(h^p)\) | asymptotic error order as \(h\to 0\) |
| \(\defeq\) | “is defined as” |
| \(e_k,\ e(h)\) | error at step \(k\) / at mesh size \(h\) |
Part I: Interpolation & approximation of functions (Ch. 1–2)
| Symbol | Meaning |
|---|---|
| \(x_0,\dots,x_n\) | interpolation / quadrature nodes (\(n+1\) nodes, degree \(n\)) |
| \(P_n(x),\ p_n(x)\) | interpolating polynomial of degree \(\le n\) |
| \(L_{n,k}(x)\) | Lagrange basis polynomial |
| \(f[x_i,\dots,x_j]\) | divided difference |
| \(S(x),\ S_j(x)\) | spline / its \(j\)-th piece |
| \(h,\ h_j\) | step size / subinterval width |
| \(\phi_k,\ P_k,\ T_k\) | orthogonal polynomials; Legendre; Chebyshev |
| \(w(x)\) | quadrature / inner-product weight function |
“Order” is overloaded. Order of accuracy \(p\) means error \(\bigO(h^p)\) (quadrature, ODE methods, Ch. 2–3). Order of convergence \(q\) means \(\norm{e_{k+1}}\le C\norm{e_k}^q\) (iterations, Ch. 8). The notes always name which one. See def-8-order-convergence.
Part I: ODE initial-value problems (Ch. 3)
| Symbol | Meaning |
|---|---|
| \(y'=f(t,y),\ y(a)=\alpha\) | initial-value problem |
| \(w_j \approx y(t_j)\) | numerical approximation at mesh point \(t_j\) |
| \(h=(b-a)/N,\ t_j=a+jh\) | step size, mesh |
| \(\tau_j(h)\) | local truncation error (LTE), see def-3-lte |
| \(L\) | Lipschitz constant of \(f\) in \(y\) |
Part I/II: Linear systems and matrices (Ch. 4–5)
| Symbol | Meaning |
|---|---|
| \(A\in\R^{n\times n},\ b\in\R^n\) | system matrix, right-hand side of \(Ax=b\) |
| \(A=LU,\ PA=LU\) | LU factorization; with partial-pivot permutation \(P\) |
| \(A=LL^\top,\ A=LDL^\top\) | Cholesky / \(LDL^\top\) of an SPD matrix |
| \(\spec(A)=\rho(A)\) | spectral radius \(\max_i\abs{\lambda_i}\) (kept distinct from any norm) |
| \(\cond(A)=\norm{A}\norm{A^{-1}}\) | condition number, see def-4-cond |
| \(T=M^{-1}N,\ x^{(k+1)}=Tx^{(k)}+c\) | iteration matrix and splitting (Jacobi/GS/SOR) |
| \(r^{(k)}=b-Ax^{(k)}\) | residual |
\(\rho(A)\) is the spectral radius, never a norm, though \(\rho(A)\le\norm{A}\) for any induced norm, and \(\rho(A)<1\) is the convergence test for stationary iterations (thm-5-jacobi-gs-conv).
Part II: Eigenvalues, SVD, nonlinear (Ch. 7–8)
| Symbol | Meaning |
|---|---|
| \(\lambda_i,\ v_i\) | eigenvalue, eigenvector |
| \(A=U\Sigma V^\top\) | singular value decomposition; \(\sigma_i\) singular values |
| \(Q,\ R\) | orthonormal factor, upper-triangular factor (Rayleigh–Ritz, subspace iteration) |
| \(g(x)=x\) | fixed-point form; \(g\) a contraction with constant \(k<1\) (thm-8-banach) |
| \(q\) | order of convergence of an iteration (def-8-order-convergence) |
The full glossary of words (Reconstruction, source-of-truth per part, etc.) is in CONTEXT.md.