Symbols
List of Symbols and Notation
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This list collects the notation used throughout the book. Some symbols are standard and may be reused locally when the context is clear. Chapter-specific notation is introduced again where it first appears. Boldface letters usually denote vectors or matrices, while calligraphic letters often denote operators, spaces, or residuals.
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Symbols table
| Symbol | Meaning |
|---|---|
| General mathematical notation | |
| \N,\Z,\R,\C | Natural numbers, integers, real numbers, complex numbers |
| i | Imaginary unit, i^2=-1, when used in complex analysis or Fourier formulas |
| x,y,z | Scalar variables, unless otherwise stated |
| \mathbf x,\mathbf y,\mathbf z | Vectors in finite-dimensional spaces |
| A,B,M,N | Matrices or linear operators, depending on context |
| I | Identity matrix or identity operator |
| 0 | Zero scalar, zero vector, or zero matrix, according to context |
| \delta_{ij} | Kronecker delta, equal to 1 if i=j and 0 otherwise |
| \abs{x} | Absolute value or modulus |
| \norm{x} | Norm of a vector, matrix, function, or operator |
| \norm{x}_p | Vector p-norm |
| \norm{A}_p | Induced matrix p-norm |
| \norm{f}_{L^p} | L^p-norm of a function |
| \norm{u}_{H^m} | Sobolev norm of order m |
| \ip{x}{y} | Inner product |
| x^T y | Euclidean inner product for real vectors |
| x^\ast y | Hermitian inner product for complex vectors |
| A^T | Transpose of A |
| A^\ast | Conjugate transpose of A |
| A^{-1} | Inverse of a nonsingular matrix or operator |
| A^\dagger | Moore--Penrose pseudoinverse |
| \rank(A) | Rank of A |
| \range(A) | Range or column space of A |
| \nullspace(A) | Null space of A |
| \spann\{v_1,\ldots,v_k\} | Linear span of the listed vectors |
| \diag(d_1,\ldots,d_n) | Diagonal matrix with diagonal entries d_i |
| \tr(A), \trace(A) | Trace of A |
| \det(A) | Determinant of A |
| \lambda(A) | Eigenvalue of A, when a generic eigenvalue is meant |
| \sigma(A) | Spectrum of A, or singular values when context indicates SVD |
| \rho(A) | Spectral radius, \rho(A)=\max_{\lambda\in\sigma(A)}|\lambda| |
| \kappa(A), \cond(A) | Condition number of A |
| O(h^p) | Quantity bounded by C h^p as h\to0 |
| o(h^p) | Quantity smaller than h^p asymptotically |
| \dd x, \dd t | Differential element in integration |
| Floating-point arithmetic and error analysis | |
| \fl(x) | Floating-point representation or rounded value of x |
| \beta | Base of a floating-point system |
| p | Precision, usually the number of digits in the significand |
| e | Floating-point exponent |
| u | Unit roundoff |
| \epsmach | Machine epsilon |
| \delta | Relative rounding error in the model \fl(x)=x(1+\delta) |
| \theta_n | Accumulated floating-point perturbation after n operations |
| \gamma_n | Bound \gamma_n=nu/(1-nu) for accumulated roundoff |
| e_k | Error at iteration k, usually e_k=x_k-x_\ast |
| r | Residual, often r=b-A\widetilde x or r=f-Au_h |
| \eta | Backward error, residual indicator, or tolerance parameter |
| \tau | Tolerance, time-like variable, or fractional clock depending on context |
| \mathrm{tol} | Numerical stopping tolerance |
| Nonlinear equations | |
| f(x)=0 | Scalar nonlinear equation |
| F(x)=0 | Nonlinear system |
| x_\ast | Exact root or solution |
| x_k | Approximation at iteration k |
| J_F(x) | Jacobian matrix of F at x |
| s_k | Newton correction or search step |
| \alpha_k | Damping or step-length parameter |
| \Phi(x) | Merit function, often \Phi(x)=\frac12\norm{F(x)}^2 |
| m | Multiplicity of a root, or dimension depending on context |
| Interpolation and approximation | |
| x_j | Interpolation node |
| f_j | Function value f(x_j) |
| p_n(x) | Interpolating polynomial of degree at most n |
| \ell_j(x) | Lagrange cardinal polynomial |
| f[x_0,\ldots,x_k] | Divided difference |
| w_j | Barycentric interpolation weight or quadrature weight |
| T_k(x) | Chebyshev polynomial of degree k |
| P_k(x) | Legendre polynomial of degree k |
| B_{i,p}(x) | B-spline basis function of degree p |
| \Pi_n f | Polynomial projection or interpolation of f |
| \omega_n(x) | Nodal polynomial \prod_{j=0}^{n}(x-x_j) |
| Linear systems and matrix factorizations | |
| Ax=b | Linear system |
| \widetilde x | Computed approximation to x |
| r=b-A\widetilde x | Linear-system residual |
| A=LU | LU factorization |
| P A=LU | LU factorization with row pivoting |
| A=QR | QR factorization |
| A=LL^T | Cholesky factorization for symmetric positive definite A |
| L | Lower triangular matrix |
| U | Upper triangular matrix, or solution variable in PDE discretization |
| Q | Orthogonal or unitary matrix |
| R | Upper triangular matrix, residual, or objective residual depending on context |
| M | Preconditioner or matrix splitting component |
| Iterative methods and Krylov spaces | |
| A=M-N | Matrix splitting |
| G=M^{-1}N | Stationary iteration matrix |
| D,L,U | Diagonal, strictly lower, and strictly upper parts of a matrix |
| \omega | Relaxation parameter in SOR, or quadrature/convolution weight depending on context |
| \mathcal K_k(A,r_0) | Krylov subspace \spann\{r_0,Ar_0,\ldots,A^{k-1}r_0\} |
| r_k | Residual at iteration k |
| p_k | Search direction in CG or nonlinear conjugate gradient |
| \alpha_k,\beta_k | CG or conjugate-gradient recurrence coefficients |
| V_k | Matrix whose columns form an orthonormal Krylov basis |
| \overline H_k | Upper Hessenberg matrix in Arnoldi/GMRES |
| P | Preconditioner, projection, or polynomial depending on context |
| Eigenvalue problems | |
| Ax=\lambda x | Eigenvalue problem |
| \lambda | Eigenvalue |
| x | Eigenvector |
| \rho(x) | Rayleigh quotient, \rho(x)=x^TAx/(x^Tx) |
| \sigma | Shift in inverse iteration, or spectrum depending on context |
| Q_kR_k | QR factorization at iteration k |
| A_k | Matrix produced by QR iteration |
| H | Hessenberg matrix |
| T | Tridiagonal matrix, final time, or Chebyshev polynomial depending on context |
| Least squares and regularization | |
| \min_x\norm{Ax-b}_2 | Linear least-squares problem |
| A^TAx=A^Tb | Normal equations |
| A=U\Sigma V^T | Singular value decomposition |
| \sigma_i | Singular value |
| \lambda | Regularization parameter or eigenvalue depending on context |
| A^\dagger | Pseudoinverse |
| \lambda\|x\|^2 | Tikhonov regularization term |
| r(x) | Nonlinear least-squares residual vector |
| J(x) | Jacobian of a nonlinear residual |
| Numerical differentiation | |
| h | Step size or mesh spacing |
| D_h f | Finite-difference approximation to a derivative |
| \delta_x | Difference operator in the x-direction |
| \delta_{xx} | Second-difference operator |
| D | Differentiation matrix, derivative operator, or diagonal matrix depending on context |
| D^{(m)} | m-th derivative matrix |
| x_j=\cos(j\pi/N) | Chebyshev--Lobatto node |
| D_{ij} | Entry of a differentiation matrix |
| Numerical integration and quadrature | |
| I(f) | Exact integral of f |
| Q(f) | Quadrature approximation to I(f) |
| w_j | Quadrature weight |
| \xi_j | Reference quadrature node |
| x_j | Physical quadrature node |
| T(h) | Trapezoidal approximation with step h |
| S(h) | Simpson approximation with step h |
| R_{k,j} | Romberg extrapolation entry |
| E | Error estimate |
| B(p,q) | Beta function |
| B_z(p,q) | Incomplete beta function |
| Initial-value problems for ODEs | |
| y'(t)=f(t,y(t)) | Ordinary differential equation in first-order form |
| t_n | Time node |
| y_n | Numerical approximation to y(t_n) |
| h | Time step |
| k_i | Runge--Kutta stage slope |
| A=(a_{ij}), b_i, c_i | Butcher tableau coefficients |
| R(z) | Stability function of a one-step method |
| z=h\lambda | Scaled eigenvalue for stability analysis |
| \lambda | Test-equation eigenvalue in y'=\lambda y |
| \Delta t | Time step in time-dependent problems |
| \mathrm{LTE} | Local truncation error |
| Boundary-value problems and Galerkin notation | |
| [a,b] | Spatial interval for one-dimensional BVPs |
| \alpha,\beta | Boundary data or fractional orders depending on context |
| V | Function space, often H_0^1(a,b) or H_0^1(\Omega) |
| V_h | Finite-dimensional approximation space |
| v | Test function |
| u_h,y_h | Galerkin or finite-element approximation |
| \phi_i | Basis or test function |
| K_{ij} | Stiffness matrix entry |
| M_{ij} | Mass matrix entry |
| F_i | Load vector entry |
| a(u,v) | Bilinear form |
| \ell(v) | Linear functional |
| R(u;v) | Weak residual |
| J(U) | Jacobian of nonlinear algebraic system |
| Partial differential equations | |
| \Omega | Spatial domain |
| \partial\Omega | Boundary of the domain |
| u(x,t) | PDE solution |
| \nabla u | Gradient of u |
| \Delta u | Laplacian of u |
| \kappa | Diffusion coefficient |
| c | Wave speed |
| a | Advection velocity or interval endpoint depending on context |
| \mu=\kappa\Delta t/\Delta x^2 | Heat-equation mesh ratio |
| \nu=a\Delta t/\Delta x | Courant number for advection |
| \lambda=c\Delta t/\Delta x | CFL number for wave equation |
| U_i^n | Approximation to u(x_i,t^n) |
| F_{j+1/2} | Numerical flux at a cell interface |
| \overline U_j^n | Cell average in finite-volume method |
| \widehat f | Numerical flux function or Fourier coefficient depending on context |
| Spectral and pseudospectral methods | |
| \widehat u_k | Fourier coefficient of mode k |
| e^{ikx} | Fourier basis function |
| k | Fourier wavenumber or iteration index depending on context |
| \operatorname{FFT} | Fast Fourier transform |
| \operatorname{IFFT} | Inverse fast Fourier transform |
| T_k(x) | Chebyshev polynomial |
| P_k(x) | Legendre polynomial |
| D | Spectral differentiation matrix |
| D^2 | Spectral second-derivative matrix |
| \eta | Normalized modal coordinate in filtering |
| \sigma(\eta) | Spectral filter |
| K | Maximum retained Fourier wavenumber, or stiffness matrix depending on context |
| N | Polynomial degree, grid size, or number of time steps depending on context |
| Optimization and nonlinear systems | |
| f(x) | Objective function |
| \nabla f(x) | Gradient of f |
| \nabla^2 f(x) | Hessian of f |
| g_k | Gradient \nabla f(x_k), or constraint function depending on context |
| H_k | Approximate inverse Hessian or Hessian-related matrix |
| B_k | Hessian approximation in quasi-Newton methods |
| s_k=x_{k+1}-x_k | Step displacement |
| y_k=\nabla f(x_{k+1})-\nabla f(x_k) | Gradient difference in quasi-Newton methods |
| \rho_k | Trust-region ratio or reciprocal curvature 1/(y_k^Ts_k) depending on context |
| \Delta_k | Trust-region radius |
| m_k(s) | Local quadratic model in trust-region methods |
| c(x)=0 | Equality constraint |
| g(x)\le0 | Inequality constraint |
| \mathcal L(x,\lambda,\mu) | Lagrangian |
| \lambda | Equality multiplier, regularization parameter, or eigenvalue depending on context |
| \mu | Inequality multiplier, heat ratio, or memory parameter depending on context |
| \theta | Parameter vector in inverse problems or residual minimization |
| J(\theta), \mathcal J(\theta) | Parameter-dependent objective functional |
| J_Q(\theta), \mathcal J_Q(\theta) | Quadrature approximation of an objective |
| Fractional calculus and fractional differential equations | |
| \alpha | Fractional order, commonly 0<\alpha<1, or boundary datum depending on context |
| \rho | Katugampola deformation parameter or spectral radius depending on context |
| \Gamma(z) | Gamma function |
| B(p,q) | Beta function |
| B_z(p,q) | Incomplete beta function |
| E_\alpha(z) | One-parameter Mittag--Leffler function |
| E_{\alpha,\beta}(z) | Two-parameter Mittag--Leffler function |
| I_{a+}^{\alpha}f | Left Riemann--Liouville fractional integral |
| I_{a+}^{\alpha,\rho}f | Katugampola fractional integral |
| D_{a+}^{\alpha}f | Riemann--Liouville fractional derivative |
| {}^{C}D_{a+}^{\alpha}f | Caputo fractional derivative |
| {}^{GL}D_{a+}^{\alpha}f | Grunwald--Letnikov fractional derivative |
| {}^{C}D_{a+}^{\alpha,\rho}f | Caputo--Katugampola fractional derivative |
| {}^{CF}D_{a+}^{\alpha}f | Caputo--Fabrizio derivative |
| {}^{ABC}D_{a+}^{\alpha}f | Atangana--Baleanu Caputo-type derivative |
| \mathcal D_\rho | Katugampola differential operator t^{1-\rho}\frac{\dd}{\dd t} |
| (t-s)^{-\alpha} | Caputo weakly singular memory kernel |
| (t^\rho-s^\rho)^{-\alpha} | Caputo--Katugampola weakly singular memory kernel |
| t_n | Time node in fractional time discretization |
| h | Uniform time step |
| y=t^\rho | Stretched Katugampola time variable |
| h_y | Step size in the stretched variable y=t^\rho |
| a_k | L1 weight a_k=(k+1)^{1-\alpha}-k^{1-\alpha} |
| g_k^{(\alpha)} | Grunwald binomial weight |
| \omega_k | Lubich convolution quadrature weight |
| \delta(\zeta) | Linear multistep generating polynomial in Lubich convolution quadrature |
| \zeta | Generating-function variable |
| \tau_\alpha(t) | Caputo fractional clock t^\alpha/\Gamma(1+\alpha) |
| \tau_{\alpha,\rho}(t) | Caputo--Katugampola clock t^{\alpha\rho}/(\rho^\alpha\Gamma(1+\alpha)) |
| \mathcal R[u] | Fractional differential-equation residual |
| \theta_m^{(q)} | Chebyshev--Lobatto panel node on [0,1] |
| s_{j,m} | Physical panel node in a Chebyshev--Lobatto memory panel |
| \ell_m^{(q)} | Lagrange cardinal polynomial on a q-degree panel |
| d_{m,r}^{(q)} | Coefficient of \theta^r in (\ell_m^{(q)})'(\theta) |
| W_{j,m}^{C,q}(t) | Caputo Chebyshev--Beta panel weight |
| W_{j,m}^{CK,q}(t) | Caputo--Katugampola Chebyshev--Beta panel weight |
| I_r^C(a,b;t) | Caputo beta moment on a panel |
| I_r^{CK}(a,b;t) | Caputo--Katugampola beta moment on a panel |
| H_\ell^n | Fast-convolution history variable |
| w_\ell,\eta_\ell | Sum-of-exponentials weight and exponent |
| L | Short-memory window length or lower triangular matrix depending on context |
| Frequently overloaded symbols | |
| \alpha | Fractional order, line-search constant, boundary datum, or method coefficient depending on context |
| \beta | Fractional power exponent, CG coefficient, or method coefficient depending on context |
| \gamma | Nonlinear coefficient, accumulated roundoff bound, or method coefficient depending on context |
| \lambda | Eigenvalue, test-equation parameter, regularization parameter, or Lagrange multiplier |
| \mu | Heat-equation mesh ratio, inequality multiplier, or measure-like parameter |
| \rho | Spectral radius, trust-region ratio, or Caputo--Katugampola deformation parameter |
| D | Derivative matrix, diagonal matrix, differential operator, or domain-dependent operator |
| K | Stiffness matrix, maximum Fourier mode, or generic compact set |
| M | Mass matrix, preconditioner, or number of exponential terms in fast convolution |
| R | Residual, upper triangular factor, stability function, or reaction term |
| T | Final time, Chebyshev polynomial symbol, or tridiagonal matrix |
| u | Unit roundoff in floating-point analysis or solution variable in differential equations |