Index

A

A-stability and A(α)-stability absolute stability accuracy (relative vs. absolute) Adams--Bashforth formula Adams--Moulton formula adaptivity in integration in IVP solver, [1] adjacency matrix, [1]

adjoint of a matrix, [1] advection equation, [1] advection-diffusion equation, [1], [2], [3] Airy equation algorithm aliasing Allen–Cahn equation, [1], [2], [3], [4] argmin Arnoldi iteration, [1] asymptotic notation

B

backward difference backward differentiation formula for IVPs backward error, [1], [2] in a linear system backward substitution banded matrix, [1] bandwidth of a matrix barycentric interpolation formula barycentric weights

Bauer–Fike theorem beam equation Bessel equation, [1] Black–Scholes equation boundary conditions inflow numerical implementation of, [1], [2], [3] periodic Broyden update Burgers equation

C

C3PO, [1] cardinal function, [1], [2], [3] Carrier equation, [1] centered difference CFL condition characteristic polynomial Chebyshev equation Chebyshev points first kind second kind, [1] Chebyshev polynomials Clenshaw–Curtis integration collocation condition number of a matrix, [1], [2], [3] of a scalar function of eigenvalues of elementary functions of initial-value problems of interpolation, [1] of linear least squares of linear system of normal equations of rootfinding

conjugate gradients conservation law contraction mapping convergence rate algebraic linear, [1], [2], [3] quadratic, [1] spectral, [1] superlinear Crank–Nicolson method cubic spline

D

D'Alembert's solution Dahlquist theorems, [1] data fitting by straight line nonlinear power law degree matrix diagonal matrix, [1], [2] diagonalizable matrix

differentiation matrix, [1], [2], [3] diffusion equation see heat equation dimension reduction, [1] Dirichlet boundary condition, [1] Dirichlet boundary conditions divide and conquer domain of dependence double exponential transformation

E

eigenvalue, [1] conditioning of dominant, [1] eigenvalue decomposition, [1], [2] eigenvector

elliptic PDE Euler's method, [1] Euler–Maclaurin formula evolutionary PDE extrapolation

F

FFT (fast Fourier transform) field of values fill-in of sparse matrices finite difference for parabolic PDE finite differences, [1], [2] for boundary value problems for parabolic PDE matrix

finite element method Fisher equation fixed-point iteration fixed-point problem floating-point numbers, [1] flops forward difference forward substitution Francis QR iteration

G

Galerkin conditions Gaussian elimination Gaussian integration Gauss–Newton method generating polynomials Gibbs phenomenon global error

GMRES preconditioning in relationship to MINRES restarting Gram matrix graph graph Laplacian matrix Gregory integration formula

H

Han Solo, [1], [2], [3] hat functions heat equation, [1], [2] Helmholtz equation, [1] Hermite equation

hermitian matrix, [1], [2] hermitian positive definite matrix see symmetric positive definite matrix Horner's algorithm Householder reflector hyperbolic PDE

I

identity matrix, [1] IEEE 754, [1] image (as a matrix), [1], [2] implicit IVP solver, [1] improper integral, [1] initial-boundary-value problem initial-value problem one-step method for inner product of functions of vectors

inner product space interpolation, [1] by piecewise linear polynomials by piecewise polynomials, [1], [2], [3] by polynomials, [1], [2], [3], [4] by trigonometric polynomials inverse iteration invertible matrix

J

Jacobian matrix, [1], [2] Julia ! ', [1] += .+ .- :, [1] @. @animate @sprintf \, [1], [2], [3], [4] adjoint, [1] annotate! Boolean indexing break broadcasting, [1], [2], [3] cg cholesky collect comprehension cond, [1] destructuring diag, [1] diagm, [1], [2] DiagonalPreconditioner eachindex eigen eigs eigvals eltype end enumerate fft fill fit FNC for, [1] functions gmres graphplot I ilu Images in-place function indexing arrays, [1], [2], [3], [4] isinf isodd keyword function arguments, [1], [2] kron length, [1] LinearMap maximum minimum minres namespace NaN nlsolve nnz norm, [1] normalize, [1] ODEProblem opnorm Pair plotting functions push!, [1] quadgk range, [1] Rational reshape return scatter scientific notation size solve sortby sortperm sparse spdiagm splatting Spline1D string interpolation subplots sum, [1] summarysize svd svdvals ternary operator transpose tril triu using vec operation ÷

K

Kronecker product Krylov matrix

Krylov subspace Kuramoto–Sivashinsky equation

L

Lagrange interpolation formula Lagrange polynomial Laguerre equation Lanczos iteration Laplace equation Legendre equation Legendre polynomials Levenberg's method

linear combination linear convergence linear least-squares problem, [1], [2] linearization of an ODE Lipschitz condition logistic equation, [1] LU factorization, [1], [2] incomplete Luke Skywalker

M

machine epsilon, [1], [2] in double precision mantissa see significand mass matrix matrix factorization Cholesky EVD LU, [1] pivoted LU QR, [1] SVD

matrix inverse, [1] Maxwell's equations method of lines, [1], [2], [3] MINRES multistep method, [1] implementation of

N

NaN Neumann boundary condition, [1] Newton's method, [1], [2] multidimensional nonlinear least squares norm Frobenius matrix, [1] vector, [1]

normal equations, [1], [2] normal matrix numerical integration, [1]

O

Obi-Wan Kenobi, [1] ONC matrix, [1] order of accuracy of a finite-difference formula, [1] of a one-step IVP method of an approximation of numerical integration of the finite element method

Oregonator orthogonal functions orthogonal matrix, [1], [2], [3] orthogonal polynomials orthogonal vectors orthonormal vectors outer product, [1], [2]

P

parabolic PDE parameter continuation pendulum piecewise linear interpolant pivoting, [1], [2]

PLU factorization Poisson equation power iteration preconditioning predator–prey model pseudoinverse

Q

quadratic convergence quadrature see numerical integration

quasi-Newton method quasimatrix

R

Rayleigh quotient residual, [1] of a linear system of rootfinding Return of the Jedi Robin boundary condition

rootfinding problem, [1], [2] multidimensional roots multiplicity of, [1] Runge phenomenon Runge–Kutta method, [1], [2]

S

secant method semidiscretization see method of lines shifted inverse iteration see inverse iteration shock wave significand significant digits similarity transformation Simpson's formula, [1] sine integral function sine–Gordon equation singular matrix singular value decomposition, [1] thin form soap film sparse matrix, [1], [2], [3], [4] SPD matrix see symmetric positive definite matrix

spline (cubic spline) stability, [1] of collocation of IVP solvers of multistep methods of polynomial interpolation stability region Star Wars: A New Hope, [1], [2], [3], [4], [5], [6] steepest descent stiff differential equation, [1], [2] stiffness matrix subtractive cancellation, [1], [2], [3], [4] superlinear convergence SVD see singular value decomposition Sylvester equation symmetric matrix, [1], [2], [3] symmetric positive definite matrix, [1], [2]

T

tensor-product domain The Empire Strikes Back, [1], [2], [3], [4] transpose, [1] trapezoid formula for an IVP, [1], [2] for integration, [1] triangular matrix, [1]

tridiagonal matrix trigonometric polynomial truncation error of a finite-difference formula of a multistep IVP formula of a numerical integration formula of a one-step IVP solver

U

unit lower triangular matrix unit roundoff see: machine epsilon unit vector

unitary matrix, [1], [2], [3] unvec upper Hessenberg matrix upwind direction

V

van der Pol equation

Vandermonde matrix, [1] vec

W

wave equation, [1]

weak solution Wedge Antilles

Y

Yoda, [1], [2]

Z

zero-stability