Index - Fundamentals of Numerical Computation

A

A New Hope: Han Solo, Star Wars: A New Hope, C3PO, Star Wars: A New Hope, Han Solo, Star Wars: A New Hope, C3PO, Star Wars: A New Hope, Wedge Antilles, Star Wars: A New Hope, Obi-Wan Kenobi, Star Wars: A New Hope, Obi-Wan Kenobi, Star Wars: A New Hope

A-stability and A(α)-stability: Definition 11.4.2

Adams–Bashforth formula: Paragraph

Adams–Moulton formula: Paragraph

Allen–Cahn equation: Example 10.5.4, Example 13.4.4

Arnoldi iteration: Algorithm 8.4.1, (8.6.2), Arnoldi iteration

absolute stability: Definition 11.3.1

accuracy (relative vs. absolute): Paragraph

adaptivity:

in IVP solver: Paragraph, Section 6.7.4
in integration: Section 5.7.2

adjacency matrix: Definition 7.1.1, Paragraph, adjacency matrix

adjoint: adjoint

adjoint of a matrix: Paragraph, Paragraph

advection equation: Definition 12.1.1, Paragraph, advection equation

advection-diffusion equation: Paragraph, Paragraph, Example 13.2.3, Example 13.4.3

algorithm: Paragraph, algorithm

argmin: Definition 3.1.1

asymptotic: asymptotic

asymptotic notation: Definition 2.5.1

B

Bauer–Fike theorem: Paragraph

Bessel equation: List

Bessel function: Example 4.1.1

Black–Scholes equation: Paragraph

Broyden update: Paragraph

backward difference: Definition 5.4.2

backward differentiation formula for IVPs: Paragraph

backward error: Definition 1.4.1, Observation 4.1.1, Paragraph, backward error

in a linear system: Paragraph

backward substitution: Paragraph, backward substitution

banded matrix: Paragraph, Paragraph

bandwidth: bandwidth

bandwidth of a matrix: Definition 2.9.1

barycentric interpolation formula: Theorem 9.2.1, barycentric interpolation formula

barycentric weights: Definition 9.2.1

big-O: big-O

boundary conditions:

inflow: Paragraph
numerical implementation of: Paragraph, Paragraph, Paragraph, Paragraph
periodic: Paragraph

boundary-value problem: boundary-value problem

C

C3PO: C3PO, Star Wars: A New Hope, C3PO, Star Wars: A New Hope

CFL condition: Theorem 12.2.1

Carrier equation: List

Chebyshev points:

first kind: Paragraph
second kind: Paragraph, Paragraph

Chebyshev polynomials: Definition 9.4.4

Cholesky factorization: Theorem 2.9.2, Cholesky factorization

Clenshaw–Curtis integration: Paragraph

cardinal function: Definition 5.1.2, Paragraph, Paragraph, Paragraph, cardinal function

centered difference: Definition 5.4.2

characteristic polynomial: Paragraph

collocation: Paragraph, collocation

condition number: condition number

of a matrix: Paragraph, Definition 3.2.3, Paragraph, Notebook-code, Notebook-code, Notebook-code
of a scalar function: Definition 1.2.1
of eigenvalues: Paragraph
of elementary functions: Paragraph
of initial-value problems: Paragraph
of interpolation: Theorem 5.1.1, Theorem 5.2.1
of linear least squares: Definition 3.2.3
of linear system: Paragraph
of normal equations: Paragraph
of rootfinding: Theorem 4.1.1

conjugate gradients: Definition 8.6.1

conservation law: Paragraph

contraction mapping: Paragraph

convergence rate:

algebraic: Definition 5.2.2
linear: Definition 4.2.2, Paragraph, Paragraph, Paragraph
quadratic: Definition 4.3.1, Paragraph
spectral: Paragraph, Paragraph
superlinear: Paragraph

cubic spline: Definition 5.3.1, cubic spline

D

Dahlquist theorems: Theorem 6.8.2, Theorem 6.8.3, Theorem 11.4.1

Dirichlet boundary condition: Definition 10.1.2, Paragraph

Dirichlet boundary conditions: Paragraph

Dirichlet condition: Dirichlet condition

data fitting: Paragraph

by straight line: Paragraph
nonlinear: Paragraph
power law: Paragraph

diagonal matrix: Paragraph, Definition 7.2.3, Definition 7.3.1

diagonalizable matrix: Definition 7.2.3, diagonalizable matrix

differentiation matrix: Paragraph, Paragraph, Paragraph, Paragraph, differentiation matrix

diffusion equation: See heat equation

dimension reduction: Paragraph, Paragraph

divide and conquer: Section 5.7.2

domain of dependence: Paragraph

dominant eigenvalue: dominant eigenvalue

double exponential transformation: Paragraph

double precision: double precision

E

Euler's method: Paragraph, Paragraph

Euler–Maclaurin formula: Paragraph

Euler’s method: Euler’s method

eigenvalue: Definition 7.2.1, Comment, eigenvalue

conditioning of: Paragraph
dominant: Paragraph, Paragraph

eigenvalue decomposition: Definition 7.2.3, Paragraph, Paragraph

eigenvalue decomposition (EVD): eigenvalue decomposition (EVD)

eigenvector: Definition 7.2.1, eigenvector

elliptic PDE: Paragraph

evolutionary PDE: Paragraph, evolutionary PDE

extrapolation: Paragraph, extrapolation

F

FFT (fast Fourier transform): Paragraph

Francis QR iteration: Paragraph

Frobenius norm: Frobenius norm

fill-in of sparse matrices: Paragraph

finite differences: Definition 5.4.1, Paragraph, Paragraph, finite differences

for boundary value problems: Paragraph
for parabolic PDE: Paragraph, Paragraph
matrix: Paragraph

finite element method: Paragraph

finite element method (FEM): finite element method (FEM)

fixed point problem: fixed point problem

fixed-point iteration: Algorithm 4.2.1, fixed-point iteration

fixed-point problem: Definition 4.2.1

floating-point numbers: Definition 1.1.1, Paragraph, floating-point numbers

flops: Paragraph, flops

forward difference: Definition 5.4.2

forward substitution: Paragraph, forward substitution

G

GMRES: Algorithm 8.5.1, GMRES

preconditioning in: Definition 8.8.1
relationship to MINRES: Paragraph
restarting: Paragraph

Galerkin conditions: (10.6.10)

Gaussian elimination: Paragraph, Gaussian elimination

Gaussian integration: Paragraph

Gauss–Newton method: Algorithm 4.7.1, Gauss–Newton method

Gram matrix: Definition 9.4.2

Gregory integration formula: (5.6.19)

generating polynomials: Paragraph, generating polynomials

global error: Definition 6.2.3, global error

graph: Definition 7.1.1, graph

H

Han Solo: Han Solo, Star Wars: A New Hope, Han Solo, The Empire Strikes Back, Han Solo, Star Wars: A New Hope, Han Solo, The Empire Strikes Back

Horner's algorithm: Example 1.3.1

Householder reflector: Definition 3.4.1

hat functions: Paragraph, hat functions

heat equation: Definition 11.1.1, Paragraph, Example 13.2.2, heat equation

hermitian: hermitian

hermitian matrix: Paragraph, Paragraph, Theorem 7.3.2

hermitian positive definite matrix: hermitian positive definite matrix See symmetric positive definite matrix

homogeneous boundary condition: Definition 10.1.2, Paragraph, homogeneous boundary condition

hyperbolic PDE: Definition 12.1.1

I

IEEE 754: Paragraph

identity matrix: Paragraph, Paragraph, identity matrix

ill-conditioned: ill-conditioned

image (as a matrix): Paragraph, Example 7.5.1, Paragraph

implicit: implicit

implicit IVP solver: Definition 6.6.1, Paragraph

improper integral: Paragraph

induced matrix norm: induced matrix norm

initial-boundary-value problem: Paragraph

initial-value problem: Definition 6.1.1, initial-value problem

one-step method for: Definition 6.2.1

inner product: Paragraph, inner product

of functions: Definition 9.4.1
of vectors: Paragraph

inner product space: Definition 9.4.1

interpolation: Definition 5.1.1, Paragraph, interpolation

by piecewise linear polynomials: Paragraph
by piecewise polynomials: Paragraph, Paragraph, Paragraph, Paragraph
by polynomials: Paragraph, Paragraph, Paragraph, Paragraph, Paragraph
by trigonometric polynomials: Paragraph

inverse iteration: Algorithm 8.3.1, inverse iteration

invertible matrix: Paragraph

J

Jacobian matrix: Paragraph, Paragraph, Paragraph, Jacobian matrix

Julia:

+=: Paragraph
.+: Paragraph
.-: Paragraph
:: Paragraph, Paragraph
@.: Paragraph
@animate: Paragraph
@sprintf: Paragraph, Paragraph
Boolean indexing: Paragraph
DiagonalPreconditioner: Paragraph, Paragraph
FNC: Paragraph
I: Paragraph
Images: Paragraph, Paragraph, Paragraph
LinearMap: Paragraph
NaN: Paragraph
ODEProblem: Notebook-code, Notebook-code
Pair: Paragraph
Rational: Paragraph
Spline1D: Paragraph, Paragraph
\!: Paragraph
\': Paragraph, Paragraph
\\: Paragraph, TabSet, Paragraph, Paragraph, Paragraph
adjoint: Paragraph, Paragraph
annotate!: Notebook-code, Notebook-code, Notebook-code
anonymous functions: Paragraph
break: Algorithm 4.3.2
broadcasting: Grid, Paragraph, Paragraph, Paragraph
cg: Paragraph
cholesky: Paragraph
collect: Paragraph
comprehension: Paragraph
cond: Paragraph, Paragraph
destructuring: Paragraph
diag: Paragraph, Paragraph
diagm: Paragraph, Paragraph, Paragraph
eachindex: Algorithm 9.5.1
eigen: Paragraph
eigs: Paragraph
eigvals: Paragraph
eltype: Paragraph
end: Paragraph
enumerate: Paragraph, Paragraph
fft: Paragraph
fill: Paragraph
fit: Paragraph, Paragraph, Paragraph
for: About the code, Paragraph
functions: Paragraph
gmres: Paragraph
graphplot: Paragraph
ilu: Paragraph
in-place function: Paragraph, Paragraph
indexing arrays: Paragraph, Attention, Paragraph, Paragraph
isinf: Algorithm 9.2.1
isodd: Algorithm 9.5.1
keyword function arguments: Algorithm 4.3.2, Algorithm 5.7.1, Notebook-code
kron: Paragraph
length: About the code, Paragraph
maximum: Paragraph
minimum: Paragraph
minres: Notebook-code
namespace: Paragraph
nnz: Paragraph
norm: Paragraph
normalize: Paragraph, Paragraph
opnorm: Paragraph
plotting functions: Paragraph
push!: Notebook-code, Notebook-code
push\!: Paragraph
quadgk: Paragraph, Paragraph
range: Paragraph, Grid
rank: Paragraph
reshape: Paragraph
return: Paragraph
scatter: Paragraph
scientific notation: Paragraph
size: Paragraph
solve: Notebook-code, Notebook-code
sortby: Paragraph
sortperm: Paragraph
sparse: Paragraph
spdiagm: Paragraph, Paragraph
splatting: Paragraph
sprandsym: Paragraph
string interpolation: Paragraph
subplots: Paragraph
sum: Paragraph, About the code
summarysize: Paragraph
svd: Paragraph
svdvals: Paragraph
ternary operator: Algorithm 9.5.1
transpose: Paragraph
tril: Paragraph, Paragraph
triu: Paragraph, Paragraph
using: Paragraph
vec operation: Paragraph
÷: Paragraph

K

Kronecker product: Definition 13.3.2, Kronecker product

Krylov matrix: Definition 8.4.1

Krylov subspace: Definition 8.4.1, Krylov subspace

L

LU factorization: Definition 2.4.1, Paragraph, Paragraph, LU factorization

incomplete: Paragraph

Lagrange formula: Lagrange formula

Lagrange interpolation formula: Theorem 9.1.2

Lagrange polynomial: Definition 9.1.1

Lanczos iteration: Paragraph, (8.6.2), Lanczos iteration

Laplace equation: Paragraph, Laplace equation

Legendre polynomials: Definition 9.4.3

Levenberg's method: Algorithm 4.6.2

Lipschitz condition: Definition 4.2.3

Luke Skywalker: Luke Skywalker, Return of the Jedi

linear combination: Paragraph

linear convergence: Definition 4.2.2, linear convergence

linear least-squares problem: Paragraph, Paragraph, Paragraph, linear least-squares problem

linearization of an ODE: Definition 11.4.1

local truncation error: local truncation error

logistic equation: Paragraph

M

MATLAB:

:: Paragraph, Paragraph
Boolean indexing: Paragraph
NaN: Paragraph
\': Paragraph, Paragraph
\\: Paragraph, Paragraph, Paragraph
\~: Paragraph
adjoint: Paragraph, Paragraph
animation: Paragraph
anonymous functions: Paragraph
broadcasting: Paragraph
chol: Paragraph
cond: Paragraph
diag: Paragraph, Paragraph, Paragraph, Paragraph
diagm: Paragraph
eig: Paragraph
eigs: Paragraph
end: Paragraph
eye: Paragraph, Paragraph
fill: Paragraph
format: Tip
fplot: Notebook-code
functions: Paragraph
gmres: Paragraph
graph (network): Paragraph
hold on: Paragraph
ilu: Paragraph
indexing arrays: Paragraph
integral: Paragraph
kron: Paragraph
length: Paragraph
linspace: Paragraph, Paragraph
max: Paragraph, Paragraph
minres: Notebook-code
nnz: Paragraph
norm: Notebook-code, Paragraph
ode: Notebook-code
ones: Paragraph
pcg: Paragraph
plot: Paragraph
polyval: Notebook-code
scatter: Paragraph
scientific notation: Paragraph
size: Paragraph
solve: Notebook-code
sparse: Paragraph
spdiags: Paragraph
subplot: Notebook-code
sum: Paragraph
svd: Notebook-code
tic and toc: Paragraph
transpose: Paragraph
tril: Paragraph, Paragraph
triu: Paragraph, Paragraph
vander: Paragraph
zeros: Paragraph

MINRES: Paragraph

Maxwell's equations: Paragraph

machine epsilon: Definition 1.1.2, Paragraph, Paragraph, machine epsilon

in double precision: Paragraph

mantissa: See significand

mass matrix: Paragraph

matrix condition number: matrix condition number

matrix factorization: Theorem 7.3.2

matrix inverse: Paragraph, Paragraph

matrix multiplication: Paragraph

method of lines: Paragraph, Paragraph, Paragraph, Paragraph, method of lines

multistep: multistep

multistep method: Definition 6.6.1, Paragraph

implementation of: Paragraph

N

NaN: Paragraph

Neumann boundary condition: Definition 10.1.2, Paragraph

Neumann condition: Neumann condition

Newton's method: Algorithm 4.3.1, Paragraph, Paragraph

multidimensional: Algorithm 4.5.1

Newton’s method: Newton’s method

nodes: nodes

nonlinear least squares: Definition 4.7.1

nonlinear least-squares problem: nonlinear least-squares problem

norm: norm

Frobenius: Paragraph
matrix: Definition 2.7.2, Theorem 7.3.3
vector: Paragraph, Paragraph

normal: normal

normal equations: Theorem 3.2.1, Theorem 3.2.2, Paragraph, normal equations

normal matrix: Definition 7.2.5

numerical integration: Paragraph, Paragraph, numerical integration

O

ONC matrix: Definition 3.3.2, Paragraph, ONC matrix

Obi-Wan Kenobi: Obi-Wan Kenobi, Star Wars: A New Hope, Obi-Wan Kenobi, Star Wars: A New Hope

Oregonator: Example 11.4.1

one-step IVP method: one-step IVP method

order of accuracy: Paragraph, order of accuracy

of a finite-difference formula: Definition 5.5.2, Paragraph
of a one-step IVP method: Paragraph
of an approximation: Definition 5.2.2
of numerical integration: Definition 5.6.3
of the finite element method: Paragraph

orthogonal functions: Definition 9.4.1

orthogonal matrix: Paragraph, Paragraph, Definition 3.4.1, Definition 7.2.2, orthogonal matrix

orthogonal polynomials: Paragraph, orthogonal polynomials

orthogonal vectors: Paragraph, orthogonal vectors

orthonormal vectors: Paragraph, orthonormal vectors

outer product: Paragraph, Paragraph, Paragraph, outer product

overdetermined: overdetermined

P

PLU factorization: Definition 2.6.1, PLU factorization

Poisson equation: Paragraph

Python:

:: Paragraph
LinearOperator: Paragraph
NaN: Paragraph
adjoint: Paragraph
animation: Paragraph
arange: Paragraph
argsort: Paragraph
broadcast: Paragraph
broadcasting: Paragraph, Paragraph
cg: Paragraph
cholesky: Notebook-code
cond: Paragraph, Notebook-code, Paragraph
destructuring: Paragraph
diag: Paragraph, Paragraph
eig: Paragraph
eigs: Paragraph
elementwise multiplication: Paragraph
enumerate: Notebook-code
gmres: Paragraph
hstack: Paragraph
indexing arrays: Paragraph
kron: Paragraph
linspace: Paragraph
lstsq: Notebook-code
matrix_power: Notebook-code
max: Paragraph
minres: Notebook-code, Paragraph
networkx: Notebook-code
nnz: Notebook-code
norm: Paragraph, Paragraph
plotting functions: Paragraph
return: Paragraph
scientific notation: Paragraph
slice: Paragraph
spilu: Paragraph
sum: Paragraph
svd: Paragraph
transpose: Paragraph
tril: Paragraph, Paragraph
triu: Paragraph, Paragraph, Notebook-code
vstack: Paragraph

parabolic PDE: Definition 11.1.1

parameter continuation: Paragraph

piecewise linear: piecewise linear

piecewise linear interpolant: Paragraph

pivoting: Paragraph, Paragraph, Theorem 2.9.2

power iteration: Algorithm 8.2.1, power iteration

preconditioning: Definition 8.8.1, preconditioning

predator–prey model: Paragraph

pseudoinverse: Paragraph, pseudoinverse

Q

QR factorization: QR factorization

quadratic convergence: Definition 4.3.1, quadratic convergence

quadrature: See numerical integration

quasi-Newton method: Paragraph

quasi-Newton methods: quasi-Newton methods

quasimatrix: Definition 9.4.2, quasimatrix

R

Rayleigh quotient: Paragraph, Rayleigh quotient

Return of the Jedi: Luke Skywalker, Return of the Jedi

Robin boundary condition: Definition 10.1.2

Runge phenomenon: Paragraph, Runge phenomenon

Runge–Kutta: Runge–Kutta

Runge–Kutta method: Paragraph, Paragraph, Paragraph

reduced QR factorization: reduced QR factorization

reduced SVD: reduced SVD

residual: Paragraph, Paragraph, Paragraph, Paragraph, Paragraph, residual

of a linear system: Definition 2.8.2
of rootfinding: Paragraph

restarting: restarting

rootfinding problem: Definition 4.1.1, Paragraph, (11.5.6), rootfinding problem

multidimensional: Definition 4.5.1

roots:

multiplicity of: Paragraph, Paragraph

row pivoting: row pivoting

S

SPD matrix: See symmetric positive definite matrix

SVD: See singular value decomposition

Simpson's formula: Paragraph, Paragraph

Sylvester equation: Paragraph

secant method: Algorithm 4.4.1, secant method

semidiscretization:

see method of lines: Paragraph

shifted inverse iteration: See inverse iteration

shock wave: Paragraph

shooting: shooting

significand: Paragraph

significant digits: Paragraph

similarity transformation: Paragraph

simple root: simple root

singular matrix: Paragraph

singular value decomposition: Definition 7.3.1, Paragraph

thin form: Paragraph

singular value decomposition (SVD): singular value decomposition (SVD)

sparse matrix: Paragraph, Paragraph, Paragraph, Paragraph, Paragraph, sparse matrix

spectral convergence: spectral convergence

spline (cubic spline): Definition 5.3.1

stability: Paragraph, Paragraph

of IVP solvers: Paragraph
of collocation: Paragraph
of multistep methods: Definition 6.8.1
of polynomial interpolation: Paragraph

stability region: Definition 11.3.2, stability region

steepest descent: Paragraph

step size: step size

stiff differential equation: Paragraph, Paragraph, Example 6.7.2, stiff differential equation

stiffness matrix: Paragraph

subtractive cancellation: Paragraph, Paragraph, Paragraph, Paragraph, Paragraph, subtractive cancellation

superlinear convergence: Paragraph, superlinear convergence

symmetric matrix: Paragraph, Definition 2.9.2, Paragraph, Paragraph, symmetric matrix

symmetric positive definite matrix: Definition 2.9.3, Paragraph, Definition 8.6.1, symmetric positive definite matrix

T

The Empire Strikes Back: Yoda, The Empire Strikes Back, Han Solo, The Empire Strikes Back, Yoda, The Empire Strikes Back, Yoda, The Empire Strikes Back, Han Solo, The Empire Strikes Back

tensor-product domain: Paragraph, tensor-product domain

thin QR factorization: thin QR factorization

thin SVD: thin SVD

transpose: Paragraph, Paragraph

trapezoid formula: trapezoid formula

for an IVP: Table 6.6.1, Paragraph, Paragraph
for integration: Definition 5.6.2, Paragraph

triangular matrix: Paragraph, Paragraph, triangular matrix

tridiagonal matrix: Definition 2.9.1, tridiagonal matrix

trigonometric interpolation: trigonometric interpolation

trigonometric polynomial: Definition 9.5.1

truncation error: truncation error

of a finite-difference formula: Definition 5.5.1
of a multistep IVP formula: Paragraph
of a numerical integration formula: Definition 5.6.3
of a one-step IVP solver: Paragraph

U

unit lower triangular matrix: Paragraph

unit roundoff: Definition 1.1.2

unit triangular matrix: unit triangular matrix

unit vector: Paragraph, unit vector

unitary matrix: Definition 7.2.2, Definition 7.2.5, Definition 7.3.1, Theorem 7.4.2, unitary matrix

unstable: unstable

unvec: Paragraph

upper Hessenberg matrix: Definition 8.4.2, upper Hessenberg matrix

upwind direction: Paragraph

V

Vandermonde matrix: Definition 2.1.2, Paragraph, Vandermonde matrix

vec: Paragraph

W

Wedge Antilles: Wedge Antilles, Star Wars: A New Hope

wave equation: Paragraph, Paragraph

weak solution: Definition 10.6.1

weights: weights

Y

Yoda: Yoda, The Empire Strikes Back, Yoda, The Empire Strikes Back, Yoda, The Empire Strikes Back

Z

zero-stability: Definition 6.8.1, zero-stability

Python codes

Glossary