Index - Fundamentals of Numerical Computation

A

Adams--Bashforth formula: Paragraph

Adams--Moulton formula: Paragraph

Arnoldi iteration: Arnoldi iteration

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, adjacency matrix

adjoint: adjoint

adjoint of a matrix: Paragraph, Paragraph

advection equation: advection equation

algorithm: Paragraph, algorithm

argmin: Definition 3.1.1

asymptotic: asymptotic

asymptotic notation: Section 2.5.1

B

Bauer–Fike theorem: Paragraph

Broyden update: Paragraph

backward difference: Paragraph

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

bandwidth: bandwidth

bandwidth of a matrix: Definition 2.9.1

barycentric formula: barycentric formula

big-O: big-O

boundary-value problem: boundary-value problem

C

Cholesky factorization: Cholesky factorization

cardinal function: Definition 5.1.2, Paragraph, cardinal function

centered difference: Paragraph

characteristic polynomial: Paragraph

collocation: collocation

condition number: condition number

of a matrix: Paragraph, Definition 3.2.3, 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

contraction mapping: Paragraph

convergence rate:

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

cubic spline: Definition 5.3.1, cubic spline

D

Dahlquist theorems: Section 6.8.2

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: diagonalizable

diagonalizable matrix: Definition 7.2.3

differentiation matrix: differentiation matrix

dimension reduction: Paragraph

divide and conquer: Section 5.7.2

dominant eigenvalue: dominant eigenvalue

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, eigenvalue

conditioning of: Paragraph

eigenvalue decomposition: Definition 7.2.3

eigenvalue decomposition (EVD): eigenvalue decomposition (EVD)

eigenvector: Definition 7.2.1, eigenvector

evolutionary PDE: evolutionary PDE

extrapolation: Paragraph, extrapolation

F

Francis QR iteration: Paragraph

Frobenius norm: Frobenius norm

finite difference: finite difference

finite differences: Definition 5.4.1, Paragraph, Paragraph

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

fixed point iteration: fixed point iteration

fixed point problem: fixed point problem

fixed-point iteration: Algorithm 4.2.1

fixed-point problem: Definition 4.2.1

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

flops: Paragraph, flops

forward difference: Paragraph

forward substitution: Paragraph, forward substitution

G

GMRES: GMRES

Gaussian elimination: Paragraph, Gaussian elimination

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

Gregory integration formula: MystDirectiveError

generating polynomials: Paragraph, generating polynomials

global error: Definition 6.2.3, global error

graph: Definition 7.1.1, graph

H

Han Solo: Han Solo, The Empire Strikes Back

Horner's algorithm: Example 1.3.1

Householder reflector: Paragraph

hat functions: Paragraph, hat functions

heat equation: heat equation

hermitian: hermitian

hermitian matrix: Paragraph, Paragraph, Theorem 7.3.2

hermitian positive definite (HPD): hermitian positive definite (HPD)

hermitian positive definite matrix: See symmetric positive definite matrix

I

IEEE 754: Paragraph

identity matrix: Paragraph, Paragraph, identity matrix

ill-conditioned: ill-conditioned

image (as a matrix): Paragraph, Example 7.5.1

implicit: implicit

implicit IVP solver: Definition 6.6.1, Paragraph

induced matrix norm: induced matrix norm

initial-value problem: Definition 6.1.1

one-step method for: Definition 6.2.1

initial-value problem (IVP): initial-value problem (IVP)

inner product: Paragraph, inner product

of vectors: Paragraph

interpolation: Definition 5.1.1, Paragraph, interpolation

by piecewise polynomials: Paragraph, Paragraph, Paragraph, Paragraph
by polynomials: Paragraph, Paragraph, Paragraph, Paragraph

inverse iteration: inverse iteration

invertible matrix: Paragraph

J

Jacobian matrix: Paragraph, Paragraph, Jacobian matrix

Julia:

+=: Paragraph
.+: Grid
.-: Grid
:: Paragraph, Paragraph
@.: Grid
@sprintf: Grid, Grid
Boolean indexing: Grid
FNC: Grid
I: Grid
Images: Paragraph, Paragraph, Paragraph
NaN: Paragraph
ODEProblem: Notebook-code, Notebook-code
Pair: Grid
Rational: Paragraph
Spline1D: Paragraph, Paragraph
\!: Grid
\': Paragraph, Paragraph
\\: Grid, TabSet, Paragraph, Grid
adjoint: Paragraph, Paragraph
annotate!: Notebook-code, Notebook-code, Notebook-code
anonymous functions: Paragraph
break: Algorithm 4.3.2
broadcasting: Grid, Grid, Grid, Paragraph
cholesky: Grid
comprehension: Grid
cond: Grid, Paragraph
destructuring: Grid
diag: Grid, Grid
diagm: Grid, Grid, Grid
eigen: Paragraph
eigvals: Paragraph
eltype: Paragraph
end: Grid
enumerate: Grid, Grid
fill: Grid
fit: Paragraph, Paragraph, Paragraph
for: Grid
functions: Paragraph
graphplot: Paragraph
indexing arrays: Grid, Attention
keyword function arguments: Algorithm 4.3.2, Algorithm 5.7.1
length: Grid
maximum: Grid
minimum: Grid
namespace: Grid
nlsolve: Paragraph
norm: Paragraph
normalize: Paragraph, Grid
opnorm: Paragraph
plotting functions: Grid
push!: Notebook-code, Notebook-code
push\!: Grid
quadgk: Paragraph, Paragraph
range: Paragraph, Grid
return: Paragraph
scatter: Grid
scientific notation: Grid
size: Grid
solve: Notebook-code, Notebook-code
sortby: Paragraph
splatting: Grid
string interpolation: Grid
subplots: Paragraph
sum: Grid, About the code
svd: Paragraph
svdvals: Paragraph
transpose: Paragraph
tril: Grid, Paragraph
triu: Grid, Paragraph
using: Grid

K

Kronecker product: Kronecker product

Krylov subspace: Krylov subspace

L

LU factorization: Definition 2.4.1, LU factorization

Lagrange formula: Lagrange formula

Lanczos iteration: Lanczos iteration

Laplace equation: Laplace equation

Levenberg's method: Algorithm 4.6.2

Lipschitz condition: Definition 4.2.3

linear combination: Paragraph

linear convergence: Definition 4.2.2, linear convergence

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

local truncation error: local truncation error

logistic equation: Paragraph

M

MATLAB:

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

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

in double precision: Paragraph

mantissa: See significand

matrix condition number: matrix condition number

matrix factorization: Theorem 7.3.2

matrix inverse: Paragraph, Paragraph

matrix multiplication: Paragraph

method of lines: method of lines

multistep: multistep

multistep method: Definition 6.6.1, Paragraph

implementation of: Paragraph

N

NaN: Paragraph

Neumann condition: Neumann condition

Newton's method: Algorithm 4.3.1, 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: Paragraph, Paragraph, ONC matrix

one-step IVP method: one-step IVP method

order of accuracy: order of accuracy

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

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

orthogonal polynomials: 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

Python:

:: Paragraph
NaN: Paragraph
adjoint: Paragraph
arange: Paragraph
broadcast: Paragraph
broadcasting: Paragraph, Paragraph
cholesky: Notebook-code
cond: Paragraph, Notebook-code, Paragraph
destructuring: Grid
diag: Paragraph, Paragraph
diags: Notebook-code
eig: Paragraph
elementwise multiplication: Paragraph
enumerate: Notebook-code
hstack: Paragraph
indexing arrays: Paragraph
length: Example 1.3.2
linspace: Paragraph
lstsq: Notebook-code
max: Paragraph
networkx: Notebook-code
norm: Paragraph, Paragraph
plotting functions: Paragraph
return: Paragraph
scientific notation: Grid
slice: Paragraph
splu: Notebook-code
sum: Paragraph
svd: Paragraph
transpose: Paragraph
tril: Paragraph, Paragraph
triu: Paragraph, Paragraph, Notebook-code
vstack: Paragraph

piecewise linear: piecewise linear

piecewise linear interpolant: Paragraph

pivoting: Paragraph, Paragraph, Theorem 2.9.2

power iteration: power iteration

preconditioning: 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 methods: quasi-Newton methods

quasimatrix: quasimatrix

R

Rayleigh quotient: Paragraph, Rayleigh quotient

Runge phenomenon: Runge phenomenon

Runge--Kutta: Runge--Kutta

Runge–Kutta method: Paragraph, Paragraph, Paragraph

reduced QR factorization: reduced QR factorization

reduced SVD: reduced SVD

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

of a linear system: Definition 2.8.2
of rootfinding: Paragraph

restarting: restarting

rootfinding problem: Definition 4.1.1, Paragraph, 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

secant method: Algorithm 4.4.1, secant method

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: sparse

sparse matrix: Paragraph

spectral convergence: spectral convergence

spline (cubic spline): Definition 5.3.1

stability: Paragraph, Paragraph

of multistep methods: Definition 6.8.1

stability region: stability region

steepest descent: Paragraph

step size: step size

stiff: stiff

stiff differential equation: Paragraph, Example 6.7.2

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

superlinear convergence: Paragraph, superlinear convergence

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

symmetric positive definite (SPD) matrix: symmetric positive definite (SPD) matrix

symmetric positive definite matrix: Definition 2.9.3, Paragraph

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

tensor-product domain: 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
for integration: Definition 5.6.2

triangular matrix: Paragraph, Paragraph, triangular matrix

tridiagonal matrix: Definition 2.9.1, tridiagonal matrix

trigonometric interpolation: trigonometric interpolation

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: unitary

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

unstable: unstable

upper Hessenberg: upper Hessenberg

V

Vandermonde matrix: Paragraph, Paragraph, Vandermonde matrix

W

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-stable: zero-stable

Python codes

Chapter 7