Mathematics: Index (Mathematics)

analytical partial derivatives (BVP)

additional parameters

distance between nodes

B

ballode demo

bandwidth of sparse matrix, reducing

Basic Fitting interface

batonode demo

bicubic interpolation

bilinear interpolation

boundary conditions

BVP example

Boundary Value Problems. See BVP

PDE example

Brusselator system (ODE example)

brussode demo

Buckminster Fuller dome

Bucky ball

burgersode demo

rewriting problem as first-order system

BVP solver

additional known parameters

basic syntax

evaluate solution at specific points

examples

boundary condition at infinity (shockbvp)

Mathieu's Equation (mat4bvp)

rapid solution changes (shockbvp)

initial guess

analytical partial derivatives

unknown parameters

BVP solver properties

modifying property structure

querying property structure

singular BVPs

solution statistics

vectorization

C

cat

characteristic polynomial of matrix

characteristic roots of matrix

chol <1> <2>

Cholesky factorization

for sparse matrices

closest point searches

Delaunay triangulation

colamd

colmmd <1> <2>

colperm

comparing

sparse and full matrix storage

complex values in sparse matrix

computational functions

applying to sparse matrices

computational geometry

contents of sparse matrix

two-dimensional

condest

convex hulls

cubic spline interpolation

two-dimensional

convolution

correlation coefficients

creating

cubic interpolation

curve fitting

Basic Fitting interface

curves

Cuthill-McKee, reverse ordering

D

DAE

solution of

data analysis

column-oriented

data filtering. See filtering

data fitting. See curve fittimg

data gridding

data. See also

rewriting problem as first-order system

DDE solver

additional known parameters

basic syntax

discontinuities

evaluating solution at specific points

examples

cardiovascular model (ddex2)

straightforward example (ddex1)

modifying property structure

DDE solver properties

ddeset function

error tolerance

event location

querying property structure

solver output

step size

ddephas2 output function

ddephas3 output function

ddeplot output function

ddeprint output function

ddex1 demo

ddex2 demo

decomposition

Delaunay multidimensional tessellations

Delaunay triangulation

closest point searches

Delay Differential Equations. See DDE

density of sparse matrix

derivatives

polynomial

determinant of matrix

diag

diagonal

creating sparse matrix from

difference equations

differential equations

boundary value problems for ODEs

initial value problems for DDEs

initial value problems for ODEs and DAEs

partial differential equations

differential-algebraic equations. See DAE

direct methods for systems of equations

discontinuities (DDE)

discrete Fourier transform. See Fourier transforms

displaying

sparse matrices

distance between nodes

dot product

E

Emden's equation

error bounds

error tolerance

effects of too large (ODE)

examples

adjacency matrix (sparse)

airflow modeling

Bucky ball

second difference operator

sparse matrix <1> <2>

theoretical graph (sparse)

examples (BVP)

boundary condition at infinity (fsbvp)

Mathieu's Equation (mat4bvp)

rapid solution changes (shockbvp)

two solutions (twobvp)

examples (DAE)

electrical circuit (amp1dae)

Robertson problem (hb1dae)

examples (DDE)

cardiovascular model (ddex2)

straightforward example (ddex1)

examples (MATLAB PDE)

electrodynamics problem (pdex4)

simple PDE (pdex1)

examples (ODE)

advanced event location (orbitode)

Brusselator system (brussode)

constant mass matrix (fem2ode)

finite element discretization (fem1ode)

nonstiff

rigid body (rigidode)

Robertson problem (hb1ode)

simple event location (ballode)

stiff

strongly state-dependent mass matrix (burgersode)

time-, state-dependent mass matrix (batonode)

van der Pol (vdpode)

exponential curve fitting

eye

derivation of the name

F

Hermitian positive definite

incomplete

partial pivoting

positive definite

fast Fourier transform. See Fourier transforms

fem1ode demo

fem2ode demo

fill-in of sparse matrix

filtering

difference equations

find function

and sparse matrices

finite differences

finite element discretization (ODE example)

first-order differential equations

representation for BVP solver

representation for DDE solver

Fourier analysis

concepts

Fourier transforms

calculating sunspot periodicity

FFT-based interpolation

length vs. speed

phase and magnitude of transformed data

fsbvp demo

full <1> <2>

function functions

functions

mathematical. See mathematical functions

optimizing

G

geometric analysis

gplot

graph

characteristics

Hermitian positive definite matrix

theoretical

H

hb1dae demo

hb1ode demo

higher-order ODEs

rewriting as system of first-order ODEs

I

identity matrix

importing

sparse matrix

incomplete factorization

infeasible optimization problems

initial conditions

initial guess (BVP)

initial value problems

initial-boundary value PDE problems

ODE and DAE

inline objects

representing mathematical functions

inner product

integration

double

numerical

J

Jacobian matrix (BVP)

Jacobian matrix (ODE)

generating sparse numerically

specifying

vectorizing ODE function

K

Kronecker tensor matrix product

L

least squares

length of curve, computing

linear algebra

linear equations

minimal norm solution

overdetermined systems

rectangular systems

underdetermined systems

linear interpolation

for sparse matrices and reordering

one-dimensional

linear systems of equations

linear transformation

load

Lobatto IIIa BVP solver

lu <1> <2>

LU factorization

M

mathematical functions

as function input arguments

finding zeros

minimizing

numerical integration

plotting

representing in MATLAB

mathematical operations on sparse matrices

Mathieu's equation (BVP example)

matrices

as linear transformation

characteristic polynomial

characteristic roots

creation

determinant

full to sparse conversion <1> <2>

matrix

matrix operations

addition and subtraction

matrix products

max

M-files

representing mathematical functions

minimizing mathematical functions

of one variable

of several variables

options

minimum degree ordering

Moore-Penrose pseudoinverse

multidimensional data gridding

multidimensional interpolation

scattered data

multistep solver (ODE)

multivariate data

matrix representation

vehicle traffic sample data

N

NaNs

propagation

removing from data

nearest neighbor interpolation

nnz <1> <2>

nonstiff ODE examples

rigid body (rigidode)

nonzero elements

number of

nonzero elements of sparse matrix

maximum number in sparse matrix

storage <1> <2>

values

visualizing with spy plot

nonzeros

normalizing data

norms

vector and matrix

numerical integration

computing length of curve

double

nzmax <1> <2>

O

ODE

overspecified systems

solution of

ODE solver

evaluate solution at specific points

ODE solver properties

modifying property structure

ode15s

odeset function

querying property structure

solver output

step size

ODE solvers

algorithms

Adams-Bashworth-Moulton PECE

Bogacki-Shampine

Dormand-Prince

modified Rosenbrock formula

numerical differentiation formulas

backward differentiation formulas

basic example

minimizing output storage

minimizing startup cost

multistep solver

nonstiff problem example

nonstiff problems

numerical differentiation formulas

obtaining solutions at specific times

one-step solver

overview

passing additional parameters

problem size

one-dimensional interpolation

sampled data

stiff problems <1> <2>

troubleshooting

variable order solver

odephas2 output function

odephas3 output function

odeplot output function

odeprint output function

ones

one-step solver (ODE)

operator

second difference

optimization

calling sequence changes (Version 5)

helpful hints

options parameters

troubleshooting

optimization code

updating to MATLAB Version 5 syntax

orbitode demo

Ordinary Differential Equations. See ODE

orthogonal matrix

outer product

outliers

removing from statistical data

output points (ODE)

increasing number of

output properties (DDE)

output properties (ODE)

increasing number of output points

output storage

minimizing for ODE problems

overdetermined

rectangular matrices

overspecified ODE systems

P

Partial Differential Equations. See PDE

partial fraction expansion

additional known parameters

discretized

PDE solver (MATLAB)

basic syntax

evaluate solution at specific points

examples

electrodynamics problem

simple PDE

PDE solver (MATLAB) properties

pdex1 demo

pdex2 demo

pdex3 demo

pdex4 demo

pdex5 demo

performance

de-emphasizing an ODE solution component

improving for BVP solver

improving for DDE solver

improving for ODE solvers

improving for PDE solver

permutations

plotting

mathematical functions

polynomial

curve fitting

regression

polynomial interpolation

polynomials

basic operations

calculating coefficients from roots

multiplying and dividing

partial fraction expansion

representing as vectors

preconditioner for sparse matrix

property structure (BVP)

creating

modifying

querying

property structure (DDE)

creating

modifying

querying

property structure (ODE)

pseudoinverse

Q

QR factorization <1> <2>

quad, quadl functions

differ from ODE solvers

quadrature. See numerical integration

R

rand

rank deficiency

overdetermined systems

pseudoinverse

QR factorization

rank deficient

singular value decomposition

underdetermined systems

regression

linear-in-the-parameters

multiple

polynomial

relative accuracy

for sparser factorizations

minimum degree ordering

to reduce bandwidth

representing

mathematical functions

residuals

analyzing

for exponential data fit

rigid body (ODE example)

Robertson problem

roots

S

sampled data

with ODE solvers

save

scalar

as a matrix

scalar product

scattered data

multidimensional interpolation

multidimensional tessellation

triangulation and interpolation

Schur decomposition

seamount data set

second difference operator, example

shockbvp demo

singular value matrix decomposition

size

solution changes, rapid (BVP example) <1> <2>

solution statistics (BVP)

solving linear systems of equations

full

sparse

sort

sparse <1> <2>

sparse matrix

advantages

and complex values

Cholesky factorization

computational considerations

contents

conversion from full <1> <2>

creating

directly

from diagonal elements

linear systems of equations

density

distance between nodes

LU factorization

and reordering

mathematical operations

nonzero elements

maximum number

specifying when creating matrix

storage <1> <2>

values

nonzero elements of sparse matrix

propagation through computations

QR factorization

reordering <1> <2>

storage

for various permutations

viewing

theoretical graph

triangular factorization

viewing contents graphically

viewing storage

visualizing

working with

sparse ODE examples

Brusselator system (brussode)

spconvert

spdiags

speye <1> <2> <3>

spones

spparms

sprand

spy

spy plot

startup cost

minimizing for ODE solvers

statistical data

See also multivariate data

See also univariate data

statistics

descriptive

step size (DDE)

initial step size

upper bound

step size (ODE) <1> <2>

initial step size

upper bound

stiff ODE examples

Brusselator system (brussode)

differential-algebraic problem (hb1dae)

finite element dicretization (fem1ode)

van der Pol (vdpode)

stiffness (ODE), defined

storage

for various permutations of sparse matrix

of sparse matrix

sparse and full, comparison

viewing for sparse matrix

sum <1> <2>

sunspot periodicity

calculating using Fourier transforms

symamd

symmetric

matrix

symmmd <1> <2>

symrcm <1> <2>

systems of equations. See linear systems of equations

T

tessellations, multidimensional

three-dimensional interpolation

transfer functions

using partial fraction expansion

transpose

complex conjugate

unconjugated complex

triangular factorization

for sparse matrices

triangular matrix

triangulation

closest point searches

Delaunay

scattered data

Voronoi diagrams

See also tessellation

tricubic interpolation

trilinear interpolation

troubleshooting (ODE)

twobvp demo

two-dimensional interpolation

comparing methods graphically

U

underdetermined

rectangular matrices

unitary matrices

QR factorization

univariate data

unknown parameters (BVP)

example

updating optimization code to MATLAB Version 5 syntax

V

van der Pol example

simple, nonstiff

simple, stiff

variable-order solver (ODE)

vdpode demo

vector products

dot or scalar

outer and inner

vectorizing ODE function (BVP)

vectors

column and row

multiplication

vehicle traffic sample data

visualizing

sparse matrix

spy plot

visualizing solver results

Voronoi diagrams