Glossary for Differential Equations with MATLAB, 3rd edition

For the updated version of this glossary for MATLAB 2019a, please go here.

This glossary is divided into the following sections:

1. MATLAB Operators: the special symbols used by MATLAB,
2. MATLAB Commands: commands that manipulate data or expressions, or that initiate a process,
3. Built-in Functions: basic mathematical functions that can be evaluated or plotted,
4. Graphics Commands: commands using in creating or modifying figures,
5. Built-in Constants,
6. MATLAB Programming: commands used for programming and structuring M-files,
7. Simulink Blocks: the Simulink blocks most useful for the Simulink problems in this book. For a more extensive listing of Simulink blocks, see the Simulink documentation.
8. Useful MuPAD Commands: for using MuPAD directly (rather than through the Symbolic Math Toolbox interface).

The distinction among these various categories, especially among commands and programming constructs, is somewhat artificial.

Each item is followed by a brief description of its effect and then (in most cases) one or more examples. To get more information on a command, you can use the help command or the Help Browser. This glossary is not a comprehensive list of MATLAB commands, but it includes the commands most useful for studying differential equations.

MATLAB Operators

@

Marker for a function handle, used to refer to a built-in function or to create an anonymous function.

  f = @(x, y) x.^2.*y + y.^2

  quadl(@atan, 0, 1)

\

Left matrix division. X = A\B is the solution of A*X = B. Type help slash for more information.

    A = [1 0; 2 1]; B = [3; 5]; A\B

/

Ordinary division or right matrix division. Type help slash for more information.

./

Element-by-element division of arrays. Type help rdivide for more information.

*

Scalar or matrix multiplication, depending on context. Type help mtimes for more information.

.*

Element-by-element multiplication of arrays. Type help times for more information.

^

Scalar or matrix power, depending on context. Type help mpower for more information.

.^

Element-by-element power. Type help power for more information.

:

Range operator, used for defining vectors. Type help colon for more information.

'

Single quote mark, used for beginning and ending strings. Also used to denote the conjugate transpose of a matrix (see ctranspose).

.'

Denotes the transpose of a matrix (see also transpose).

;

Suppresses output of a MATLAB command. Also used to separate rows of a matrix.

    X = 0:0.1:30;

    A = [1 2 3; 4 5 6; 7 8 9]

...

Line continuation operator. Cannot be used inside strings.

    1 + 3 + 5 + 7 + 9 + 11 ...

        + 13 + 15 + 17

    ['This is a way to create very long strings ', ...

        'that span more than one line. Note the ', ...

        'square brackets.']

MATLAB Commands

addpath

Adds the specified directory to MATLAB's file search path.

    addpath('C:\my_mfiles')

    addpath C:\my_mfiles

ans

A variable holding the value of the most recent unassigned output.

bvp4c

Boundary value problem solver. Usually requires use of bvpinit to set the ``initial guess'' argument. The example below solves and plots the solution of y''+y=0 with boundary values y(0)=0, y(π/2)=1.

    solinit = bvpinit((0:0.1:1)*pi/2, [1, 0]);

    sol = bvp4c(@(x, y) [y(2); –y(1)], ...

        @(ya, yb) [ya(1); yb(1) – 1], solinit);

    plot(sol.x, sol.y(1,:))

cd

Makes the specified directory the current (working) directory.

    cd C:\mydocs\mfiles

char

Converts a symbolic expression to a string. Useful for defining inline functions or for defining input to dsolve.

    syms x y

    f = inline(char(sin(x)*sin(y)), 'x', 'y')

clear

Clears values and definitions for variables and functions. If you specify one or more variables, then only those variables are cleared.

    clear

    clear f, g

collect

Collects coefficients of powers of the specified symbolic variable in a given symbolic expression.

    syms x y

    collect(x^2 – 2*x^2 + 3*x + x*y, x)

conj

Gives the complex conjugate of a complex number.

    conj(2 + 3*i)

ctranspose

Conjugate transpose of a matrix. Usually invoked with the ' operator. Equivalent to transpose for real matrices.

    A = [1 3 i]; A'

D

Not a true MATLAB command. Used in the dsolve command to denote differentiation. See diff.

    dsolve('t*Dy + y = sin(t)')

deal

Distributes its arguments to different variables.

    [r, t] = deal(sqrt(x^2 + y^2), atan2(y, x))

delete

Deletes a file from the disk.

    delete filename

det

The determinant of a matrix.

    det([1 3; 4 5])

deval

Evaluates a numerical solution to an ODE at specified points.

    sol = ode45(@(t, y) t.*y, [0, 5], 1)

    deval(sol, 1)

diff

Symbolic differentiation operator (and difference operator).

    syms x; diff(x^3)

    syms x y; diff(x*y^2, y)

dir

Lists the files in a directory.

disp

Displays a string or the value of a variable.

    disp('The answer is:'), disp(a)

doc

Displays details about a specific command in the Help Browser.

    doc print

double

Gives a double precision value for either a numeric or symbolic quantity. Applied to a string, double returns a vector of ASCII codes for the characters in the string.

    z = sym('pi'); double(z)

dsolve

Symbolic ODE solver. By default, the independent variable is t, but a different variable can be specified as the last argument. The input must be a string, not a symbolic expression.

    dsolve('D2y – t*y = 0')

    dsolve('Dy + y^2 = 0', 'y(0) = 1', 'x')

    [x, y] = dsolve('Dx = 2*x + y', 'Dy = – x')

echo

Turns on or off the echoing of commands inside script M-files.

edit

Opens an M-file in the Editor/Debugger.

    edit mymfile

eig

Computes eigenvalues and eigenvectors of a square matrix.

    eig([2, 3; 4, 5])

    [e, v] = eig([1, 0, 0; 1, 1, 1; 1, 2, 4])

end

Last entry of a vector. (Also a programming construct.)

    v(end)

    v(3:end)

eval

Evaluates a string or symbolic expression. Useful in M-files or in defining anonymous functions.

    sol = dsolve('Dy = t^2 + y', 'y(0) = 2', 't');

    fsol = @(t) eval(vectorize(sol))

eye

The identity matrix of the specified size.

factor

Factors a polynomial or integer.

    syms x y; factor(x^4 – y^4)

feval

Evaluates a function specified by a string. Useful in function M-files.

    feval('exp', 1)

format

Specifies the output format for numerical variables.

    format short

    format long

fzero

Tries to find a zero of the specified function or expression (given as a string) near a given starting point or on a specified interval. If the first argument is a string, the independent variable must be x.

    fzero(@cos, [–pi 0])

    fzero('cos(x) – x', 1)

help

Asks for documentation for a MATLAB command. See also lookfor.

    help factor

ilaplace

Inverse Laplace Transform.

    syms s t; ilaplace(1/s, s, t)

inline

Constructs a MATLAB inline function from a string expression. Useful in MATLAB 6, but discouraged in later versions, in which it is better to use anonymous functions.

    sol = dsolve('Dy = t^2 + y', 'y(0) = 2', 't')

    fsol = inline(vectorize(sol), 't')

int

Integration operator for both definite and indefinite integrals.

    int('1/(1 + x^2)', 'x')

    syms x; int(exp(–x), x, 0, Inf)

inv

Inverse of a square matrix.

    inv([1 2; 3 5])

jacobian

Computes the Jacobian matrix, i.e., the matrix of partial derivatives with respect to the indicated variables.

    syms x y; jacobian([x^2*y y–x], [x y])

laplace

Computes the Laplace Transform.

    syms t s; laplace(t^5, t, s)

length

Returns the number of elements in a vector or string.

    length('abcde')

limit

Finds a two-sided limit, if it exists. Use right or left for one-sided limits.

    syms x; limit(sin(x)/x, x, 0)

    syms x; limit(1/x, x, Inf, 'left')

load

Loads Workspace variables from a disk file with a .mat suffix.

    load filename

lookfor

Searches for a specified string in the first line of all M-files found in the MATLAB path. Can be slow if many toolboxes are installed.

    lookfor ode

more

Turns on (or off) page-by-page scrolling of MATLAB output. Use the space bar to advance to the next page, the ENTER or RETURN key to advance line-by-line, and q to abort the output.

    more on

    more off

num2str

Converts a number to a string. Useful in programming.

    constant = ['a' num2str(1)]

ode45

Numerical ODE solver for first order equations. See MATLAB's online help for ode45 for a list of other MATLAB ODE solvers.

    [t, y] = ode45(@(t, y) t^2 + y, [0 10], 1);

    plot(t, y)

odeset

Used to set options for ode45, such as Events, AbsTol, and RelTol.

    options = odeset('Events', @eventfile)

ones

Creates a matrix of ones.

    ones(3), ones(3,1)

path

Without an argument, displays the search path. With an argument, sets the search path.

pause

Suspends execution of an M-file until the user presses a key.

pretty

Displays a symbolic expression in a more readable format.

    syms x y; expr = x/(x – 3)/(x + 2/y)

    pretty(expr)

publish

Runs the specified M-file and assembles input, comments, and output into a finished document in specified format. (The default is a web page, i.e., html format.)

    publish('mymfile', 'doc')

pwd

Shows the name of the current (working) directory.

quadl

Numerical integrator. The input function must be ``vectorized.''

    quadl(@(x) (1–x.^2).^(1/2), –1, 1)

quit

Terminates a MATLAB session.

roots

Finds the roots of a polynomial whose coefficients are given by the elements of the vector argument of roots.

    roots([1 1 –2])

save

Saves Workspace variables to a specified file. See load.

    save filename

sim

Runs a Simulink model. Specifying the time interval is optional.

    sim('mymodel', [0, 5])

simplify

Simplifies an algebraic expression.

    syms x; simplify((x+1)^2 – x^2)

size

Returns the number of rows and the number of columns in a matrix.

    A = [1 3 2; 4 1 5]

    [r, c] = size(A)

solve

Solves an algebraic equation. Will guess what the independent variable is if it is not specified.

    solve('x^2 + 3*x + 1')

strcat

Concatenates two or more strings.

    strcat('This ', 'is ', 'a ', 'long ', 'string')

str2num

Converts a string to a number. Useful in programming.

    constant = 'a7'

    index = str2num(constant(2))

struct

Used to create a ``structure.''

    x = struct; x.first = 'a'; x.second = 'b'; x

subs

Substitutes for parts of an expression.

    subs('x^3 – 4*x + 1', 'x', 2)

    subs('sin(x)^2 + cos(x)', 'sin(x)', 'z')

sum

Sums a vector, or sums the columns of a matrix.

    k = 1:10; sum(k)

sym

Creates a symbolic variable or number.

    pi = sym('pi')

    x = sym('x')

    constant = sym('1/2')

syms

Shortcut for creating symbolic variables. The command syms x is equivalent to x = sym('x').

    syms x y z

symsum

Evaluates a symbolic sum in closed form.

    syms x n; symsum(x^n/n, n, 1, inf)

taylor

Gives a Taylor polynomial approximation of order one less than a specified number (the default is 6) around a specified point (default 0). Will guess what the independent variable is if it is not specified.

    syms x; taylor(cos(x), 8, 0)

    taylor(exp(1/x), 10, x, inf)

transpose

Transpose of a matrix. Converts a column vector to a row vector, and vice versa. Usually invoked with the .' operator. See also ctranspose.

    A = [1 3 4]

    A.'

type

Displays the contents of a specified file.

    type myfile.m

vectorize

Vectorizes a symbolic expression. Useful in defining functions; see also eval and inline.

    f = inline(vectorize('x^2 – 1/x'))

vpa

Evaluates an expression to the specified degree of accuracy using variable precision arithmetic.

    vpa('1/3', 20)

whos

Lists current information on all the variables in the Workspace.

zeros

Creates a matrix of zeros.

    zeros(10), zeros(1,10)

Built-in MATLAB Functions

abs

absolute value

acos

inverse cosine

airy

Airy functions; airy(0, x) and airy(2, x) are linearly independent solutions of Airy's equation

asin

inverse sine

atan

inverse tangent

atan2

atan2(y,x) is the polar coordinate θ of the point (x, y).

besselj, bessely

Bessel functions. besselj(n, x) and bessely(n, x) are linearly independent solutions of Bessel's equation of order n.

cos

cosine

cosh

hyperbolic cosine

dirac

the unit impulse "function"

erf

the error function, the integral from 0 to x of (2/√π)exp(−t²).

exp

the exponential function e^x

expm

the matrix exponential function, defined like e^x, but using matrix multiplication

gamma

the gamma function Γ(x) = ∫₀^∞t^x−1e^−tdt

heaviside

the unit step function

hypgeom

the hypergeometric function of Gauss, ₂F₁(a,b;c;z)

imag

the imaginary part of a complex number

log

the natural logarithm

real

the real part of a complex number

sin

the sine

sinh

the hyperbolic sine

sinint

the sine integral ∫₀^x (sin t/t) dt

sqrt

the square root

tan

the tangent

tanh

the hyperbolic tangent

MATLAB Graphics Commands

axis

Sets axis scaling and appearance.

    axis([xmin xmax ymin ymax]) -- sets ranges for the axes.

    axis tight -- sets the axis limits to the full range of the data.

    axis equal -- makes the horizontal and vertical scales equal.

    axis square -- makes the axis box square.

    axis off -- hides the axes and tick marks.

close

Closes the current figure window; close all closes all figure windows.

contour

Plots the level curves of a function of two variables; usually used with the meshgrid command.

    [X, Y] = meshgrid(–3:0.1:3, –3:0.1:3);

    contour(X, Y, X.^2 – Y.^2)

ezplot

Basic plot command for symbolic expressions. Can also be used for implicit plots of f(x,y)=0.

    ezplot('exp(–x^2)', [–5, 5])

    syms x; ezplot(sin(x))

    syms x y; ezplot(x^3 – x – y^2, [–2, 2])

figure

Creates a new figure window.

gca

Get current axes. Returns the identifier (``handle'') of the axes properties of the active figure window.

gcf

Get current figure. Returns the number of the active figure window.

ginput

Gathers coordinates from a figure using the mouse (press the Enter or Return key to finish).

    [X, Y] = ginput

gtext

Places a text label using the mouse.

    gtext('Region of instability')

hold

Holds the current graph. Superimpose any new graphics generated by MATLAB on top of the current figure.

    hold on

    hold off

legend

Creates a legend for a figure.

    t = (0:0.1:2)*pi;

    plot(t, cos(t), t, sin(t))

    legend('cos(t)', 'sin(t)')

meshgrid

Creates a vector array that can be used as input to commands such as contour or quiver.

    [X, Y] = meshgrid(0:0.1:1, 0:0.1:2);

    contour(X, Y, X.^2 + Y.^2)

plot

Plots vectors of data. Specifying the plot style is optional.

    X = [0:0.1:2];

    plot(X, X.^3)

    plot(X, X.^3), 'rx–', 'LineWidth', 2)

plot3

Creates 3-dimensional (line) plots.

    t = [0:0.1:30];

    plot3(t, t.*cos(t), t.*sin(t))

print

Sends the contents of the current figure window to the printer or to a file. With the –s option, prints the current Simulink model.

    print

    print –deps picture.eps

    print –s

quiver

Plots a vector field for a pair of functions of two variables. It takes numerical, rather than symbolic, data. It is often useful to normalize vectors so that they have length close to 1 and to scale them by a factor of about 1/2.

    [X, Y] = meshgrid(–2:0.2:2, –2:0.2:2);

    U = X.*(X – Y – 2); V = Y.*(–X + Y + 1);

    L = sqrt(1 + U.^2 + V.^2));

    quiver(X, Y, U./L, V./L, 0.5)

set

Sets a property of a figure window.

    set(gca, 'FontSize', 14)

simplot

Similar to plot, but uses the style of a Simulink Scope window.

    t = [0:0.1:10]'; simplot(t, [sin(pi*t), cos(pi*t)])

subplot

Breaks the figure window into a grid of smaller plots.

    subplot(2, 2, 1), ezplot('x^2')

    subplot(2, 2, 2), ezplot('x^3')

    subplot(2, 2, 3), ezplot('x^4')

    subplot(2, 2, 4), ezplot('x^5')

text

Annotates a figure, by placing text at specified coordinates.

    text(x, y, 'string')

title

Assigns a title to the current figure window.

    title 'Nice Picture'

xlabel

Assigns a label to the horizontal coordinate axis.

    xlabel 'Year'

ylabel

Assigns a label to the vertical coordinate axis.

    ylabel 'Population'

zoom

Rescales a figure by a specified factor; zoom by itself enables use of the mouse for zooming in or out.

    zoom

    zoom(4)

Built-in MATLAB Constants

i

the imaginary unit

Inf

∞

pi

the number π

MATLAB Programming

end

Terminates an if, for, while, or switch statement.

else

Alternative in a conditional statement. See if.

elseif

Nested alternative in a conditional statement. See the online help for if.

error

Displays an error message and aborts execution of an M-file.

for

Repeats a block of expressions a specified number of times. Must be terminated by end.

    figure, hold on

    t = –1:0.05:1;

    for k = 0:10

        plot(t, t.^k)

    end

function

Always used at the top of a function M-file to define a new function.

    function y = myfunction(x)

if

Conditional execution of MATLAB statements. Must be terminated by end.

    if (x >= 0)

        sqrt(x)

    else

        error('Invalid input')

    end

input

Used in a script M-file to prompt for user input.

    answer = input('Please enter [x, y] coordinates: ')

isa

Used in programs to check whether an object is of a given class (double, sym, etc.).

    isa(x, 'sym')

keyboard

Used in an M-file to return control to the keyboard. Useful for debugging M-files.

nargin

In M-files, returns the number of arguments passed to the function.

    if (nargin 2); error('Wrong numberof arguments'); end

MATLAB has many other programming constructs, including while, switch, case, otherwise, break, nargout, and return. Looking at MATLAB's built-in M-files is a good way to learn how to use these.

Useful Simulink Blocks

Continuous Library

• Integrator: the most important block for differential equations, computes the integral. Also used to set initial conditions.

Math Operations Library

• Sum, Add, Subtract: all can be used for addition and subtraction
• Gain: used to multiply by a constant
• Trigonometric Function: used for sin, cos, sinh, tan, etc.
• Sqrt: square root
• Abs: absolute value
• MinMax: takes min or max
• Polynomial: can compute any polynomial function
• Product: self-explanatory
• Divide: self-explanatory
• Math Function: can compute many mathematical functions, such as exp and log

Sources Library

• Constant: used to input a constant
• Sine Wave: used to input a trigonometric signal
• Step: used for a Heaviside function
• Ramp: used for a ramp signal (Heaviside × linear)

Sinks Library

• To Workspace: sends Simulink output back to MATLAB
• Scope: used for plotting a signal
• XY Graph: used for plotting one signal against another

Useful MuPAD Commands

• MuPAD notebook
• PDF version of the MuPAD notebook
• The official MuPAD documentation