matlab symbolic math toolbox

8/6/2019 MATLAB Symbolic Math Toolbox

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Computation

Visualization

Programming

For Use with MATLAB

Users Gu ideVersion 2

Symbolic MathToolbox


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S ymb olic Mat h T oolbox Users Guid e

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O ct ob er 1 99 4 S econ d p r in t i n g

M a y 1 99 7 T h ir d p r in t i n g for S ym b ol ic M a t h T ool box 2 .0

September 1998 Updated for Release 11 (online only)


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i

Contents

1

T u t o r i a l

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

G e t t i n g H e l p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

G e t t i n g S t a r t e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5

Symbolic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5Creating Symbolic Variables an d Expressions . . . . . . . . . . . . . 1-6

Symbolic and N um eric Conversions . . . . . . . . . . . . . . . . . . . . . . 1-7

Constr ucting Real a nd Complex Variables . . . . . . . . . . . . . . 1-9

Creating Abstr act Fu nctions . . . . . . . . . . . . . . . . . . . . . . . . 1-10

Using sym to Access Ma ple Fu nctions . . . . . . . . . . . . . . . . . 1-11

Exam ple: Crea ting a Symbolic Mat rix . . . . . . . . . . . . . . . . . 1-11

The Default Symbolic Var iable . . . . . . . . . . . . . . . . . . . . . . 1-13

Creating Symbolic Math Fu nctions . . . . . . . . . . . . . . . . . . . . . 1-15

Using S ymbolic Expr essions . . . . . . . . . . . . . . . . . . . . . . . . . 1-15

Creating a n M-File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16

C a l c u l u s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17

Differentia tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17

Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21

Integra tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23Integra tion with Real Consta nts . . . . . . . . . . . . . . . . . . . . . 1-26

Real Varia bles via sym . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-28

Symbolic Summ at ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-30

Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-31

Extended Calculus E xample . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-33

Plotting Symbolic Fun ctions . . . . . . . . . . . . . . . . . . . . . . . . . 1-33


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S i m p l i f i c a t i o n s a n d S u b s t i t u t i o n s . . . . . . . . . . . . . . . . . . . . . 1-47

Simplificat ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-47

collect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-48

expand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-49horner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-49

factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-50

simplify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-52

simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-52

Substitut ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-56

subexpr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-56

subs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-59

V a ri a bl e-P reci s i o n A ri thm eti c . . . . . . . . . . . . . . . . . . . . . . . . . 1-64

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-64

Example: Using th e Different Kinds of Arithm etic . . . . . . . . . 1-65

Rat iona l Arithm etic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-65

Var iable-Pr ecision Num bers . . . . . . . . . . . . . . . . . . . . . . . . . 1-66

Converting to Floating-Point . . . . . . . . . . . . . . . . . . . . . . . . 1-67

Another E xample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-67

Li nea r A l g ebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-69

Basic Algebra ic Opera tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-69

Linear Algebraic Operat ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-70

Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-74

J orda n Canonical F orm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-81

Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 1-82

Eigenvalue Tr ajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-86

S o l v i n g E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-96

Solving Algebra ic Equ at ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-96

Several Algebra ic Equ at ions . . . . . . . . . . . . . . . . . . . . . . . . . . 1-104

Single Differen tial E qua tion . . . . . . . . . . . . . . . . . . . . . . . . . . 1-107

Exam ple 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-108

Exam ple 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-108Exam ple 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-109

Several Differential E quat ions . . . . . . . . . . . . . . . . . . . . . . . . 1-109

I n t e g r a l T r a n s f o r m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-112

The Fourier and In verse Fourier Tra nsforms . . . . . . . . . . . . 1-112

The Laplace and In verse Laplace Transforms . . . . . . . . . . . . 1-120


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1

Tutorial

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . 1-2

G e t t i n g H e l p . . . . . . . . . . . . . . . . . . . . 1-4

Ge tt i n g S t a rte d . . . . . . . . . . . . . . . . . . 1-5

C a l c u l u s . . . . . . . . . . . . . . . . . . . . . . 1-17

S i m p l i f i c a t i o n s a n d S u b s t i t u t i o n s . . . . . . . . . . 1-47

Va ri a b le -P re ci s io n Ari th m e tic . . . . . . . . . . . 1-64

Li nea r A l g ebra . . . . . . . . . . . . . . . . . . . 1-69

S o l v i n g E q u a t i o n s . . . . . . . . . . . . . . . . . 1-96

I n t e g r a l T r a n s f o r m s . . . . . . . . . . . . . . . 1-112

S p e c i a l M a t h e m a t i c a l F u n c t i o n s . . . . . . . . . . 1-130

U s i n g M a p l e F u n c t i o n s . . . . . . . . . . . . . . 1-136

E x t e n d e d S y m b o l i c Ma t h T o o l b o x . . . . . . . . . 1-143


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IntroductionThe Symbolic Math Toolboxes incorpora te symbolic computa tion int o

MATLABs nu mer ic envir onmen t. Th ese t oolboxes su ppleme nt MATLABs

numer ic and gra phical facilities with several oth er t ypes of mat hema tical

computation:

The computational engine underlying the toolboxes is the kernel of Maple, asystem developed primarily at the University of Waterloo, Canada, and, more

recently, at the Eidgenssiche Technische Hochschule, Zrich, Switzerland.Maple is mar keted a nd supported by Waterloo Maple, Inc.

These versions of the Symbolic Mat h Toolboxes a re designed to work with

MATLAB 5 a nd Maple V Release 4.

There are two toolboxes. The basic Symbolic Math Toolbox is a collection of

more than one-hundred MATLAB functions that provide access to the Maple

Facility Covers

C alcu lu s Diffe ren t ia t ion , in t egr a t ion , lim it s, s um m a tion ,

and Taylor series

L in e a r Alg eb r a I n ve r s es , d e t er m i n a n t s, e ige n va l u es , s in gu l a r

value decomposition, an d canonical form s of

symbolic ma tr ices

Sim plification M ethods of sim plifying algebraic expressions

Solution of

Equations

Symbolic and numerical solutions to algebraic and

differential equations

Variable-Precision

Arithmetic

Numer ical evaluation of ma thema tical expressions

to a ny specified a ccur acy

Tr an sfor ms F ou rier , La pla ce, z-tran sform, an d corr esponding

inverse t ra nsforms

Special

Mathematical

Functions

Special functions of classical a pplied ma th ema tics


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Introduction

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kernel using a synta x and st yle tha t is a nat ura l extension of the MATLABlan gua ge. The ba sic toolbox also allows you t o access fun ctions in Ma ples

linear algebra package. The Extended Symbolic Math Toolbox augments this

functiona lity to include a ccess t o all nongra phics Maple packa ges, Maple

program ming featu res, a nd user -defined pr ocedures. With both toolboxes, you

can wr ite your own M-files t o access Maple fun ctions an d t he Ma ple work space.

The following sections of th is Tut orial pr ovide explan at ion an d exam ples on

how to u se t he toolboxes.

Cha pter 2, Referen ce, provides det ailed descriptions of each of the functions

in the toolboxes.

Section Covers

Gettin g Help How to get online h elp for Symbolic Mat h

Toolbox functions

Getting Sta rt ed Basic symbolic ma th opera tions

Calculus How to differentiate and integrate symbolicexpressions

Simplifications and

Substitutions

How to simplify an d su bstitute values into

expressions

Variable-Precision

Arithmetic

How to cont rol th e pr ecision of

computations

Linear Algebra Exam ples usin g th e t oolbox functions

Solving Equ at ions How to solve symbolic equations

Integral Tran sforms Four ier, Lapla ce, an d z-transforms

Special Mathematical

Functions

How to a ccess Ma ples special ma th

functions

Using Maple Functions How to us e Ma ple s h elp, debugging, and

user-defined procedur e fun ctions

Exten ded S ymbolic Mat h

Toolbox

Featur es of the Extended Symbolic Math

Toolbox


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Getting HelpTher e ar e two ways t o find inform at ion on usin g Symbolic Mat h Toolbox

fun ctions. On e, of cour se, is to rea d t his m an ua l! The oth er is t o use MATLABs

comman d line help syst em. Gener ally, you can obta in help on MATLAB

fun ctions simply by typing

help function

where function is the name of the MATLAB function for which you need help.This is n ot su fficient , however, for s ome Symbolic Mat h Toolbox functions. Th e

rea son? The Symbolic Ma th Toolbox overloads ma ny of MATLABs n um er ic

fun ctions. Tha t is, it provides symbolic-specific implement at ions of th e

functions, using the same function name. To obtain help for the symbolic

version of an overloaded function, type

help sym/function

where function is th e overloaded functions n am e. For exam ple, to obta in h elpon the symbolic version of the overloaded function, diff, type

help sym/diff

To obtain information on the numeric version, on the other hand, simply type

help diff

How can you tell whether a function is overloaded? The help for the numeric

version t ells you so. For example, th e help for th e diff function contains thesection

Overloaded methods

help char/diff.m

help sym/diff.m

This tells you t hat ther e ar e two other diff comma nds t hat operat e on

expressions of class char and class sym, respectively. See t he next section for

inform at ion on class sym. For m ore inform at ion on overloaded comm an ds, see

Chapt er 14 of the U s i n g M A T L A B guide.

You can use the mhelp command to obtain help on Maple commands. For

example, to obtain help on the Maple diff comma nd, type mhelp diff. This

retur ns t he help page for t he Maple diff fun ction. For m ore inform at ion on th e


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Getting Started

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mhelp command, type help mhelp or rea d th e section Using Ma ple Fu nctionsin th is Tutorial.

Getting Started

This section describes how to crea te an d u se symbolic objects. It also describes

the default symbolic variable. If you are familiar with version 1 of the Symbolic

Math Toolbox, please note that version 2 uses substantially different and

simpler syntax.

(If you already have a copy of the Maple V Release 4 library, please see the

reference pa ge for mapleinit before proceeding.)

To get a quick online introduction to the Symbolic Math Toolbox, type demos a t

th e MATLAB comm an d line. MATLAB displays t he MATLAB Dem os dialog

box. Select Sy m bo l i c Ma t h (in t he left list box) an d th en Int r o duc t i o n (in the

right list box).

Symbolic ObjectsThe S ymbolic Math Toolbox defines a new MATLAB da ta type called a

symbolic object or sym (see Chapter 14 in U s in g M A T L A B for a n int roduction


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to MATLAB classes a nd objects). Int ern ally, a sym bolic object is a dat astr uctu re th at s tores a str ing represen ta tion of th e symbol. The Symbolic Mat h

Toolbox us es symbolic objects to re pr esen t symbolic var iables, express ions , an d

matrices.

Creating Symbolic Variables and ExpressionsTh e sym command lets you construct symbolic variables and expressions. For

example, th e comma nds

x = sym('x')

a = sym('alpha')

create a symbolic variable x t h a t p r i n t s a s x and a symbolic variable a t h a t

prints as alpha.

Suppose you want to use a symbolic variable to represent the golden ra tio

The comma nd

rho = sym('(1 + sqrt(5))/2')

achieves this goal. Now you can perform various mathematical operations on

rho. For example

f = rho^2 rho 1f =

(1/2+1/2*5^(1/2))^2-3/2-1/2*5^(1/2)

simplify(f)

r e t u r n s

0

Now suppose you want to study t he qu adra tic fun ction f = ax2 + bx + c. The

statem ent

f = sym('a*x^2 + b*x + c')

assigns th e symbolic expression ax2 + bx + c to the variable f. Observe tha t in

th is case, th e Symbolic Mat h Toolbox does n ot creat e var iables corr esponding

1 5+2-----------------=


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to the term s of the expression, a, b, c, a n d x. To perform symbolic mat hopera tions (e.g., integra tion, different iation, subst itut ion, et c.) on f, you need

to create the variables explicitly. You can do this by typing

a = sym('a')

b = sym('b')

c = sym('c')

x = sym('x')

or simplysyms a b c x

In genera l, you can use sym or syms to creat e symbolic var iables. We

recommend you use syms because it requires less typing.

Symbolic and Numeric Conversions

Lets ret urn to the golden ra tio

Th e sym function has four options for returning a symbolic representation of

the num eric value rho = (1 + sqrt(5)/2). The 'f' option

sym(rho,'f')

returns a symbolic floating-point representation

'1.9e3779b97f4a8'*2^(0)

Th e 'r' option

sym(rho,'r')

returns the rational form

7286977268806824*2^(52)

1 5+

2-----------------=


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This is the default setting for sym. That is, calling sym without a secondargum ent is the sam e as using sym with t he 'r' option.

sym(rho)

ans =

7286977268806824*2^(52)

sym(.25)

ans =

1/4

The third option 'e' returns the rational form ofrho plus t he d ifference

between the theoretical rational expression for rho and its actual (machine)

float ing-point value in ter ms ofeps (th e float ing-point rela tive a ccur acy)

sym(rho,'e')

ans =

7286977268806824*2^(52)

In this case, the theoretical and actual floating-point values for rho a r e t h e

sam e. For 1/3, however, we obta in

sym(1/3,'e')

ans =1/3-eps/12

The four th option 'd' retu rns the decimal expansion ofrho up to the n um ber

of significant digits specified by digits.

sym(rho,'d')

ans =

1.6180339887498949025257388711907


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The default va lue ofdigits is 32 (hence, sym(rho,'d') return s a num ber with32 significant digits), but if you pr efer a s hort er r epresen ta tion, use the digits

command as follows.

digits(7)

sym(rho,'d')

ans =

1.618034

A par ticularly effective u se ofsym is to convert a mat rix from nu meric to

symbolic form . Th e comm an d

A = hilb(3)

generates the 3-by-3 Hilbert m atr ix.

A =

1.0000 0.5000 0.33330.5000 0.3333 0.2500

0.3333 0.2500 0.2000

By applying sym t oA

A = sym(A)

you can obtain the (infinitely precise) symbolic form of the 3-by-3 Hilbert

m atrix.

A =

[ 1, 1/2, 1/3]

[ 1/2, 1/3, 1/4]

[ 1/3, 1/4, 1/5]

Constructing Real and Complex VariablesTh e sym command allows you to specify the mathematical properties ofsymbolic varia bles by using t he 'real' option. That is, the st at ements

x = sym('x','real'); y = sym('y','real');


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or more efficientlysyms x y real

z = x + i*y

crea te symbolic var iables x a n d y tha t h ave the added m ath em atical property

of being real variables. Specifically this means that the expression

f = x^2 + y^2

is strictly nonnegative. Hence, z is a (formal) complex var iable an d can bemanipulated as such. Thus, the commands

conj(x), conj(z), expand(z*conj(z))

return the complex conjugates of the variables

x, x i*y, x^2 + y^2

Th e conj comm an d is th e complex conjugat e operat or for th e t oolbox. If

conj(x) == x return s 1, then x is a r eal variable.

To clear x of its rea l propert y, you mu st type

syms x unreal

or

x = sym('x','unreal')

The comma nd

clear x

does n ot m a k e x a n onr eal variable.

Creating Abstract FunctionsIf you want to creat e a n abstr act (i.e., indetermina nt) function f(x), type

f = sym('f(x)')

Then f acts like f(x) and can be ma nipulated by t he toolbox comma nds. To

constr uct t he first difference ra tio, for example, type

df = (subs(f,'x','x+h') f)/'h'


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orsyms x h

df = (subs(f,x,x+h)f)/h

which returns

df =

(f(x+h)-f(x))/h

This a pplicat ion ofsym is useful when comput ing Fourier, Laplace, andz-transforms (see the section Int egra l Tran sforms).

Using sym to Access Maple FunctionsSimilarly, you can access Maples factorial function k!, using sym.

kfac = sym('k!')

To comp ut e 6! or n!, t ype

syms k n

subs(kfac,k,6), subs(kfac,k,n)

ans =

720

ans =

n!

Or, if you wan t to compu te, for example, 12!, simply use t he prod function

prod(1:12)

Example: Creating a Symbolic MatrixA circulan t ma tr ix ha s the pr opert y tha t each r ow is obta ined from th e previous

one by cyclically permuting the entries one step forward. We create the

circulant matrixA whose elements are a, b, a n d c, using the comma nds

syms a b c

A = [a b c; b c a; c a b]


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which returnA =

[ a, b, c ]

[ b, c, a ]

[ c, a, b ]

SinceA is circulant , the s um over ea ch row and column is the sa me. Lets check

this for the first row and second column. The command

sum(A(1,:))

r e t u r n s

ans =

a+b+c

The comma nd

sum(A(1,:)) == sum(A(:,2)) % This is a logical test.

r e t u r n s

ans =

1

Now replace th e (2,3) entr y ofA with beta and the variable b with alpha. The

commands

syms alpha beta;A(2,3) = beta;

A = subs(A,b,alpha)

r e t u r n

A =

[ a, alpha, c]

[ alpha, c, beta]

[ c, a, alpha]

From this example, you can see that using symbolic objects is very similar to

using regular MATLAB n um eric objects.


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The Default Symbolic VariableWhen m an ipulating ma thema tical functions, th e choice of the independentvariable is often clear from context. For example, consider the expressions in

the table below.

If we ask for th e derivat ives of th ese expressions, with out specifying th e

independent var iable, then by mat hemat ical convention we obtain f' = nxn ,

g' = a cos(a t + b), an d h ' = Jv(z)(v/z)Jv+1(z). Lets a ssume t ha t the independent

variables in t hese th ree expressions a re x, t, a n d z , resp ectively. The other

symbols, n , a , b, and v , are usu ally regar ded as consta nts or para meters. If,

however, we wanted to differentiate the first expression with respect to n , for

example, we could wr ite

or

to get xn ln x.

By mathematical convention, independent variables are often lower-case

letters found near the end of the Latin alphabet (e.g., x, y, or z). This is th e idea

behind findsym, a utility function in the toolbox used to determine default

symbolic varia bles. Defau lt symbolic varia bles ar e u tilized by t he calculus,

simplificat ion, equ at ion-solving, a nd tr an sform functions. To apply th is ut ility

to the example discussed above, type

syms a b n nu t x z

f = x^n; g = sin(a*t + b); h = besselj(nu,z);

This creates the symbolic expressions f, g, a n d h to mat ch t he example. To

differentiate these expressions, we use diff.

diff(f)

Mathematical Function MATLAB Command

f = xn f = x^n

g = sin(at+b) g = sin(a*t + b)

h = J v(z) h = besselj(nu,z)

d

d n-------f x( )

d

d n-------x

n


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r e t u r n sans =

x^n*n/x

See t he section Differentiation for a more det ailed discussion of

different iation an d th e diff command.

Her e, as a bove, we did not specify the varia ble with respect t o different iation.

How did the toolbox determine that we wanted to differentiate with respect to

x? The answer is the findsym command.

findsym(f,1)

which returns

ans =

x

Similarly, findsym(g,1) a n d findsym(h,1) r e t u r n t a n d z, respectively. Her e

the second a rgumen t offindsym denotes the number of symbolic variables we

wan t to find in t he symbolic object f, using the findsym rule (see below). The

absence of a second argument in findsym resu lts in a list of all symbolic

varia bles in a given symbolic expression. We see th is demonstr at ed below. The

command

findsym(g)

returns t he result

ans =

a, b, t

findsym Rule: The defau lt symbolic var iable in a symbolic expression is th eletter tha t is closest t o 'x' alph abet ically. If ther e ar e t wo equa lly close, th e

lett er later in the a lphabet is chosen.


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Here a re some exam ples:

Creating Symbolic Math FunctionsThere a re two ways to creat e functions:

Use symbolic expressions

Create an M-file

Using Symbolic ExpressionsThe sequ ence of comman ds

syms x y z

r = sqrt(x^2 + y^2 + z^2)

t = atan(y/x)

f = sin(x*y)/(x*y)

generates the symbolic expressions r, t, and f. You can use diff, int, subs,an d other Symbolic Mat h Toolbox fun ctions to ma nipulat e such express ions.

Expression Variable Returned Byfindsym

x^n x

sin(a*t+b) t

besselj(nu,z) z

w*y + v*z y

exp(i*theta) theta

log(alpha*x1) x1

y*(4+3*i) + 6*j y

sqrt(pi*alpha) alpha


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Creating an M-FileM-files perm it a more gener al u se of functions. Su ppose, for example, you wa ntto create the sinc function sin(x)/x. To do this, create an M-file in the @sym

directory.

function z = sinc(x)

%SINC The symbolic sinc function

% sin(x)/x. This function

% accepts a sym as the input argument.

if is equal(x,sym(0))z = 1;

else

z = sin(x)/x;

end

You can extend such examples to functions of several variables. For a more

detailed discussion on object-oriented programming, see Chapter 14 of the

U s in g M A T L A B guide.


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CalculusThe Symbolic Math Toolboxes pr ovide functions to do t he basic opera tions of

calculus; differentiation, limits, integration, summation, and Taylor series

expansion. The following sections out line t hese functions.

DifferentiationLet s crea te a symbolic expre ssion.

syms a x

f = sin(a*x)

Then

df = diff(f)

differentiates f with respect to its symbolic variable (in this case x), as

determined by findsym.df =

cos(a*x)*a

To differentiat e with respect to th e var iable a, type

dfa = diff(f,a)

which returns d f / d a

dfa=

cos(a*x)*x

Mathematical Function MATLAB Command

f = xn

f' = nxn 1

f = x^n

diff(f) or diff(f,x)

g = acos(at+b)

g' = acos(at+b)

g = acos(a*t + b)

diff(g) or diff(g,t)

h = Jv(z)

h ' = Jv(z)(v/z) Jv+1 (z)

h = besselj(nu, z)

diff(h) or diff(h,z)


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To calculate the second derivatives with respect to x a n d a, r espectively, t ypediff(f,2) % or diff(f,x,2)

which returns

ans =

sin(a*x)*a^2

a n d

diff(f,a,2)

which returns

ans =

sin(a*x)*x^2

Define a, b, x, n, t, and thet a in th e MATLAB workspace, using the sym

comma nd. The table below illustr at es th e diff command.

To differentiate the Bessel function of the first kind, besselj(nu,z), with

respect t o z, type

syms nu z

b = besselj(nu,z);

db = diff(b)

which returns

db =besselj(nu+1,z)+nu/z*besselj(nu,z)

f diff(f)

x^n x^n*n/x

sin(a*t+b) a*cos(a*t+b)

exp(i*theta) i*exp(i*theta)


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Th e diff fun ction can a lso ta ke a symbolic ma tr ix as its inpu t. In th is case, th edifferentiation is done element-by-element. Consider the example

syms a x

A = [cos(a*x),sin(a*x);sin(a*x),cos(a*x)]

which returns

A =

[ cos(a*x), sin(a*x)]

[ sin(a*x), cos(a*x)]

The comma nd

dy = diff(A)

r e t u r n s

dy =

[ sin(a*x)*a, cos(a*x)*a]

[ cos(a*x)*a, sin(a*x)*a]

You can a lso perform differentia tion of a column vector with respect t o a r ow

vector. Consider t he tr an sform at ion from Euclidean (x, y, z) to spherical (r, , )coordina tes a s given by x = r cos cos , y = r cos si n , and z = r si n . Notet h a t corresponds to elevation or latitude while denotes azimuth orlongitude.

z

y

x

(x,y,z)

r


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To calculate the J acobian m at rix,J

, of this tr ansforma tion, use the jacobianfunction. The mathematical notation for J is

=

For t he pur poses of toolbox synta x, we use l for a n d f for . The comma nds

syms r l f

x = r*cos(l)*cos(f); y = r*cos(l)*sin(f); z = r*sin(l);

J = jacobian([x; y; z], [r l f])

return the J acobian

J =

[ cos(l)*cos(f), r*sin(l)*cos(f), r*cos(l)*sin(f)]

[ cos(l)*sin(f), r*sin(l)*sin(f), r*cos(l)*cos(f)]

[ sin(l), r*cos(l), 0]

and t he com m an d

detJ = simple(det(J))

r e t u r n s

detJ =

cos(l)*r^2

Notice that the first argument of the jacobian function must be a column

vector an d th e second a rgum ent a row vector. Moreover, since th e determ ina nt

of the Jacobian is a rather complicated trigonometric expression, we used the

simple comma nd to ma ke tr igonometric substitutions an d r eductions

(simplifications). Th e se ction Simplifications and Substitutions discusses

simplificat ion in m ore deta il.

J x y x, ,( )r , ,( )-----------------------


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A table summar izing diff a n d jacobian follows.

LimitsThe fundamental idea in calculus is to make calculations on functions as a

varia ble gets close to or a pproaches a cert ain va lue. Recall th at th e definition

of th e der ivative is given by a limit

provided this limit exists. The Symbolic Math Toolbox allows you to computethe limits of functions in a direct man ner. The comma nds

syms h n x

dc = limit( (cos(x+h) cos(x))/h,h,0 )

Mathematical Operator MATLAB Command

f(x) = exp(ax + b)

syms a b x

f = exp(a*x + b)

diff(x) or diff(f,x)

diff(f,a)

diff(f,b,2)

r = u 2 + v2

t = arctan (v/ u )

syms r t u v

r = u^2 + v^2

t = atan(v/u)

J = jacobian([r:t],[u,v])

xd

df

ad

df

b2

2

d

d f

Jr t,( )u v,( )-----------------=

f x( )f x h+( ) f x( )

h----------------------------------

h 0lim=


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while th e comma nd

limit(1/x,x,0,'right')

r e t u r n s

ans =

inf

Observe tha t t he defau lt case, limit(f) is the sam e as limit(f,x,0). Explore

the options for the limit command in this table. Here, we assume that f is afunction of t he symbolic object x.

IntegrationIff is a symbolic expression, th en

int(f)

attempts to find another symbolic expression, F, so that diff(F) = f. That is,

int(f) returns the indefinite integral or antiderivative off (provided one exist s

in closed form ). Similar to differen tiat ion,

int(f,v)

Mathematical Operation MATLAB Command

limit(f)

limit(f,x,a) or

limit(f,a)

limit(f,x,a,'left')

limit(f,x,a,'right')

f x( )x 0lim

f x( )x alim

f x( )x a

lim

f x( )x a

+lim


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uses the symbolic object v as t he variable of integration, ra ther than the

variable determined by findsym. See h ow int works by looking at this table.

In cont ra st t o differen tiat ion, symbolic integr at ion is a m ore complicat ed ta sk.

A number of difficulties can arise in computing the integral. The

antiderivative, F, ma y not exist in closed form ; it ma y define an un familiar

fun ction; it ma y exist, but t he softwa re can t find the a nt iderivative; th e

softwa re could find it on a la rger compu ter , but r un s out of time or m emory onthe available machine. Nevertheless, in many cases, MATLAB can perform

symbolic integra tion successfully. For exa mple, creat e t he symbolic varia bles

syms a b theta x yn x1 u

This table illustrates integration of expressions containing those variables.

Mathematical Operation MATLAB Command

int(x^n) or

int(x^n,x)

int(sin(2*x),0,pi/2) orint(sin(2*x),x,0,pi/2)

g = cos(a*t + b)

int(g) or

int(g,t)

int(besselj(1,z)) or

int(besselj(1,z),z)

f int(f)

x^n x^(n+1)/(n+1)

y^(1) log(y)

n^x 1/log(n)*n^x

xn

xd xn 1+

n 1+-------------=

2x( )sin xd

0

2

1=

g a t b+( )cos=

g t( ) td a t b+( )si n a=

J1 z( ) d z J 0 z( )=


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The last example shows wha t h appen s if th e toolbox can t find th e

an tiderivat ive; it simply return s t he comma nd, including th e var iable of

integration, unevaluated.

Definite int egrat ion is also possible. The comm an ds

int(f,a,b)

a n d

int(f,v,a,b)

ar e u sed t o find a symbolic expression for

a n d

respectively.

Here ar e some additiona l exam ples.

sin(a*theta+b) cos(a*theta+b)/a

exp(x1^2) 1/2*pi^(1/2)*erf(x1)

1/(1+u^2) atan(u)

besselj(nu,z) int(besselj(nu,z),z)

f a, b int(f,a,b)

x^7 0, 1 1/8

1/x 1, 2 log(2)

log(x)*sqrt(x) 0, 1 4/9exp(x^2) 0, inf 1/2*pi^(1/2)

bessel(1,z) 0, 1 besselj(0,1)+1

f int(f)

f x( )a

b

d x f v( ) vda

b


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For the Bessel function (besselj) example, it is possible to compu te a

numer ical a pproximation to the value of the integra l, using the double

function. The command

a = int(besselj(1,z),0,1)

r e t u r n s

a =

besselj(0,1)+1

and t he com m an d

a = double(a)

r e t u r n s

a =

0.23480231344203

Integration with Real ConstantsOne of the subt leties involved in symbolic integra tion is t he value of various

par ameter s. For example, it would seem evident t ha t t he expression

is the positive, bell shaped curve t hat tends to 0 as x tends to for a ny real

number k. An exam ple of th is curve is depicted below with

and genera ted, using these comma nds.

syms x

k = sym(1/sqrt(2));

f = exp((k*x)^2);

ezplot(f)

ek x( ) 2

k1

2-------=


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result in the output

Definite integration: Can't determine if the integral is

convergent.

Need to know the sign of --> k^2

Will now try indefinite integration and then take limits.

Warning: Explicit integral could not be found.

ans =

int(exp(k^2*x^2),x= inf..inf)

In t he n ext section, you well see how to make k a r eal variable an d ther efore k2

positive.

Real Variables via symNotice th at Maple is not able to determ ine th e sign of the expression k^2. How

does one sur mount t his obsta cle? The an swer is to ma ke k a r eal variable, using

t h e sym comma nd. One part icularly u seful featur e ofsym, nam ely the realoption, allows you to declare k to be a real var iable. Consequently, the integra l

above is compu ted, in th e t oolbox, using t he sequence

syms k real % Be sure that x has been declared a sym.

int(f,x,inf,inf)

which returns

ans =

signum(k)/k*pi^(1/2)

Notice t ha t k is now a symbolic object in t he MATLAB workspa ce and a rea l

variable in th e Maple kern el workspa ce. By typing

clear k

you only clear k in the MATLAB workspace. To ensure that k ha s no forma l

propert ies (tha t is, to ensure k is a purely formal variable), type

syms k unreal

This variation of the syms command clears k in t he Ma ple work space. You can

also declar e a sequen ce of symbolic var iables w , y, x, z to be real, using

syms w x y z real


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In this case, all of the var iables in between t he words syms a n d real a r e

assigned the property real. That is, they are r eal variables in th e Maple

workspace.

Mathematical Operation MATLAB Commands

f(x) = ekx syms k x

f = exp(k*x)

int(f) or int(f,x)

int(f,k)

int(f,x,0,1) or int(f,0,1)

syms k real

g = exp((k*x)^2)

int(g,x,inf,inf) or

int(g,inf,inf)

f x( ) xd

f k( ) kd

f x( ) xd0

1

g x( ) e k x( )2

=

g x( ) xd

1


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Symbolic SummationYou can compute symbolic summations, when they exist, by using the symsumcommand. For example, the p-series

adds to 2/6, while th e geometric series 1 + x +x2 + ... adds to 1/(1-x), pr ovided| x| < 1. Three summ at ions a re demonstra ted below.

syms x k

s1 = symsum(1/k^2,1,inf)

s2 = symsum(x^k,k,0,inf)

s1 =

1/6*pi^2

s2 =

-1/(x-1)

11

22

------1

32

------ + + +


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Taylor SeriesThe statem ent

T = taylor(f,8)

r e t u r n s

T =

1/9+2/81*x^2+5/1458*x^4+49/131220*x^6

which is all the t erm s up t o, but n ot including, order eight (O(x8)) in t he Ta ylorseries for f(x).

Technically, T is a MacLaurin series, since its basepoint is a = 0.

These comma nds

syms x

g = exp(x*sin(x))

t = taylor(g,12,2)

generate the first 12 nonzero terms of the Taylor series for g about x = 2.

x a( )n

n 0=

fn( )

a( )n !

-----------------

1


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Lets plot th ese functions together to see how well this Ta ylor appr oximation

compar es t o the actua l function g.

xd = 1:0.05:3; yd = subs(g,x,xd);

ezplot(t, [1,3]); hold on;

plot(xd, yd, 'r-.')

title('Taylor approximation vs. actual function');

legend('Function','Taylor')

Special thanks to Professor Gunnar Bckstrm of UMEA in Sweden for this

example.

Then the command

pretty(T)

prints T in a forma t r esembling typeset ma thema tics.

2 4 49 6

1/9 + 2/81 x + 5/1458 x + ------ x

131220

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

1

2

3

4

5

6

x

Taylor approximation vs. actual function

FunctionTaylor


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Extended Calculus ExampleThe function

provides a starting point for illustrating several calculus operations in the

toolbox. It is also an interesting function in its own right. The statements

syms x

f = 1/(5+4*cos(x))

store t he symbolic expression defining t he fun ction in f.

Plotting Symbolic FunctionsThe Symbolic Math Toolbox offers a set of easy-to-use commands for plotting

symbolic expressions, including plan ar curves (ezplot), cont our s (ezcontour

a n d ezcontourf), surfaces (ezsurf, ezsurfc, ezmesh, and ezmeshc), polar

coordin at es (ezpolar), and parametrically defined curves (ezplot a n d

ezplot3) and sur faces (ezsurf). See Chapter 2, Reference for a detailed

description of these functions. The rest of this section illustrates the use of

ezplot to gra ph functions of th e form y=f(x).

f x( )1

5 4 x( )cos+------------------------------=

1
http://-/?-http://-/?-


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The fun ction ezplot(f) produces th e plot off(x) as shown below.

Th e ezplot fun ction t ries to ma ke r easona ble choices for t he r an ge of the x-axis

and for the resulting scale of the y-axis. Its choices can be overr idden by a n

additional input argum ent, or by subsequentaxis

commands. The defaultdomain for a function displayed by ezplot is 2 x 2. To alter the domain,type

ezplot(f,[a b])

This produces a graph off(x) for a x b.

Lets n ow look at th e second derivat ive of th e function f.

f2 = diff(f,2)

f2 =

32/(5+4*cos(x))^3*sin(x)^2+4/(5+4*cos(x))^2*cos(x)

6 4 2 0 2 4 6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

1/(5+4*cos(x))


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compute f' ' '(x) an d display it in a more reada ble forma t.

3

sin(x) sin(x) cos(x) sin(x)

384 --------------- + 96 --------------- 4 ---------------

4 3 2

(5 + 4 cos(x)) (5 + 4 cos(x)) (5 + 4 cos(x))

We can simplify and th is expression using the stat ements

f3 = simple(f3);

pretty(f3)

2 2

sin(x) (96 sin(x) + 80 cos(x) + 80 cos(x) 25)

4 -------------------------------------------------

4

(5 + 4 cos(x))

Now use th e solve function to find th e zeros off' ' '(x).

z = solve(f3)

returns a 5-by-1 symbolic matrix

z =

[ 0]

[ atan((25560*19^(1/2))^(1/2),10+3*19^(1/2))][ atan((25560*19^(1/2))^(1/2),10+3*19^(1/2))]

[ atan((255+60*19^(1/2))^(1/2)/(103*19^(1/2)))+pi]

[ atan((255+60*19^(1/2))^(1/2)/(103*19^(1/2)))pi]

each of whose en tr ies is a zero off'''(x). The comma nd

format; % Default format of 5 digits

zr = double(z)


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converts the zeros to double form.

zr =

0

0+ 2.4381i

0 2.4381i

2.4483

2.4483

So far , we have foun d th ree r eal zeros and two complex zeros. However, a graph

off3 shows that we have not yet found all its zeros.

ezplot(f3)

hold on;

plot(zr,0*zr,'ro')

plot([2*pi,2*pi], [0,0],'g-.');

title('Zeros of f3')

6 4 2 0 2 4 6

3

2

1

0

1

2

3

x

Zeros of f3

1 T t i l


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This occurs becau se f'''(x) cont ains a factor of sin (x), which is zero at integer

mult iples of. The fun ction, solve(sin(x)), however, only reports t he zer o atx = 0.

We can obtain a complete list of the real zeros by translating zr

zr = [0 zr(4) pi 2*pizr(4)]

by mu ltiples of 2

zr = [zr2*pi zr zr+2*pi];

Now lets plot th e tr an sform ed zr on our graph for a complete picture of the

zeros off3.

plot(zr,0*zr,'kX')

The first zero off'''(x) found by solve is at x = 0. We substitute 0 for the

symbolic variable in f2

f20 = subs(f2,x,0)

6 4 2 0 2 4 6

3

2

1

0

1

2

3

x

Zeros of f3

l l


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to compute the corresponding value off ''(0).

f20 =

0.0494

A look at the graph off''(x) shows that this is only a local minimum, which we

demonstr at e by replott ing f2.

clf

ezplot(f2)

axis([2*pi 2*pi 4.25 1.25])ylabel('f2');

title('Plot of f2 = f''''(x)')

hold on

plot(0,double(f20),'ro')

text(1,0.25,'Local minimum')

The resulting plot

6 4 2 0 2 4 6

4

3.5

3

2.5

2

1.5

1

0.5

0

0.5

1

x

Plot of f2 = f(x)

f2

Local minimum

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indicates tha t the global minima occur near x = - a n d x = . We candemonstr at e th at they occur exactly at x= , u sing t he following sequen ce ofcomma nds. First we tr y substituting - a n d into f'''(x).

simple([subs(f3,x,pi),subs(f3,x,pi)])

The result

ans =

[ 0, 0]

shows tha t - a n d ha ppen t o be crit ical points off'''(x). We can see th at - a n d are global minima by plotting f2(pi) a n d f2(pi) against f2(x).

m1 = double(subs(f2,x,-pi)); m2 = double(subs(f2,x,pi));

plot(-pi,m1,'go',pi,m2,'go')

text(-1,-4,'Global minima')

The actual m inim a a re m1, m2

ans =[ 4, 4]

as shown in the plot on the following page.

C l l


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The foregoing analysis confirms part of our original guess that the range of

f''(x) is [-4, 1]. We can confirm t he other pa rt by exam ining th e four th zero of

f'''(x) found by solve. First extract th e fourt h zero from z and assign it to a

separate variable

s = z(4)

to obtain

s =

atan((255+60*19^(1/2))^(1/2)/(103*19^(1/2)))+pi

Executing

sd = double(s)

displays the zeros corresponding numeric value.

sd =

2.4483

6 4 2 0 2 4 6

4

3.5

3

2.5

2

1.5

1

0.5

0

0.5

1

x

Plot of f2 = f(x)

f2

Local minimum

Global minima

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Plotting the point (s, f2(s)) against f2, using

M1 = double(subs(f2,x,s));

plot(sd,M1,'ko')

text(-1,1,'Global maximum')

visually confirms that s is a maximum.

The maximum is M1 = 1.0051.

Therefore, our guess that the m aximum off''(x) is [4, 1] was close, but

incorr ect. The a ctu al r an ge is [4, 1.0051].

Now, lets s ee if integr at ing f''(x) twice with respect to x recovers our originalfunction f(x) = 1/(5 + 4 cos x). The command

g = int(int(f2))

6 4 2 0 2 4 6

4

3.5

3

2.5

2

1.5

1

0.5

0

0.5

1

x

Plot of f2 = f(x)

f2

Local minimum

Global minima

Global maximum

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r e t u r n s

g =

8/(tan(1/2*x)^2+9)

This is certa inly not th e origina l expression for f(x). Lets look a t t he d ifferen ce

f(x) g(x).

d = f g

pretty(d)

1 8

[ + ]

5 + 4 cos(x) 2

tan(1/2 x) + 9

We can simplify this u sing simple(d) or simplify(d). Either command

produces

ans =1

This illustrates the concept that differentiating f(x) twice, then integrat ing the

result twice, produces a function that may differ from f(x) by a linear function

of x.

Finally, integrat e f(x) once more.

F = int(f)

The result

F =

2/3*atan(1/3*tan(1/2*x))

involves the arctangent function.

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Though F(x) is the antiderivative of a continuous function, it is itself

discontinuous as the following plot shows.

ezplot(F)

Note th at F(x) has jum ps at x = . This occurs because tan x is singular atx = .

6 4 2 0 2 4 6

1

.8

.6

.4

.2

0

.2

.4

.6

.8

1

x

2/3*atan(1/3*tan(1/2*x))

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In fact, as

ezplot(atan(tan(x)))

shows, th e n umerical value ofatan(tan(x))differ s from x by a piecewise

consta nt function t ha t h as jumps at odd multiples of/2.

To obtain a representation ofF(x) th at does not h ave jumps at these points, we

must introduce a second function, J (x), tha t compensat es for t he

discontinu ities. Then we add the appr opriate multiple ofJ(x) t o F(x).

J = sym('round(x/(2*pi))');

c = sym('2/3*pi');

F1 = F+c*J

F1 =

2/3*atan(1/3*tan(1/2*x))+2/3*pi*round(1/2*x/pi)

6 4 2 0 2 4 6

1.5

1

0.5

0

0.5

1

1.5

x

atan(tan(x))

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and plot the result.

ezplot(F1,[6.28,6.28])

This representa tion does have a continu ous gra ph.

Notice that we use the domain [6.28, 6.28] in ezplot rath er than the default

doma in [2, 2]. The reason for this is to prevent an evaluation ofF1 = 2/3 atan(1/3 tan 1/2 x) at th e singular points x = a n d x = where the

jumps in F a n d J do not can cel out one an oth er. The pr oper h an dling of bra nch

cut discont inuities in mu ltivalued fun ctions like ar ctan x is a deep and difficult

problem in sym bolic compu ta tion. Although MATLAB a nd Maple cann ot do

this entirely automatically, they do provide the tools for investigating such

questions.

6 4 2 0 2 4 6

2.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

x

2/3*atan(1/3*tan(1/2*x))+2/3*pi*round(1/2*x/pi)


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This toolbox provides several functions that apply various algebraic and

trigonometric identities to transform one representation of a function into

another, possibly simpler, r epresentat ion. These functions ar e collect,

expand, horner, factor, simplify, a n d simple.

collectThe statem ent

collect(f)

views f as a polynomial in its symbolic variable, say x, an d collects all th e

coefficients with th e sa me power ofx. A second argument can specify the

var iable in which to collect t erm s if th ere is more tha n one can didat e. Here ar e

a few examples:

f collect(f)

(x1)*(x2)*(x3) x^36*x^2+11*x6

x*(x*(x6)+11)6 x^36*x^2+11*x6

(1+x)*t + x*t 2*x*t+t

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expandThe statem ent

expand(f)

distr ibutes pr oducts over su ms a nd a pplies oth er ident ities involving functions

of sums. For example,

hornerThe statem ent

horner(f)

tr an sform s a symbolic polynomial f into its H orner , or nested, representa tion.

For example,

f expand(f)

a(x + y) ax + ay

(x1)(x2)(x3) x^36x^2+11x6

x(x(x6)+11)6 x^36x^2+11x6

exp(a+b) exp(a)exp(b)

cos(x+y) cos(x)*cos(y)sin(x)*sin(y)

cos(3acos(x)) 4x^3 3x

f horner(f)

x^36x^2+11x6 6+(11+(6+x)*x)*x

1.1+2.2x+3.3x^2 11/10+(11/5+33/10*x)*x

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factorIff is a polynomial with rational coefficients, the statement

factor(f)

expresses f as a product of polynomials of lower degree with r at iona l

coefficients. Iff can not be factored over t he r at iona l numbers, th e result is f

itself. For example,

Here is another example involving factor. It factors polynomials of the form

x^n + 1. This code

syms x;

n = 1:9;

x = x(ones(size(n)));

p = x.^n + 1;

f = factor(p);

[p; f].'

retur ns a mat rix with the polynomials in its first column a nd t heir factoredforms in its second:

[ x+1, x+1 ]

[ x^2+1, x^2+1 ]

[ x^3+1, (x+1)*(x^2x+1) ]

[ x^4+1, x^4+1 ]

[ x^5+1, (x+1)*(x^4x^3+x^2x+1) ]

[ x^6+1, (x^2+1)*(x^4x^2+1) ]

[ x^7+1, (x+1)*(1x+x^2x^3+x^4x^5+x^6) ]

[ x^8+1, x^8+1 ]

[ x^9+1, (x+1)*(x^2x+1)*(x^6x^3+1) ]

f factor(f)

x^36x^2+11x6 (x1)(x2)(x3)

x^36x^2+11x5 x^36x^2+11x5

x^6+1 (x^2+1)(x^4x^2+1)



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As an aside at t his point, we mention t hat factor can also factor symbolic

objects containing integers. This is an alternative to using the factor functionin MATLABs specfun directory. F or exa mple, t he following code segmen t

one = '1'

for n = 1:11

N(n,:) = sym(one(1,ones(1,n)));

end

[N factor(N)]

displays t he factors of symbolic integers consistin g of 1s.

[ 1, 1 ]

[ 11, (11) ]

[ 111, (3)*(37) ]

[ 1111, (11)*(101) ]

[ 11111, (41)*(271) ]

[ 111111, (3)*(7)*(11)*(13)*(37) ][ 1111111, (239)*(4649) ]

[ 11111111, (11)*(73)*(101)*(137) ]

[ 111111111, (3)^2*(37)*(333667) ]

[ 1111111111, (11)*(41)*(271)*(9091) ]

[ 11111111111, (513239)*(21649) ]

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simplifyTh e simplify function is a powerful, general purpose tool that applies a

number of algebraic identities involving sums, integral powers, square roots

and other fractional powers, as well as a number of functional identities

involving t rig fun ctions, exponen tia l a nd log functions, Bessel functions,

hypergeometr ic functions, a nd the gamma function. H ere a re some examples.

simpleTh e simple function has the unorthodox mathematical goal of finding a

simplification of an expression that has the fewest number of characters. Of

course, there is little mathematical justification for claiming that one

expression is simpler than another just because its ASCII representation is

shorter, but this often proves satisfactory in practice.

Th e simple function achieves its goal by independently applying simplify,

collect, factor, and other simplification functions to an expression and

keeping tr ack of the length s of the results. The simple function then returns

the shortest result.

f simplify(f)

x(x(x6)+11)6 x^36x^2+11x6

(1x^2)/(1x) x + 1

(1/a^3+6/a^2+12/a+8)^(1/3) ((2*a+1)^3/a^3)^(1/3)

syms x y positive

log(xy) log(x) + log(y)

exp(x) exp(y) exp(x+y)besselj(2,x) ...

2*besselj(1,x)/x +

besselj(0,x)

0

gamma(x+1)x*gamma(x) 0

cos(x)^2 + sin(x)^2 1



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Th e simple function ha s several forms, ea ch retu rn ing different outpu t. The

form

simple(f)

displays each tr ial simplificat ion a nd t he simplification fun ction t ha t pr oduced

it in th e MATLAB comma nd window. The simple function then returns the

shortest r esult. For example, the comma nd

simple(cos(x)^2 + sin(x)^2)

displays th e following alt ern at ive simplifications in t he MATLAB comm an d

window

simplify:

1

radsimp:

cos(x)^2+sin(x)^2

combine(trig):

1

factor:

cos(x)^2+sin(x)^2

expand:

cos(x)^2+sin(x)^2

convert(exp):

(1/2*exp(i*x)+1/2/exp(i*x))^21/4*(exp(i*x)1/exp(i*x))^2

convert(sincos):

cos(x)^2+sin(x)^2

convert(tan):

(1tan(1/2*x)^2)^2/(1+tan(1/2*x)^2)^2+4*tan(1/2*x)^2/

(1+tan(1/2*x)^2)^2

collect(x):

cos(x)^2+sin(x)^2

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a n d r e t u r n s

ans =

1

This form is useful when you wa nt to check, for example, wheth er t he sh ort est

form is indeed the simplest. If you are not interested in how simple achieves

its result, use t he form

f = simple(f)

This form simply retu rn s th e shortest expression found. F or example, the

statem ent

f = simple(cos(x)^2+sin(x)^2)

r e t u r n s

f =

1

If you want to know which simplificat ion retu rned the shortest r esult, use t he

multiple output form

[F, how] = simple(f)

This form retu rn s the shortes t resu lt in th e first var iable and the simplificat ion

method used t o achieve th e resu lt in t he second var iable. For example, the

statem ent

[f, how] = simple(cos(x)^2+sin(x)^2)

r e t u r n s

f =

1

how =

combine(trig)

Th e simple function sometimes improves on the result returned by simplify,

one of the simplifications that it tries. For example, when applied to the



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examples given for simplify, simple returns a simpler (or at least shorter)

result in two cases:

In some cases, it is advant ageous to apply simple twice to obta in t he effect of

two different simplification functions. For example, the statements

f = (1/a^3+6/a^2+12/a+8)^(1/3);

simple(simple(f))

r e t u r n

1/a+2

The first application, simple(f), uses radsimp to pr oduce (2*a+1)/a; the

second a pplicat ion uses combine(trig) to transform this to 1/a+2.

Th e simple function is particularly effective on expressions involving

trigonometric functions. Here are some examples:

f simplify(f) simple(f)

(1/a^3+6/a^2+12/a+8)^(1/3) ((2*a+1)^3/a^3)^(1/3) (2*a+1)/a

syms x y positive

log(xy) log(x)+log(y) log(x*y)

f simple(f)

cos(x)^2+sin(x)^2 1

2cos(x)^2sin(x)^2 3cos(x)^21

cos(x)^2sin(x)^2 cos(2x)

cos(x)+(sin(x)^2)^(1/2) cos(x)+isin(x)

cos(x)+isin(x) exp(ix)

cos(3acos(x)) 4x^33x

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SubstitutionsThere are two functions for symbolic substitution: subexpr a n d subs.

subexprThese comma nds

syms a x

s = solve(x^3+a*x+1)

solve th e equa tion x^3+a*x+1 = 0 for x.

s =

[ (1/2+1/18*(12*a^3+81)^(1/2))^(1/3)

1/3*a/(1/2+1/18*(12*a^3+81)^(1/2))^(1/3) ]

[ 1/2*(1/2+1/18*(12*a^3+81)^(1/2))^(1/3)+

1/6*a/(1/2+1/18*(12*a^3+81)^(1/2))^(1/3)+

1/2*i*3^(1/2)*((1/2+1/18*(12*a^3+81)^(1/2))^(1/3)+

1/3*a/(1/2+1/18*(12*a^3+81)^(1/2))^(1/3)) ]

[ 1/2*(1/2+1/18*(12*a^3+81)^(1/2))^(1/3)+

1/6*a/(1/2+1/18*(12*a^3+81)^(1/2))^(1/3)

1/2*i*3^(1/2)*((1/2+1/18*(12*a^3+81)^(1/2))^(1/3)+

1/3*a/(1/2+1/18*(12*a^3+81)^(1/2))^(1/3)) ]



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Use th e prett y fun ction to display s in a more reada ble form.

pretty(s)

s =

[ 1/3 a ]

[ %1 1/3 ----- ]

[ 1/3 ]

[ %1 ]

[ ]

[ 1/3 a 1/2 / 1/3 a \ ]

[ 1/2 %1 + 1/6 ----- + 1/2 i 3 |%1 + 1/3 -----| ]

[ 1/3 | 1/3| ]

[ %1 \ %1 / ]

[ ]

[ 1/3 a 1/2 / 1/3 a \ ]

[ 1/2 %1 + 1/6 ----- 1/2 i 3 |%1 + 1/3 -----| ]

[ 1/3 | 1/3| ]

[ %1 \ %1 / ]

3 1/2

%1 := 1/2 + 1/18 (12 a + 81)

Th e pretty command inherits the %n (n, an integer) notation from Maple to

denote subexpressions that occur multiple times in the symbolic object. The

subexpr function allows you to sa ve th ese common s ubexpressions a s well as

the symbolic object rewritten in terms of the subexpressions. Thesubexpressions are saved in a column vector called sigma.

Continu ing with the example

r = subexpr(s)

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r e t u r n s

sigma =

108+12*(12*a^3+81)^(1/2)

r =

[ 1/6*sigma^(1/3)2*a/sigma^(1/3)]

[ 1/12*sigma^(1/3)+a/sigma^(1/3)+

1/2*i*3^(1/2)*(1/6*sigma^(1/3)+2*a/sigma^(1/3))]

[ 1/12*sigma^(1/3)+a/sigma^(1/3)

1/2*i*3^(1/2)*(1/6*sigma^(1/3)+2*a/sigma^(1/3))]

Notice t ha t subexpr creat es the var iable sigma in t he MATLAB workspa ce.

You can verify t his by t yping whos, or t he comma nd

sigma

which returns

sigma =108+12*(12*a^3+81)^(1/2)



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subsLets find t he eigenvalues a nd eigenvectors of a circulant ma tr ixA.

syms a b c

A = [a b c; b c a; c a b];

[v,E] = eig(A)

v =

[ (a+(b^2b*ac*bc*a+a^2+c^2)^(1/2)b)/(ac),

(a(b^2b*ac*bc*a+a^2+c^2)^(1/2)b)/(ac), 1]

[ (bc(b^2b*ac*bc*a+a^2+c^2)^(1/2))/(ac),

(bc+(b^2b*ac*bc*a+a^2+c^2)^(1/2))/(ac), 1]

[ 1,

1, 1]

E =

[ (b^2b*ac*b

c*a+a^2+c^2)^(1/2), 0, 0]

[ 0, (b^2b*ac*b

c*a+a^2+c^2)^(1/2), 0]

[ 0, 0, b+c+a]

Suppose we wan t to replace th e ra ther lengthy expression

(b^2b*ac*bc*a+a^2+c^2)^(1/2)

throughout v a n d E. We first use subexpr:

v = subexpr(v,'S')

which returns

S =

(b^2b*ac*bc*a+a^2+c^2)^(1/2)

v =

[ (a+Sb)/(ac), (aSb)/(ac), 1]

[ (bcS)/(ac), (bc+S)/(ac), 1]

[ 1, 1, 1]

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Next, substitute the symbol S into E with

E = subs(E,S,'S')

E =

[ S, 0, 0]

[ 0, S, 0]

[ 0, 0, b+c+a]

Now suppose we wan t t o evaluat e v a t a = 10. We can do this using the subs

command:

subs(v,a,10)

This r eplaces all occurr ences ofa in v with 10

[ (10+Sb)/(10c), (10Sb)/(10c), 1]

[ (bcS)/(10c), (bc+S)/(10c), 1]

[ 1, 1, 1]

Notice, however, tha t t he symbolic express ion r epresen ted by S is unaffected bythis substitu tion. Tha t is, the symbol a in S is not replaced by 10. The subs

comman d is a lso a useful function for subst itut ing in a variety of values for

several var iables in a pa rt icular expression. Lets look at S. Suppose tha t in

addition to substituting a = 10, we also wan t to substitu te th e values for 2 an d

10 for b a n d c, respectively. The way t o do th is is to set values for a, b, and c in

the workspace. Then subs evaluat es its input using th e existing symbolic and

double variables in the current workspace. In our example, we first set

a = 10; b = 2; c = 10;

subs(S)

ans =

64^(1/2)



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To look at the contents of our workspace, type whos, which gives

Name Size Bytes Class

A 3x3 878 sym object

E 3x3 888 sym object

S 1x1 186 sym object

a 1x1 8 double array

ans 1x1 140 sym object

b 1x1 8 double array

c 1x1 8 double array

v 3x3 982 sym object

a, b, and c ar e now var iables of class doublewhileA, E, S, an d v rem ain symbolic

expressions (class sym).

If you want to preserve a, b, and c as symbolic variables, but still alter their

value within S, use t his procedure.

syms a b csubs(S,{a,b,c},{10,2,10})

ans =

64^(1/2)

Typing whos reveals that a, b, a n d c rem ain 1-by-1 sym objects.

Th e subs command can be combined with double to evaluate a symbolic

expression numerically. Suppose we have

syms t

M = (1t^2)*exp(1/2*t^2);

P = (1t^2)*sech(t);

and want to see how M a n d P differ graphically.

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One appr oach is to type

ezplot(M); hold on; ezplot(P)

but this plot

does not readily help us identify the curves.

6 4 2 0 2 4 6

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

t

(1t^2)*sech(t)


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Variable-Precision Arithmetic

OverviewThere are three different kinds of arithmetic operations in this toolbox.

Numer ic MATLABs float ing-point ar ithmetic

Rational M aples exact sym bolic ari thm etic

VP A M a ple s va r i a bl e-p r e ci si on a r i t h m et i c

For example, the MATLAB sta tement s

format long

1/2+1/3

use numeric computation to produce

0.83333333333333

With the Symbolic Math Toolbox, the statement

sym(1/2)+1/3

uses symbolic computa tion t o yield

5/6

And, also with the toolbox, the statements

digits(25)vpa(1/2+1/3)

use variable-precision ar ithmetic to ret urn

.8333333333333333333333333

The float ing-point opera tions used by numer ic ar ithm etic ar e the fast est of th e

three, and require the least computer memory, but the results are not exact.

The n um ber of digits in th e prin ted out put of MATLABs double quan tities iscontrolled by th e format statement, but the internal representation is always

the eight-byte floating-point representation provided by the particular

computer hardware.

In t he comput at ion of th e numer ic result a bove, th ere ar e actually three

roundoff errors, one in th e division of 1 by 3, one in th e a ddition of 1/2 to th e



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result of the division, and one in the binary to decimal conversion for the

printed output. On computers that use IEEE floating-point standardar ithmetic, th e resulting intern al value is th e binary expansion of 5/6,

tru ncated t o 53 bits. This is a pproximately 16 decimal digits. But , in t his

part icular case, th e print ed output shows only 15 digits.

The symbolic operat ions used by rat ional ar ithmetic are potent ially th e m ost

expensive of th e thr ee, in term s of both compu ter t ime and memory. The resu lts

ar e exact, a s long as enough time a nd m emory a re a vailable to complete th e

computations.

Variable-precision arithmetic falls in between the other two in terms of both

cost a nd a ccur acy. A global pa ra met er, set by the fun ction digits, contr ols th e

nu mber of significan t decimal digits. Increasing t he n um ber of digits in creases

the a ccur acy, but a lso increases both th e time an d memory requirements. The

default valu e ofdigits is 32, corresponding roughly to floating-point accuracy.

The Maple docum ent at ion uses th e term ha rdwa re floating-point for wha t we

ar e calling nu mer ic or float ing-point an d uses th e t erm float ing-point

ar ithm etic for wha t we ar e calling var iable-precision a rith met ic.

Example: Using the Different Kinds of Arithmetic

Rational ArithmeticBy defau lt, the Symbolic Mat h Toolbox uses ra tional ar ithm etic operat ions, i.e.,

Maples exact symbolic ar ithm etic. Rat iona l ar ithm etic is invoked wh en you

create symbolic variables using the sym function.

Th e sym function convert s a double mat rix to its symbolic form. F or example, if

the double mat rix is

A =

1.1000 1.2000 1.3000

2.1000 2.2000 2.3000

3.1000 3.2000 3.3000

its symbolic form, S = sym(A), is

S =

[11/10, 6/5, 13/10]

[21/10, 11/5, 23/10]

[31/10, 16/5, 33/10]

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For t his m atrixA, it is possible to discover tha t the elements ar e th e r atios of

small integers, so the symbolic representation is formed from those integers.On the other ha nd, the statem ent

E = [exp(1) sqrt(2); log(3) rand]

r e t u r n s a m a t r i x

E =

2.71828182845905 1.41421356237310

1.09861228866811 0.21895918632809

whose elements a re n ot the ra tios of small integers, so sym(E) reproduces th e

float ing-point repr esent at ion in a symbolic form.

[3060513257434037*2^(50), 3184525836262886*2^(51)]

[2473854946935174*2^(51), 3944418039826132*2^(54)]

Variable-Precision Numbers

Variable-precision numbers are distinguished from the exact rationalrepr esent at ion by th e pr esence of a decimal point. A power of 10 scale factor,

denoted by 'e', is allowed. To use var iable-precision inst ead of ra tional

ar ithmetic, creat e your variables using the vpa function.

For ma tr ices with pu rely double entries, the vpa function genera tes th e

representation that is used with variable-precision arithmetic. Continuing on

with our example, and u sing digits(4), applying vpa to the m atrix S

vpa(S)

generates the output

S =

[1.100, 1.200, 1.300]

[2.100, 2.200, 2.300]

[3.100, 3.200, 3.300]

and with digits(25)F = vpa(E)



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generates

F =

[2.718281828459045534884808, 1.414213562373094923430017]

[1.098612288668110004152823, .2189591863280899719512718]

Converting to Floating-PointTo convert a rational or variable-precision number to its MATLAB

floating-point representation, use the double function.

In our example, both double(sym(E)) a n d double(vpa(E)) r e t u r n E.

Another ExampleThe next example is perhaps more interesting. Start with the symbolic

expression

f = sym('exp(pi*sqrt(163))')

The statem ent

double(f)

produces the printed floating-point value

2.625374126407687e+17

Using the second a rgument ofvpa to specify th e nu mber of digits,

vpa(f,18)

r e t u r n s

262537412640768744.

whereas

vpa(f,25)

r e t u r n s

262537412640768744.0000000

We suspect t hat f might actually have an integer value. This suspicion is

reinforced by th e 30 digit value, vpa(f,30)

262537412640768743.999999999999

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Finally, the 40 digit value, vpa(f,40)

262537412640768743.9999999999992500725944

shows that f is very close to, but n ot exactly equa l to, an in teger.

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Linear Algebra

Basic Algebraic OperationsBasic algebraic operations on symbolic objects are the same as operations on

MATLAB objects of class double. This is illustrated in the following example.

The Givens transformation produces a plane rotation through the angle t. The

statem ents

syms t;G = [cos(t) sin(t); sin(t) cos(t)]

create this transformation matrix:

G =

[ cos(t), sin(t) ]

[ sin(t), cos(t) ]

Applying th e Givens t ra nsform at ion t wice should simply be a r otat ion t hr oughtwice the angle. The corresponding matrix can be computed by multiplying G

by itself or by ra ising G to th e second power. Both

A = G*G

a n d

A = G^2

produce

A =

[cos(t)^2sin(t)^2, 2*cos(t)*sin(t)]

[ 2*cos(t)*sin(t), cos(t)^2sin(t)^2]

Th e simple function

A = simple(A)

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uses a trigonometric identity to retu rn th e expected form by trying several

different ident ities a nd picking th e one tha t pr oduces th e shortestrepresentation:

A =

[ cos(2*t), sin(2*t)]

[sin(2*t), cos(2*t)]

A Givens rotat ion is an orth ogonal m at rix, so its tr an spose is its inverse.

Confirming this by

I = G.' *G

which produces

I =

[cos(t)^2+sin(t)^2, 0]

[ 0, cos(t)^2+sin(t)^2]

a n d t h e n

I = simple(I)

I =

[1, 0]

[0, 1]

Linear Algebraic OperationsLets do severa l basic linear algebraic operat ions.

The comma nd

H = hilb(3)

generates th e 3-by-3 H ilbert mat rix. With format short, MATLAB pr ints

H =

1.0000 0.5000 0.3333

0.5000 0.3333 0.25000.3333 0.2500 0.2000

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The computed elements ofH ar e float ing-point nu mbers th at ar e the r atios of

small integers. Indeed, H is a MATLAB a rr ay of class double. Converting H t oa symbolic matrix

H = sym(H)

gives

[ 1, 1/2, 1/3]

[1/2, 1/3, 1/4]

[1/3, 1/4, 1/5]

This a llows su bsequent symbolic opera tions on H to produce results tha t

correspond to the infinitely precise Hilbert matrix, sym(hilb(3)), not its

floating-point appr oximat ion, hilb(3). Ther efore,

inv(H)

produces

[ 9, 36, 30][36, 192, 180]

[ 30, 180, 180]

a n d

det(H)

yields

1/2160

We can use the backslash operator to solve a system of simultaneous linear

equations. The comma nds

b = [1 1 1]'

x = H\b % Solve Hx = b

produce th e solut ion

[ 3]

[24]

[ 30]

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All three of these r esults, the inverse, the determina nt , and t he solution to the

linear system, ar e t he exact r esults corr esponding to t he infinitely precise,ra tiona l, Hilbert m at rix. On th e other ha nd, using digits(16), the comma nd

V = vpa(hilb(3))

r e t u r n s

[ 1., .5000000000000000, .3333333333333333]

[.5000000000000000, .3333333333333333, .2500000000000000]

[.3333333333333333, .2500000000000000, .2000000000000000]The decimal points in t he repr esentat ion of the individual element s ar e the

signal to use variable-precision arithmetic. The result of each arithmetic

operation is rounded to 16 significant decimal digits. When inverting the

mat rix, these err ors ar e ma gnified by the mat rix condition num ber, which for

hilb(3) is about 500. Consequently,

inv(V)

which returns

[ 9.000000000000082, 36.00000000000039, 30.00000000000035]

[36.00000000000039, 192.0000000000021, 180.0000000000019]

[ 30.00000000000035, 180.0000000000019, 180.0000000000019]

shows t he loss of two digits. So d oes

det(V)

which gives

.462962962962958e3

a n d

V\b

which is

[ 3.000000000000041]

[24.00000000000021]

[ 30.00000000000019]

Since H is nonsingular, the null space ofH

null(H)

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and the column space ofH

colspace(H)

produce an empty matrix and a permutation of the identity matrix,

respectively. To ma ke a more inter esting exam ple, lets t ry t o find a value s for

H(1,1) t h a t m a k e s H singular . The comma nds

syms s

H(1,1) = s

Z = det(H)

sol = solve(Z)

produce

H =

[ s, 1/2, 1/3]

[1/2, 1/3, 1/4]

[1/3, 1/4, 1/5]

Z =

1/240*s1/270

sol =

8/9

Then

H = subs(H,s,sol)

substitut es th e comput ed value ofsol for s in H to give

H =

[8/9, 1/2, 1/3]

[1/2, 1/3, 1/4]

[1/3, 1/4, 1/5]

Now, th e comma nd

det(H)

r e t u r n s

ans =

0

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a n d

inv(H)

produces an error m essage

??? error using ==> inv

Error, (in inverse) singular matrix

because H is singular. For th is matr ix, Z = null(H) a n d C = colspace(H) a r e

nontrivial.

Z =

[ 1]

[ 4]

[10/3]

C =

[ 0, 1]

[ 1, 0][6/5, 3/10]

It should be pointed out that even though H is singular , vpa(H) is not. For an y

integer value d, setting

digits(d)

and then computing

det(vpa(H))inv(vpa(H))

results in a determina nt of size 10^(d) and an inverse with elements on the

order of10^d.

EigenvaluesThe symbolic eigenvalu es of a squ ar e ma tr ixA or t he symbolic eigenvalues a nd

eigenvectors ofA are computed, respectively, using the commands

E = eig(A)

[V,E] = eig(A)


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r e t u r n

Td =

1.0000 1.6497 -1.2837

-4.0000 1.0000 1.0000

3.3333 0.7051 1.5851

Ed =

0 0 0

0 1.3344 0

0 0 0.0878

The first eigenvalue is zero. The corresponding eigenvector (the first column of

Ts) is th e sam e as t he basis for t he nu ll space found in th e last section. The

other two eigenvalues a re t he r esult of applying the quadr atic formu la to

x^264/45*x+253/2160

which is the quadratic factor in factor(poly(H)).

syms xg = simple(factor(poly(H))/x);

solve(g)

Closed form symbolic expressions for t he eigenvalu es a re possible only when

the characteristic polynomial can be expressed as a product of rational

polynomials of degree four or less. The Rosser m at rix is a class ic nu mer ical

analysis test m atr ix that happens to i llustrate th is requirem ent. The

statem ent

R = sym(gallery('rosser'))

generates

R =

[ 611 196 192 407 8 52 49 29]

[ 196 899 113 192 71 43 8 44]

[192 113 899 196 61 49 8 52]

[ 407 192 196 611 8 44 59 23][ 8 71 61 8 411 599 208 208]

[ 52 43 49 44 599 411 208 208]

[ 49 8 8 59 208 208 99 911]

[ 29 44 52 23 208 208 911 99]

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The comma nds

p = poly(R);

pretty(factor(p))

produce

2 2 2

[x (x 1020) (x 1020 x + 100)(x 1040500) (x 1000) ]

The cha ra cteristic polynomial (of degree 8) factors nicely into th e product of two

linear terms and three quadratic terms. We can see immediately that four ofth e eigenvalues a re 0, 1020, an d a double root a t 1000. The oth er four roots a re

obtained from t he r emaining quadra tics. Use

eig(R)

to find a ll these values

[ 0]

[ 1020][510+100*26^(1/2)]

[510100*26^(1/2)]

[ 10*10405^(1/2)]

[ 10*10405^(1/2)]

[ 1000]

[ 1000]

The Rosser ma tr ix is not a typical example; it is rar e for a full 8-by-8 mat rix to

have a characteristic polynomial that factors into such simple form. If we

change t he t wo corn er elemen ts ofR from 29 to 30 with the comma nds

S = R; S(1,8) = 30; S(8,1) = 30;

a n d t h e n t r y

p = poly(S)

we findp =

40250968213600000+51264008540948000*x

1082699388411166000*x^2+4287832912719760*x^3

5327831918568*x^4+82706090*x^5+5079941*x^6

4040*x^7+x^8

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We also find that factor(p) is p itself. That is, the characteristic polynomial

cannot be factored over the rationals.

For this modified Rosser matrix

F = eig(S)

r e t u r n s

F =

[ 1020.0532142558915165931894252600]

[ .17053529728768998575200874607757][ .21803980548301606860857564424981]

[ 999.94691786044276755320289228602]

[ 1000.1206982933841335712817075454]

[ 1019.5243552632016358324933278291]

[ 1019.9935501291629257348091808173]

[ 1020.4201882015047278185457498840]

Notice that these values are close to the eigenvalues of the original Rossermat rix. Fur ther , the numer ical values ofF ar e a r esu lt of Map les float ing-point

ar ithm etic. Consequ ent ly, differen t set tings ofdigits do not alter th e num ber

of int egers t o the r ight of th e decima l place.

It is a lso possible to try t o compu te eigenva lues of symbolic ma tr ices, but closed

form solutions a re r ar e. The Givens tr ansforma tion is generat ed as t he ma tr ix

exponential of the elementar y mat rix

The Symbolic Math Toolbox commands

syms t

A = sym([0 1; 1 0]);

G = expm(t*A)

r e t u r n

[ cos(t), sin(t)]

[ sin(t), cos(t)]

A 0 1

1 0=

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Next, the comma nd

g = eig(G)

produces

g =

[ cos(t)+(cos(t)^21)^(1/2)]

[ cos(t)(cos(t)^21)^(1/2)]

We can use simple to simplify this form ofg. Indeed, repeated application of

simple

for j = 1:4

[g,how] = simple(g)

end

produces th e best r esult

g =

[ cos(t)+(sin(t)^2)^(1/2)][ cos(t)(sin(t)^2)^(1/2)]

how =

simplify

g =

[ cos(t)+i*sin(t)]

[ cos(t)i*sin(t)]

how =

radsimp

g =

[ exp(i*t)]

[ 1/exp(i*t)]

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how =

convert(exp)

g =

[ exp(i*t)]

[ exp(i*t)]

how =

simplify

Notice t he first applicat ion ofsimple uses simplify to produce a sum of sines

an d cosines. Next, simple invokes radsimp to pr oduce cos(t) + i*sin(t) for

the first eigenvector. The third application ofsimple uses convert(exp) t o

cha nge t he sines an d cosines t o complex exponen tia ls. The last applicat ion of

simple uses simplify to obtain the final form.

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Jordan Canonical FormThe J orda n canonical form results from at tempts to diagona lize a m atr ix by a

similarity tr ansforma tion. For a given mat rixA, find a n onsingular m atr ix V,

so tha t inv(V)*A*V, or, more succinctly, J = V\A*V, is as close to diagonal as

possible. For almost all matrices, the Jordan canonical form is the diagonal

mat rix of eigenvalues and th e columns of the tr an sforma tion m atr ix are th e

eigenvectors. This always ha ppens if the mat rix is symmetric or if it ha s

distinct eigenvalues. Some nonsymmetric matrices with multiple eigenvalues

cannot be diagonalized. The Jordan form has the eigenvalues on its diagonal,

but some of th e superdia gona l element s ar e one, instea d of zero. The sta tem ent

J = jordan(A)

computes the Jordan canonical form ofA. The statem ent

[V,J] = jordan(A)

also computes the similarity transformation. The columns ofV a r e t h e

genera lized eigenvectors ofA.

The J ordan form is extrem ely sensitive to pert ur bat ions. Almost an y cha nge in

A causes its J ordan form to be diagona l. This mak es it very difficult to comput e

the Jordan form reliably with floating-point arithmetic. It also implies thatA

mu st be k nown exactly (i.e., without r oun d-off err or, etc.). Its element s mu st be

integers, or ratios of small integers. In particular, the variable-precision

calculation, jordan(vpa(A)), is not allowed.

For example, let

A = sym([12,32,66,116;25,76,164,294;

21,66,143,256;6,19,41,73])

A =

[ 12, 32, 66, 116]

[ 25, 76, 164, 294]

[ 21, 66, 143, 256]

[ 6, 19, 41, 73]

Then

[V,J] = jordan(A)

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produces

V =

[ 4, 2, 4, 3]

[ 6, 8, 11, 8]

[ 4, 7, 10, 7]

[ 1, 2, 3, 2]

J =

[ 1, 1, 0, 0]

[ 0, 1, 0, 0][ 0, 0, 2, 1]

[ 0, 0, 0, 2]

ThereforeA has a double eigenvalue at 1, with a single Jordan block, and a

double eigenvalue at 2, also with a single Jordan block. The matrix has only

two eigenvectors, V(:,1) a n d V(:,3). They satisfy

A*V(:,1) = 1*V(:,1)

A*V(:,3) = 2*V(:,3)

The other two columns ofV are generalized eigenvectors of grade 2. They

satisfy

A*V(:,2) = 1*V(:,2) + V(:,1)

A*V(:,4) = 2*V(:,4) + V(:,3)

In m ath emat ical notation, with vj = v(:,j), th e column s ofV and eigenvalues

satisfy t he relationships

Singular Val

matlab symbolic math toolbox

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