Symbolic Mathematics Chapter 12

MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and

Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

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Symbolic MathematicsChapter 12

ezcontour

-2 0 2

ezcontourf

-2 0 2

ezpolar

r = sin(x)-5 0 5

ezplot

MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system,

or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

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Objectives

After studying this chapter you should be able to:

• Create and manipulate symbolic variables• Factor and simplify mathematical expressions• Solve symbolic expressions• Solve systems of equations • Determine the symbolic derivative of an

expression• Integrate an expression symbolically

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• MATLAB’s symbolic capability is based on the MuPad software, • Originally produced by Sciface

Software and purchased by the Mathworks in 2008.

• Previous versions of MATLAB, before 2007b, used Maple as the symbolic engine

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Symbolic Manipulation

• Available in the “Symbolic Math Toolbox” – which is an option in the professional version

• Included with the Student Edition• There are some differences between the

MuPad and Maple implementations of the symbolic toolbox

• If you have Maple installed on your computer it has an interface to MATLAB, which may over-ride the default MuPad engine

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Capabilities

• Manipulate symbolic expressions to • Simplify• Solve symbolically• Evaluate numerically• Take derivatives• Integrate • Perform linear algebraic manipulations

• More advanced features include • LaPlace transforms• Fourier transforms• Variable precision arithmetic

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Notebook Environment

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Section 12.1Symbolic Algebra

• Used regularly in math, engineering and science classes

• Often preferable to manipulate equations symbolically before you substitute in values for variables

)3(*22

Consider this equation

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)3(*22

This looks like a fairly complicated function of x

)96(*2

)3(*22

If you expand it, it simplifies dramatically

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However, when you simplify you may lose information

undefinedy

)33(*2 2

When x is equal to -3, the equation is undefined

)3(*22

Let x equal -3

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You can choose

• MATLAB’s symbolic capability allows you perform the simplification, or to manipulate the numerator and denominator separately

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Relationships are not always easy to solve

If we know• k0

• Q• R• T

Its easy to solve for k

It’s not easy to solve for T!

Q RTk k e0

ln( ) ln( )

ln( / )

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MATLAB’s symbolic capability makes it easy to solve this problem

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Creating Symbolic Variables

• Two approaches• Use the sym command to create

• Single variable• Expression• Equation

• Use the syms command to create • Single variables• Compose expressions and equations

from the variables you’ve defined

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Here’s an example

• Define x as a symbolic variable• x=sym('x') or• syms x

• Use x to create a more complicated expression• y = 2*(x+3)^2/(x^2+6*x+9)

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x and y are both symbolic variables

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The syms command can create multiple variables

• syms Q R T k0

• Use these variables to create another symbolic variables

k=k0*exp(-Q/(R*T))

Notice that we used standard algebraic operators – the array operators (.*, ./ and .^) are not used in symbolic algebra

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Create an entire expression with the sym command

• E=sym('m*c^2')

• Since m and c have not been specifically defined as symbolic variables, they are not stored

• E was set equal to a character string, defined by the single quotes inside the function.

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Workspace

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Equations vs Expressions

• We can create an entire equation, and give it a name

• ideal_gas_law=sym('P*V=n*R*Temp')

This is an algebraic equationThis is an assignment statement

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Reserved Variable Names

One idiosyncrasy of the implementation of MuPad inside MATLAB is that a number of commonly used variables are reserved. They can be overwritten, however it you try to use them inside expressions or equations you may run into problems.

D, E, I, O, beta, zeta, theta, psi, gamma, Ci, Si, Ei

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Manipulating Symbolic Expressions and Equations

• Equations are set equal to something

• Expressions are not

• Both expressions and equations can be assigned a name, using the MATLAB assignment operator

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The variable ideal_gas_law has been assigned to an equation

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The variable E as been assigned to an expression

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Working with Equations and Expressions

• Some MATLAB functions work on both expressions and equations

• Others are limited in their scope

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Extracting Numerators and Denominators

• These functions work on expressions

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The numden function extracts the numerator and denominator from an expression

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You can combine symbolic variables using standard algebraic operators

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Expanding and Factoring

num is an expression

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w is an equation

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The factor function used with an expression, den

The factor function used with an equation, w

The collect function is similar, and collects common terms

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Simplifying

• The expand, factor and collect functions can be used to “simplify” an expression and sometimes an equation

• What constitutes a simplification is not always obvious

• The simplify function uses a set of built in rules

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simplify used on an expression

simplify used on an equation

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Simple

• The simple function is different from simplify

• It tries all of the different simplification techniques, and chooses the result that is the shortest

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All of the possibilities evaluated are reported, however there is only one actual answer

Both simple and simplify work on expressions and equations

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• Use the poly2sym function as a shortcut to create a polynomial

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• Extract the coefficients from a polynomial, using the sym2poly function

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Section 12.2Solving Equations and Expressions

• Use the solve function

• Automatically sets expressions equal to 0 and solves for the roots

• Uses the equality specified in equations

• Solves for the variables in systems of equations

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You can define your expression or equation in the solve function

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If x has previously been defined as a symbolic, the quotes aren’t

necessary

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The answer from the solve function is not necessarily a number

You can specify what variable you want to solve for

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r Remember, if you have defined variables as symbolics previously – you can use them in expressions or equations without the single quotes

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Example using solve

Sometimes it’s useful to redefine a variable for latter use

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Solving Systems of Equations

This result is a structure array. These arrays are described in Chapter 11

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There are several different approaches to find the actual values of x, y, and z

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Give the result a name, such as answer, and then specify the field name inside the structure array to retrieve the values for x, y, and z

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Assign individual variable names. Notice that x, y and z are symbolic variables

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If you need to use the value of x, y or z in a function that needs a double as input, you’ll need to change the variable type from symbolic to double

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• Using the solve function for multiple equations has both advantages and disadvantages over using the linear algebra techniques described in chapter 9.

• In general, if a problem can be solved using matrices the matrix solution will take less computer time.

• However, linear algebra is limited to first order equations.

• The solve function may take longer, but it can solve non-linear problems and can solve problems with symbolic variables.

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Example 12.1

• Use MATLAB’s symbolic capability to solve an equation

• k = k0*exp(-Q/RT)• Solve for Q

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Hand solution

0 expQ

expk Q

Solutions

MATLAB Solution

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Substitution

• Once we have a symbolic expression we’ll probably want to substitute numbers into it.

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We could substitute in a new variable – in this case we’ll put a y everywhere there used to be an x

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We could substitute in a number – in this case 3 for x

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If the variables inside the expression have been explicitly defined as symbolics we don’t need the single quotes

To substitute into multiple variables group them with curly braces

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Example 12.2

• Using Symbolic Math to solve a ballistics problem

vertical distance

horizontal distance

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Equations of Motion

)cos(**0_ tvxdist

2**2/1)sin(**0_ tgtvydist

Find the equation for the distance traveled when the projectile hits the ground

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Strategy

• Find the time the projectile hits the ground

• Substitute back into the horizontal distance equation

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MATLAB solution M-file

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Results

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Section 12.3Symbolic Plotting

• The symbolic toolbox includes a group of functions to create symbolic plots

• The most basic is the ezplot

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EZPlot

• Allows you to plot symbolic expressions

• ezplot(S)• Defaults to a range of -2to +2

• ezplot(S, [xmax, xmin])

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Note this plot was created with the student version –The symbolic functionality is included in the student version

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Add your own titles, axis labels and other annotations using the same functions described for numeric plotting in Chapter 5

Notice that ezplot creates a title and axis labels automatically

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ezplot supports implicit plotting

• The equation for a circle can be expressed implicitly as:• x2 + y2 = 1

• You could solve for y, but it’s not necessary with ezplot

• ezplot('x^2 + y^2 =1',[-1.5,1.5])

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Ezplot supports parametric equation graphs

• The equation for a circle can be expressed parametrically as:• x=sin(t)• y=cos(t)

• To create the graph use…• ezplot(‘sin(x)’,’cos(x)’)

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x2 + y2 =1 = 0

-1 0 1-1.5

-1 -0.5 0 0.5 1

x = sin(x), y = cos(x)

Implicit and Parametric plots of a circle

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Most symbolic functions will allow you to either enter a symbolic variable that represents a function, or to enter the function itself enclosed in single quotes. For exampley=sym(‘x^2-1’)ezplot(y)is equivalent to ezplot(‘x^2-1’)

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Other Symbolic Plots

• Additional symbolic plotting functions are available, which mirror the functions used in numeric MATLAB plotting options

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Symbolic Plot Types

ezplot Function plotter if z is a function of x ezplot(z)

ezmesh Mesh plotter if z is a function of x and yezmesh(z)

ezmeshc Combined mesh and contour plotter

if z is a function of x and yezmeshc(z)

ezsurf Surface plotter if z is a function of x and yezsurf(z)

ezsurfc Combined surface and contour plotter

if z is a function of x and yezsurfc(z)

ezcontour Contour plotter if z is a function of x and yezcontour(z)

ezcontourf Filled contour plotter if z is a function of x and yezcontourf(z)

ezplot3 3-D parametric curve plotter

if x is a function of tif y is a function of tif z is a function of tezplot3(x,y,z)

ezpolar Polar Coordinate plotter if r is a function of ezpolar(r)

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To demonstrate these plot types create a symbolic version of “peaks”

We broke this function up into three parts to make it easier to enter into the computer. Notice that there are no “dot” operators used in these expressions, since they are all symbolic.

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When we created the same plots using a standard MATLAB approach it was necessary to define an array of both x and y values, mesh them together, and calculate the values of z based on the two dimensional arrays. The symbolic plotting capability contained in the symbolic toolbox makes creating these graphs much easier.

All of these graphs can be annotated using the standard MATLAB functions such as title, xlabel, text, etc.

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These contour plots are a two-dimensional representation of the three-dimensional peaks function

The polar graph requires us to define a new function

Any of these ezplot graphs can handle parameterized equations

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Example 12.3

• Plot the trajectory of a ballistic projectile• Assume 100 m/sec initial velocity• Assume g = 9.8 m/sec2

vertical distance

horizontal distance

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12.4 Calculus

• MATLAB’s symbolic toolbox supports• Symbolical differentiation• Symbolic integration

• This makes it possible to find analytical solutions for many problems, instead of numeric approximations.

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Differentiation

• Concept introduced in Calculus I

• However… a derivative is really just the slope of an equation

• A common application of derivatives is to find velocities and accelerations

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Consider a race car…

• Assume that during a race the car starts out slowly, and reaches its fastest speed at the finish line

• To avoid running into the stands, the car must then slow down until it finally stops

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• We might model the position of the car using a sine wave

)20/)10(*sin(*2020 tdist

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Create a plot of position vs time using ezplot

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Car position

Finish Line

ezplot of position

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diff function

• The diff function finds a symbolic derivative

• The velocity is the derivative of the position, so to find the equation of the velocity of the car we’ll use the diff function, then plot the result

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Find the symbolic derivative, which corresponds to the velocity

Create a plot of velocity and time

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The velocity is the derivative of the position with respect to time

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time, sec

Race Car Velocity

Finish Line

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Acceleration

• The acceleration is the derivative of the velocity, so to find the equation of the acceleration of the car we’ll use the diff function, then plot the result

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Determine the equation for the acceleration

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Acceleration is the derivative of the velocity

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Race Car Acceleration

Finish Line

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Symbolic Differentiationdiff(f) Returns the derivative of the

expression f with respect to the default independent variable

y=sym('x^3+z^2')diff(y)ans =3*x^2

diff(f,’t’)

Returns the derivative of the expression f with respect to the variable t.

y=sym('x^3+z^2')diff(y,'z')ans =2*z

diff(f,n)

Returns the nth derivative of the expression f with respect to the default independent variable

y=sym('x^3+z^2')diff(y,2)ans =6*x

diff(f,’t’,n)

Returns the nth derivative of the expression f with respect to the variable t.

y=sym('x^3+z^2')diff(y,'z',2)ans =2

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Partial Derivatives

• If you have multiple variables, MATLAB takes the derivative with respect to x – unless you specify otherwise

• All the other variables are kept constant

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To find the derivative with respect to some variable other than x, you must specify it in the diff function

Notice that t is enclosed in single quotes, since we haven’t specified it as a symbolic variable

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Higher order derivatives

• To find higher order derivatives we can either • nest the diff function• or specify the derivative order in the

diff function

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Example 12.4

• Find the angle where the maximum range occurs

• Find the maximum range

horizontal distance

Vertical distance

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Angle for Max Range

Max Height

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Integration

• Usually introduced in Calculus II

• Often visualized as the area under a curve

• MATLAB has built in symbolic integration capability.

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Consider a piston cylinder device

• Work done by a piston cylinder device as it moves up or down, can be calculated by taking the integral of P with respect to V

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To perform the integration we need to know how P changes with V

• If P is constant the problem becomes

1W P dV

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time, hour

Model of the behavior of a piston cylinder device

0 1 2 3 4 599

101Pressure Profile in a Piston Cylinder Device

Volume, cm3

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time, hour

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

Volume, cm3

Pressure Profile in a Piston Cylinder Device

Initial volume

Final volume

The area under the curve corresponds to the work

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Hand Calculation

psiacmW

psiaPif

VPPVPVPVdVPPdVW

Read this as: Work is equal to the integral of P with respect to V, from V=1 to V=4

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MATLAB Solution

Work is equal to the integral of P with respect to V, from V=1 to V=4

Substitute in 100 as the value of P

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Symbolic Integration

int(f) Returns the integral of the expression f with respect to the default independent variable

y=sym('x^3+z^2')int(y)ans =1/4*x^4+z^2*x

int(f,’t’)

Returns the integral of the expression f with respect to the variable t.

y=sym('x^3+z^2')int(y,'z')ans =x^3*z+1/3*z^3

int(f,a,b)

Returns the integral with respect to the default variable, of the expression f between the numeric bounds, a and b.

y=sym('x^3+z^2')int(y,2,3)ans =65/4+z^2

int(f,’t’,a,b)

Returns the integral with respect to the variable t, of the expression f between the numeric bounds, a and b.

y=sym('x^3+z^2')int(y,'z',2,3)ans =x^3+19/3

int(f,’t’,a,b)

Returns the integral with respect to the variable t, of the expression f between the symbolic bounds, a and b.

y=sym('x^3+z^2')int(y,'z','a','b')ans =x^3*(b-a)+1/3*b^3-1/3*a^3

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12.5 Differential Equations

• Differential equations contain both

• the derivative of the dependent variable with respect to the independent variable

• the dependent variable

dt is a differential equation

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Default variable

• Although any symbol can be used for either the independent or the dependent variable, the default independent variable is t in MATLAB (and is the usual choice for most ordinary differential equation formulations.)

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dsolve

• When we solve a differential equation, we are looking for an expression for y in terms of t

• dsolve requires the differential equation as input• use the symbol D to

specify derivatives with respect to the independent variable

dsolve is a “function function”

Using a single input results in a family of results

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Specify an initial or boundary condition in the second field

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Here’s a more complicated example

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You can specify the independent variable in the third field

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To specify a higher order derivative in the dsolve function put the order immediately after the D

Higher Order Derivatives

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• Don’t use the letter D in your variable names in differential equations.

• It will confuse the function into thinking you are trying to specify a derivative

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Use the dsolve function to solve systems of equations

• dsolve('eq1,eq2,...', 'cond1,cond2,...', 'v')

The result is a structure array

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• For complicated (or ill behaved) systems of equations you may find it easier to use MuPad • Remember that MATLAB’s symbolic

capability is based on the MuPad engine

• There are many differential equations that can’t be solved analytically at all • The numerical techniques described in

Chapter 13 can be used to solve many of these equations.

MATLAB can not solve every differential equation symbolically.

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12.6 Converting Symbolic Expressions to MATLAB functions

• It is often useful to manipulate expressions symbolically … but then to perform numeric calculations using more traditional MATLAB functions

• matlabFunction converts a symbolic expression to an anonymous function

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Symbolic Manipulation

An anonymous function

matlabFunction

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Summary

• MATLAB uses MuPad as its symbolic engine

• The symbolic toolbox is an optional component of the professional version

• A subset is included with the student version

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Summary – Variable Definition

• Use either• sym• syms

• The sym command can be used to create symbolic expressions or equations

• The syms command can create multiple symbolic variables in one step

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Summary – Composition of expressions

• Once symbolic variables have been created they can be used to create more complicated expression

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SummaryEquations vs Expressions

• Equations are set equal to something

• Expressions are not

• If you set one expression equal to another, you’ve created an equation

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Summary – Symbolic functions

• numden

• expand

• factor

• collect

• simplify

• simple

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Summary – Solve

• If the input to solve is an expression MATLAB sets it equal to 0 and solves

• If the input is an equation, MATLAB solves the equation for either the default variable, or a user defined variable

• solve can also solve systems of equations

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Summary - dsolve

• Used to solve differential equations

• D signifies a derivative

• Can be used to solve systems of equations

• Not all differential equations can be solved analytically

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Summary - Calculus

• diff - finds the derivative

• int - takes the integral

Symbolic Mathematics Chapter 12

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