topic 12 symbolic mathematics and programming · 2012. 10. 22. · using the solve function for...

CITS2401 Computer Analysis & Visualisation

SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING

FACULTY OF ENGINEERING, COMPUTING AND MATHEMATICS

Topic 12 Symbolic

Mathematics and Programming

Based on material by Paul Abbott and Holly Moore

The University of Western Australia

Objectives ◊  Create, visualize and solve symbolic expressions in

Mathematica. ◊  Understand the difference between symbolic and

numerical methods ◊  Solve systems of equations ◊  Determine the symbolic derivative of an expression ◊  Integrate an expression symbolically ◊  Use Matlab to perform symbolic computations.


Symbolic Versus Numeric Computation ◊  A computers memory is discrete. Every floating point number

is stored using 64 binary digits (bits).

◊  By the IEEE 754 standard there are 52 bits for precision, 11 bits for an exponent and 1 bit for the sign, but it is necessarily an approximation.

◊  Consider Sqrt[5] in Mathematica and sqrt(5) in Matlab:


Symbolic Computation

◊  Symbolic computation doesn’t map calculations to numbers. Instead it records the operations that were applied to the data so no information is lost:

•  We can use N to produce a numeric approximation when require, but until then the symbols can just accumulate.

•  This can allow the expressions to get very big, very quickly, and cause slow computation.


Rule based computation ◊  To manage the size of expressions, rules are used to simplify

expressions whenever possible

◊  To do this we require rules to seek out and apply simplifying substitutions.


Simplify ◊  Simplify is a function that will apply a large sequence of

algebraic rules to an expressions and return the simplest one it can find:

◊  We can also include assumptions, which are predicates:


Expand ◊  Expand works in the opposite way to simplify, applying

algebraic rules to expand an expression:

•  Expand works on polynomials

•  TrigExpand works on trigonometric expressions

•  LogicalExpand works on Boolean expressions


Solve ◊  Solve will solve an equation

symbolically. An equation requires a set of equalities or inequalities, and a set of symbols to solve for:


Solve ◊  For each member of the parameter list, Solve returns a

substitution (or list of substitutions):

◊  Use the /. notation to return apply the substitutions to the original functions. If the system cannot be completely solved, there may be parameters left in the substitution. You can also specify the domain as either Integers, Reals or Complexes


NSolve ◊  NSolve works in the same way as solve, except it seeks a

numerical approximation to the set of equations


Symbolic Visualisation ◊  Functions Plot, Plot3D and

ParametricPlot3D can be used to visualize symbolic functions:


Symbolic Computation in Matlab

◊  MuPad is a Mathematica like interface that has been incorporated into Matlab

◊  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


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


Symbolic Algebra ◊  Used regularly in math, engineering and

science classes ◊  Often preferable to manipulate equations

symbolically before you substitute in values for variables

96)3(*2

2

2

++

+=

xxxy

Consider this equation


y = 2*(x +3)2

x2 + 6x + 9

This looks like a fairly complicated function of x

y = 2*(x +3)2

x2 + 6x + 9=2*(x2 + 6x + 9)(x2 + 6x + 9)

= 2

If you expand it, it simplifies dramatically, but if you evaulate it numerically….

Let x equal -3

y = 2*(−3+3)2

9−18+ 9= 2* 0

0= undefined


Relationships are not always easy to solve

If we know k0, Q, R, T the its easy to solve for k. It’s not easy to solve for T!

/0

Q RTk k e−=0

0

0

0

ln( ) ln( )

ln

ln

ln( / )

Qk kRT

k Qk RT

k Qk RT

QTR k k

= −

⎛ ⎞= −⎜ ⎟

⎝ ⎠

⎛ ⎞ =⎜ ⎟⎝ ⎠

=


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


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)


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)) ◊  Create an entire expression with syms:

•  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.


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 equation This is an assignment statement

•  Equations are set equal to something •  Expressions are not •  Both expressions and equations can be assigned a

name, using the MATLAB assignment operator


The variable ideal_gas_law has been assigned to an equation


The variable E as been assigned to an expression


Working with Equations and Expressions ◊  Some MATLAB functions work on both expressions and

equations

◊  Others are limited in their scope


Extracting Numerators and Denominators ◊  These functions work on expressions


Expanding and Factoring

num is an expression


w is an equation


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

◊  Simple ◊  The simple function is different from simplify

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


simplify used on an expression

simplify used on an equation


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

Both simple and simplify work on expressions and equations


Hints ◊  Use the poly2sym function as a

shortcut to create a polynomial

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


Solving 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


The answer from the solve function is not necessarily a number

You can specify what variable you want to solve for


Solving Systems of Equations

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


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


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


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


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


Hint

◊  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.


Example 12.1 ◊  Use MATLAB’s symbolic capability to solve an equation

– k = k0*exp(-Q/RT),…..Solve for Q

0 expQk kRT−⎛ ⎞= ⎜ ⎟

⎝ ⎠

0

expk Qk RT

−⎛ ⎞= ⎜ ⎟⎝ ⎠

0

ln k Qk RT⎛ ⎞ −

=⎜ ⎟⎝ ⎠

0ln kQ RTk

⎛ ⎞= ⎜ ⎟⎝ ⎠


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


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


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


Example 12.2 ◊  Using Symbolic Maths to solve a ballistics

problem

vertical distance

horizontal distance

Range dist _ x = v0* t *cos(θ )dist _ y = v0* t *sin(θ )−1/ 2*g* t2


Strategy ◊  Find the time the projectile hits the ground

◊  Substitute back into the horizontal distance equation


Results


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.


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


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


Model

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

dist = 20+ 20*sin(π *(t −10) / 20)


Create a plot of position vs time using ezplot


0 5 10 15 20

05

1015

2025303540

time, sec

Car position

Dist

ance

from

Sta

rting

Lin

e

Finish Line

ezplot of position


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


Find the symbolic derivative, which corresponds to the velocity

Create a plot of velocity and time


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

0 5 10 15 20

0

0.5

1

1.5

2

2.5

3

time, sec

Race Car Velocity

velo

city

, dis

tanc

e/tim

e

Finish Line


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


Determine the equation for the acceleration


Acceleration is the derivative of the velocity

0 5 10 15 20-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time, sec

Race Car Acceleration

acce

lera

tion,

vel

ocity

/tim

eFinish Line


Symbolic Differentiation diff(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


Integration

◊  Usually introduced in Calculus II

◊  Often visualized as the area under a curve

◊  MATLAB has built in symbolic integration capability.


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

W = PdV1

2∫

To perform the integration we need to know how P changes with V. If P is constant the problem becomes: W = P dV

1

2∫


Model of the behavior of a piston cylinder device

0 1 2 3 4 599

99.5

100

100.5

101Pressure Profile in a Piston Cylinder Device

Volume, cm3

Pres

sure

, psi

a

gas


Hand Calculation

W = PdV = P1

4∫ dV = PV

1

4

1

4∫ = PV4 −PV1 = PΔV

if P =100psiaW = 3cm3 *100psia

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


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


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


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


Symbolic Manipulation

An anonymous function

Sym to Matlab Function

topic 12 symbolic mathematics and programming · 2012. 10. 22. · using the solve function for...

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