MATH15L Coursewares

8/3/2019 MATH15L Coursewares

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presents

in cooperation with


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Course Code: MATH15L

Course Title: MATLAB

Pre-requisite:

Co-requisite: MATH 15

Credit: 1 unitEquivalent Course Codes: None

Faculty:


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Course Description

The course utilizes the capability ofinformation technology to facilitate theunderstanding of basic mathematicalprinciples and operations.

A mathematical software (MATLAB)will be used to perform algebraicoperations, differentiation, integration,

matrix operations, graphic manipulationand some basic MATLAB programmingfor simulations and analysis.


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Course Objectives and

Relationship to Program ObjectivesIt is a laboratory course in

mathematics. Its goals are to give the

students sufficient understanding ofMATLAB and its application for the

purpose of developing students skill in

solving problems and to apply their acquired learning in engineering

applications.


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It prepares students to recognize

patterns and formulate rules as a firststep to develop students skill for

independent critical thinking.

The course aims to develop thestudents zest for knowledge,

application and appreciation of an

orderly and logical solution as guided bythe different principles undertaken in the

course.


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Course Coverage

Foundation and FundamentalConcepts

Arrays and Matrices

Symbolic MathGraphs

Programming


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Course Outcomes and Relationship

to Program Outcomes:

A student completing this course shouldat a minimum be able to:

perform polynomial operations such asmultiplication and division.

decompose rational polynomials into sum ofpartial fractions.

perform the four fundamental operations onmatrices as well as getting the determinant ofsquare matrix.


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solve systems of equations and polynomialequations.

perform differentiation and integration.

plot the graphs of lines, circles, ellipses,parabolas and hyperbolas.

plot the graphs of polynomial functions and

transcendental functions.

compile a simple MATLAB program.


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Course Evaluation

Points

Long Tests 2 250 25%

Classroom Exercise/Hands on

Exam

5 450 45%

Classroom Participation

Portfolio 1 50 5%

Final Examination 1 250 25%

TOTAL 1000 100%


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Long Test/Classroom Exercise/

Hands on Exam/Final Exam

Guidelines

No borrowing of terminal/ terminal account.

Use black ballpen/tech-pen.

Show your answers clearly.

Use short white/bond paper.

Avoid going out during examination.

No special exam.


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Portfolio Guidelines

No late Portfolio will be accepted.

Use short black Clear Book.


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Average Grade Average Grade

Below 70 5.00 83.21-86.5 2.00

70-73.3 3.00 86.51-89.8 1.75

73.31-76.6 2.75 89.81-93.1 1.50

76.61-79.9 2.50 93.11-96.4 1.2579.91-83.2 2.25 96.41-100 1.00


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Other course Policies

According to CHED policy, total number of

absences by the students should not be more

than 20% of the total number of meetings or

9 hrs for this one-unit-course. Student

incurring more than 9 hours of absences

automatically gets a failing grade regardlessof class standing.

Attendance


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Honor, Dress and Grooming Codes

All of us have been instructed on the

Dress and Grooming Codes of the Institute.

We have all committed to obey and sustainthese codes. It will be expected in this class

that each of us will honor the commitments

that we have made.


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For this course the HonorCode is that

there will be no plagiarizing on written work

and no cheating on exams. If a student is

caught on two exams, the student will be

referred to the Prefect of Student Affairs and

given a failing grade.


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Consultation Schedule

Consultation schedules with the Professor

are posted outside the Mathematics Faculty

Room. It is recommended that the student

first set an appointment to confirm the

instructors availability if outside the

consultation schedules.


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Computer Laboratory Guidelines

No student shall be allowed to enter thelaboratory without his or her Instructor.

EATING, DRINKING, LITTERING and

VANDALISM in any form are strictly

prohibited.

All belongings shall place in the designated

areas. Valuables and important belongings

should be brought in.The assigned personnel is not liable for any

loses or damages that would occur.


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A strict one (1) computer to one (1) studentratio shall be observed.

Students are not allowed to use externaldevices without the approval of the DeputyDirector for Systems Administration.

Students must notify the assign personnel in

charge regarding any errors and/orbreakages of facilities and/or equipment intheir designated areas.

No excessive noise.

Using mobile phone or any musicalinstrument is strictly prohibited.

Diskettes are not allowed.

Accessing the internet is prohibited.


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MATLAB BASICS


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What is MATLAB?

- The name stands for MATrix LABoratory

- MATLAB is a high-performance language for technicalcomputing. It integrates computation, visualization, and

programmingenvironment.- MATLAB is a modern programming language

environment: it has sophisticated data structures,contains built-in editing and debugging tools, andsupports object-oriented programming.


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THE MATLAB SYSTEM

- Development Environment

- The MATLAB Mathematical Function Library

- The MATLAB Language

- The MATLAB Application Program Interface (API)


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Development Environment

- Set of tools and facilities that help use MATLABfunctions and files.

- Includes:

- MATLAB Desktop- Command Window

- Command History

- Editor and Debugger and browsers for viewing help

- Workspace


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MATLAB MATHEMATICAL FUNCTION LIBRARY

- This is a vast collection of computational algorithmsranging from elementary functions

Examples:

> sum, sine, cosine, and complex arithmetic

> sophisticated functions like matrix inverse, matrixeigenvalues, Bessel functions, and fast Fouriertransforms.


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MATLAB LANGUAGE

- This is a high-level matrix/array language with controlflow statements, functions, data structures,input/output, and object-oriented programmingfeatures.

- It allows both "programming in the small" to rapidlycreate quick and dirty throw-away programs, and"programming in the large" to create complete largeand complex application programs.


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GRAPHICS

- MATLAB has extensive facilities for displaying vectorsand matrices as graphs, as well as annotating andprinting these graphs.

- It includes high-level functions for two-dimensional andthree-dimensional data visualization, image processing,animation, and presentation graphics.

- It also includes low-level functions.


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MATLAB APPLICATION PROGRAM INTERFACE

- This is a library that allows you to write C and Fortranprograms that interact with MATLAB.

- It includes facilities for calling routines from MATLAB(dynamic linking), calling MATLAB as a computationalengine, and for reading and writing MAT-files.


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Development Environment

Desktop Tools

- Command Window

- Command History

- Star Button and Launch Pad

- Help Browser

- Current Directory Browser

- Workspace Browser- Array Editor

- Editor/Debugger

- Profiler


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WORKSPACE BROWSER

- Consists of the set of variables during a MATLAB sessionand stored in memory


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ARRAY EDITOR

- Use to edit the variables in the workspace.

- Three ways to open: double click the variable in theworkspace browser, select the variable in the workspaceand click open , use the openvar syntax


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EDITOR/DEBUGGER

- Used to create and debug M-files, which are programyou write to run the MATLAB functions


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Foundation and FundamentalConcepts


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Entering Commands and

Expressions

The prompt >> is displayed in the Command

Window and when the Command Window is

active, a blinking cursor should appear to theright of the prompt.

This cursor and the MATLAB prompt signify

that MATLAB is waiting to perform amathematical operation.


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clear- Removes all the variables from theworkspace.

- Frees up system memory

who- Displays the list of variables currently in

the memory.

whos- Will display more details which include

size, space, allocation and class of variables

exist - Checks for existence of the variable.

global - Declares variable to be global.

help - Searches for help topic

lookfor - Searches help entries for keyword

quit - Stops the MATLAB

Commands for Managing a Session


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Help Features in MATLAB

helpbrowser Opens the help window

help function_nameDisplays the help document in the

command window

helpwin function_nameDisplays the help document in

separate window

doc function_name Displays detailed help document in

separate window


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Numeric Display Formats

format- Controls the display output of the

command window

Common Format Functionsshort - Four decimal digits

long - Sixteen decimal digits

short e - Five decimal digits plus exponent

long e - Sixteen digits plus exponent

bank - Two decimal digits


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Mathematical Functions

ans - Most recent answeri , j - The imaginary unit

Inf - Infinity

NaN - Undefined numerical result (not a number)

pi - The number

Special Variables and Constants


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Mathematical FunctionsElementary Functions

cos(x) Cosine abs(x) Absolute value

sin(x) Sine ceil(x) Round towards + Inf

tan(x) Tangent floor(x) Round towards - Inf

acos(x) Arc cosine round(x) Roundasin(x) Arc sine rem(x) Remainder after division

atan(x) Arc tangent angle(x) Phase

exp(x) Exponential conj(x) Complex Conjugate

sqrt(x) Square root imag (x) imaginary

log(x)Natural

Logarithmreal (x) Real number

log10(x)Common

Logarithmprimes (x) Prime Number


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Scalar Arithmetic

Symbol Operation

+ Addition

- Subtraction

* Multiplication

/ Right Division

\ Left Division

^ Exponentiation


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Exponential and Logarithmic Functions

exponential exp (x)

Natural Logarithm ln (x)

Common Logarithm log10 (x) = [ log10 (x) ]

Square root sqrt (x)


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Example1:

2 + 3 4

In Command Window>>2 + 3 4

ans =

1Example2:

In Command Window

>> 6 / 3

ans =

2

36 z


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Example3:

In Command Window

>> 6 \ 3

ans =0.5

MATLAB has assigned the answer to a

variable called ans, which is an abbreviationfor answer. A variable in MATLAB is asymbol used to contain a value.

63 z


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MATLAB does not care about spaces for themost part. Spaces in the line improve itsreadability.

When you want to calculate a more complexexpression use parentheses, in the usualway, to indicate precedence.

The mathematical operations represented bythe symbols + * / \, and ^ follow a set of rules called precedence..


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Mathematical expressions are evaluatedstarting from the left, with the exponentiationoperation having the highest order of precedence, followed by multiplication anddivision with equal precedence, followed byaddition and subtraction with equal

precedence.

Parentheses can be used to alter this order.Evaluation begins with the innermost pair of

parentheses, and proceeds outward.


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To avoid mistakes, you should feel free to

insert parentheses wherever you are unsureof the effect precedence will have on the

calculation.

Example4:

In Command Window

>> (3*(23 + 14.7 (4 / 6))) / 3.5

5.3

)6

47.1423(3


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Naming constants and variables

MATLAB allows us to give constants and

variables names of our choice. This is a

powerful facility that can reduce work and

help in avoiding input errors.

When the user begins a session in which the

same values must be used several times, theuser can define them once and then call them

by name.


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Example5:

Given:

a = 2, A = 3

Find:

a. 2a

b. w=3A

In Command Window

>>a=2

a =2

>>A=3;


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When you write the semicolon ; at the endof a statement, the computer will not displaythe result of the command, and it will notecho the input.

>>2*a

ans =

4>>w=3*A

w =

9

MATLAB does not tell you the value of all thevariables; it merely gives you their names. Tofind their values, you must enter their names

at the MATLAB prompt.


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>>a = 4

a =

4

>>2*a

ans =

8

If you reuse a variable in the precedingexample, or assign a value to one of the

special variables, its prior value is overwrittenand lost. However, any other expressionscomputed using the prior value do notchange.


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Example6:

Given:

E = 30, F = 52, K = 76

Find:

a. Sin E

b. Sin F

c. Sin K

In Command Window

>> alpha = 30;>> beta = 52;

>> gamma = 76;


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>> sin (alpha)

ans =

0.9880 A pair of parentheses is used after the

functions name to enclose the value calledthe functions argument that is operated on

by the function.>> sin (beta)

ans =

0.9866>> sin (gamma)

ans =

0.5661


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MATLAB remembered past information.

To recall previous commands, MATLAB uses

the cursor keys, n, o, p, q, on your

keyboard.

In addition, all text after a percent sign (%) is

taken as a comment statement.


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Example ofFormating:

T or pi

In Command Window

>> pi

ans =

3.1416

MATLAB uses high precision for its

computations, but by default it usually

displays its results using four decimal places.This is called the short format. Using the

format command can change this default.


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>> format long

>> pi

ans =

3.14159265358979

MATLAB uses the notation e to represent

exponentiation to a power of 10.

>> format short e

>> pi

ans =

3.1416e+000


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>> format long e

>> pi

ans =3.14159265358979e+000

>> format bank

>> pi

ans =

3.14

To return to default format

>> format

>> pi

ans =

3.1416


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Most interesting is the format rat: it yields arational approximation of a real number, thatis a fraction that approximates a givennumber.

>> format rat

>>pi

ans =355/113

>>format

>>355/113ans =

3.1416


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Practice Set:

1. Perform the indicated operation:

a. 2{ 4 [6 + 3 + (7 (1 + 8)) + 12] 3} + 5

b. 5{ [ 2 + 6 8(4 (7 4) 2) + 2] 1}

c.

d

e. Given:

)2(312

3)2(3

52

)4(8

)2(4

85

z

2

5x

T

!


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Determine:

i. y = sin x

ii. when y = 1, what is z = sin-1 yg.

h. cos (T / 2)

2. Suppose that x = 3 and y = 4. Use MATLABto compute the following.

a.

b. 3 T x2

c.

3.0cos 1e

1

5

x

11

8x4

y3


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d.

3. Assuming that the variables a, b, c, d and f are scalars, write MATLAB statements to

compute and display the following

expressions. Test your statements for the

values a = 1.12, b = 2.34, c = 0.72, d = 0.81,

f = 19.83

a.

b.

6x3

)5y(4

2f

c

b

a1x !

cd

abs

!

1 f1 2


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c. d.

4. Evaluate the following expressions inMATLAB, for the values x = 5 + 8i,

y = 6 + 7i.

a. u = x + y

b. v = xy

c. w = x / y

d. z = ex

e.

f. s = xy2

d

1

c

1

b

1

a

1

1r

!2

f

c

1aby

2

!

yr!


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Polynomial Algebra

Partial fractionRoots of Equation

Polynomial and Symbolic Conversion


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Command Description

conv(a, b) Computes the product of thetwo polynomials described by

the coefficient arrays a and b.

The two polynomials need not

be the same degree. Theresult is the coefficient array of

the product polynomial.

Consider f(x) = 12x4 3x2 + x + 7. This

function f can be written in array form calledcoefficient array, that is, f = [12, 0, -3, 1, 7].


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Example Answer

1. (2x+1)(3x-5) 6x^2 -7x -5

2. (x^2-3)(x^3+2x+1) X^5-x^3+x^2-

6x-3


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[q, r] =

deconv(num,

den)

Computes the result of dividing

a numerator polynomial,

whose coefficient array is num,by a denominator polynomial

represented by the coefficient

array den.

Example Answer

Q= 4x+11

R= 59x 41

Q= x+3

R= 9x -18

473

325

2

2

xx

xx312x

672

6

2

2

xx

xx32x


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poly(r) Computes the coefficients of thepolynomial whose roots are specified

by the vector r. The result is a rowvector that contains the polynomials

coefficients arranged in descending

order of power

Example Answer

X1=1

X2=-1

1 0 -1 => X^2-1

X1=1

X2=2

X3=3

1 -6 11 -6 => x^3-6x^2+11x-6


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[r,p,k] =

residue(a,b)

Finds the residues, poles and

direct term of a partial fraction

expansion of the ratiopolynomials a(x) / b(x)

Example Answer

2

75

23

xxx

x

22

7242

23

234

xxx

xxxx

2

1

1

1

1

2

xxx

2

1

1

1

1

22

xxxx

l l( ) E l t l i l t


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polyval(a, x) Evaluates a polynomial at

specified values of its

independent variable x, whichcan be a matrix or a vector.

The polynomials coefficients of

descending powers are stored

in the array a.Example Answer

1. f(x)=x^2-x+5. Find f(2) 7

t ( ) C t th t f


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roots(a) Computes the roots of a

polynomial specified by the

coefficient array a. The resultis a column vector that

contains the polynomials roots.

Example Answer

X^2-1 X1=1

X2=-1

x^3-6x^2

+11x-6

X1=1

X2=2

X3=3

2 l C t b li l i l


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sym2poly Convert a symbolic polynomial

to polynomial coefficient vector

Example:

sym2poly(x^3 - 2*x - 5) returns [1 0 -2 -5]

poly2sym Convert a polynomial

coefficient vector to symbolicpolynomial

Example:poly2sym([1 0 -2 -5]) is x^3-2*x-5


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SYMBOLIC PROCESSING

Process of obtaining answers in the form of

expressions

Terms used to describe how MATLAB performs

operations or evaluates mathematical expressions inthe way, for examples, that humans do algebra with

pencil and paper.


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Symbolic Object

Symbolic object is a data structure that stores astring representation of the symbol. The two ways tocreate symbolic object are:

1. Using sym function

Typing x = sym(x) creates the symbolic

variable with name x.2. Using syms command

syms x is equivalent to typing x = sym(x).

Typing syms x y z creates three symbolic

variables x, y and z.

S b li C t t


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Symbolic Constants:

To create symbolic constant, use the sym function.

>> pi = sym(pi)

>> fraction = sym(1/4)

>> sqroot2 = sym(sqrt(2))

Symbolic Variables & Expressions

The sequence of commands

>> syms x y x

>> m = sqrt (x ^2 + y^2 + z^2)

>> n = sin (x*y) / (x*y)

generates the symbolic m and n.

S b li C t t


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Symbolic Constants:

To create symbolic constant, use the sym function.

>> pi = sym(pi)

>> fraction = sym(1/4)


Symbolic Variables & Expressions

The sequence of commands

>> syms x y x

>> m = sqrt (x ^2 + y^2 + z^2)

>> n = sin (x*y) / (x*y)

generates the symbolic m and n.


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Symbolic objects can be classified as:

1. symbolic variable (ex. x, y, z)2. symbolic constant (ex. pi, sqrt(3))

3. symbolic expression (ex. x^2 + y^2)

4. symbolic matrix (ex. [a, b, c; b, c, a; c, a, b]

To create symbolic constant, use the symfunction.

Example:

>> pi = sym(pi)>> fraction = sym(1/4)



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pretty

Theprettyfunction displays symbolic output in aformat that resembles typeset mathematics.

Syntax: pretty (S)

>>m = sqrt (x^2 + y^2 + z^2)

m =

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

>> pretty (m)

2 2 2 1/2

(x + y + z )


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Examples:

>> f = x^3-6*x^2+11*x-6

>> g= (x-1) * (x-2)* (x-3)

>> pretty (f)

>> pretty (g)

Or

>> pretty (x^3-6*x^2+11*x-6)

>> pretty ((x-1) * (x-2)* (x-3))

double

The statement double (s) converts the symbolic object Sto a numeric object.

Syntax: r = double (S)

>> sqroot2 = sym (sqrt (2));

>> z = 6* sqroot2

>> double (z)

S C O


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SIMPLIFICATION:

Commands

collect (s) expand (s)

horner (s) factor (s)

simplify (s) simple (s)

ll t


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collect

collect (f) views fas a polynomial in its symbolic

variables, say x, and collects all the coefficients withthe same power of x.

Syntax: collect (S)

Ex.

1. H =x[x(x-6)+11]-6

>> h = x * (x *(x-6)+11)-6

collect (h)

ans =

x^3-6*x^2+11*x-6

Syntax : collect (S, x) the 2nd argument can specifythe variable in which to collect terms if there is morethan one candidate.

d


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expand

expand (f) distributes products over sums and applies

other identities involving transcendental functions.

Syntax: expand (S)

Ex.

1. (x-2)(x-3)(x-5)2. cos (x+y)

3. e a+b

h


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horner

horner (f) transforms a sumbolic polynomial finto its

Horner, (or nested, representation)

Syntax: horner (f)

Ex.

1. (x3

3x2

4x +5)2. 2x5 x4 + 3x2 4x +5

3. 1.1 + 2.2x + 3.3x2

factor


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factor

factor (S) factors the expression Swhich can be

positive integer, an array of symbolic expressions, oran array of symbolic integers.

Syntax: factor (S)

Ex.

1. factor (x^3 - y^3)

simplify


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simplify

simplify (E) simplify the expression E using Maples

simplification rules.

Ex.

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

ans

1

1. 1- x2

1 x

simplify ((1-x^2)/(1-x))

ans =

x+1

simple


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simple

simple(E) searches for the symbolic expressions

simplest form; that is, an expression that has thefewest character.

Syntax: r = simple (S)

Ex. Apply the simple and simplify commands

1. log (xy)2. cos (3cos-1(x))

3. (x+1)(x)(x-1)

Substitution


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Substitution

subs used for symbolic substitution in a symbolic

expression or matrix

Syntax: R = subs (S, old, new)

Ex. Given f(x) = x3+4x2-3x+5 Find:

1. f(4)

2. f(2z)

3. f(x+1)

Solving Equations


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Solving Equations

If S is a symbolic expression solve (S) attempts to find

values of the symbolic variable is S for which S is zero

Syntax: g = solve (eq)

g = solve (eq, var)

Ex. >>solve (2*x^2-x-6=0)

Solve the ff equations:

x2+9y4=0

e2x + 3ex =543. y2 + 3y + 2 = 0 solve for y

4. b2 + 8c + 2b = 0, solve for c, then solve for b.

Solving a Systems of Equations


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Solving a Systems of Equations

[a, b, c, ] = solve(eq1, eq2, eq3, )

Ex.

1. 6x + 2y = 14

3x + 7y = 31

2. x + 6y = a2 x 3 y = 9

Diff ti ti


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Differentiation

MathC

ommand MATLABC

ommanddiff(f) or diff(f, x)

diff(f, a)

diff(f, b, 2)2

2

db

fd

da

df

dx

df


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Examples:

Differentiate the following functions:1.

2.

3.

)sin( 2xdx

d

6

3

3

y

dy

d

222

2

3xyyx

dy

d

Li it f F ti


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Limit of a Function

MathC

ommand MATLABC

ommandlimit(f)

limit(f, x, a) or limit(f, a)

limit(f, x, a, right)

limit(f, x, a, left)

pax

f(x)lim

pax

f(x)lim

ax

f(x)limp

0x

f(x)limp

Example:


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Example:

Evaluate the following:

1.

2.

3.

4.

125x

102xlim 35x

p

9x

3xlim

23x

p

h

sinxh)sin(xlim

0x

p

x

sinlim

0x

x

p


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5.

6. and

7.

p x

1lim

0x

p x

1lim

0x

2

2

1x 1)sin(x

1xlim

p

h

sinxh)sin(xlim

0h

p

Integration


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Integration

MathC

ommand MATLABC

ommandint(f) or int(f, x)

int(f, y)

int(f, a, b) or int(f, x, a, b)

int(f, y, a, b)

f(x)dx

y)dyf(x,

b

a

f(x)dx

b

a

y)dyf(x,


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Examples:

Evaluate the following:1.

2.

3.

4.

dxx5

xcosxdxsin4

y)dycos(xy

2

1

dyy

1


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5.

6.dxe

0

x2

g

1

0

lnzdzz


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ARRAYS AND MATRICES

CREATING ARRAYS


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CREATING ARRAYS

Type the elements inside a pair of square

brackets, separating the elements inside with

a space or a comma.Example:

>>A = [10 3 4 6] or

>>A = [10, 3, 4, 6]

A. ROW VECTOR


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B. COLUMN VECTOR

Separate the elements by semicolons or

create a row vector first and then use the

transpose notation ( ) which converts a row

vector into a column vector, or vice versa.Example:

>>B=[5; 3; 6; 0; 2] or

>>B=[5, 3, 6, 0, 2]


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Another way is to type a left bracket and thefirst element, press ENTER, type the secondelement, press ENTER, and so on until youtype the last element followed by the rightbracket.

Example:

>>B=[5

3

6

0

2]


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C. MATRICES

Type the matrix row by row, separating the

elements in a given row with spaces or

commas and separating the rows with

semicolons.Example:

To create , type

>>A=[7, 4, 3; 1, 2, 4]

-

!

42-1

347A

APPENDING ARRAYS


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APPENDING ARRAYS

Array can be created by appending one arrayto anther.

Illustration:

1. Let array1 = [4, 6, 2] and

array2 = [2, 5]. Type

>>array3 = [array1, array2]

2. Let and

Typing >>C = [A , B]

-

! 9-07

62-4A

-

! 15-

42B

SPECIAL WAY OF CREATING


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SPECIAL WAY OF CREATING

MATRICES

1. Using :

The colon ( : ) generates a vector of

regularly spaced elements. Typing>>x = [m: q: n]

creates a vector x with first value m and last

value n with a spacing q. The last value isless than n if m n is an integer not multiple

of q.


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Example:>>x = [2: 2: 8]

>>y = [1: 3: 11]

>>z = [20: 2: 10]

If q is omitted, it is presumed to be 1.

Example:

>>w = [3: 2]

2. Using linspace


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g p

Linspace command creates a linearly

spaced row vector by specifying the numberof elements rather than the increment. The

syntax >>x = [a, b, n] means creating a

vector x where a and b are the lower and

upper limits, respectively and n is thenumber of elements.

Example:

>>linspace (5, 8, 31).It contain 31 elements. If n is omitted, the

spacing is 1.


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3. Using logspace

It creates an array of logarithmically spacedelements. The syntax is

>>logspace (a, b, n)

where n is the number of elements between

10a and 10b.

Example:

>>x = logspace (1, 1, 6)

If n is omitted, the number of elements

defaults to 50.

SOME USEFUL ARRAY


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SOME USEFUL ARRAY

FUNCTIONS

Functions Descriptions

magic(n) creates an n x n matrix where the

sum of its diagonal, rows and

columns are equal

eye(n) creates an n x n identity matrix

ones(n) creates an n x n matrix of ones

ones(m,n) creates an m x n matrix of ones

zeros(n) creates an n x n matrix of zeros

zeros(m,n) creates an m x n matrix of zeros

size(A) returns a row vector [m, n]


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( ) [ ]

containing the sizes of the m x n

array Alength(A) computes either the number of

elements of A if A is a vector or the

largest value of m or n if A is an m x

n matrixsum(A) sums the elements in each column

of the array A and returns a row

vector containing the sumssort(A) sorts each column of the array A in

ascending order and returns an

array the same size as A


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max(A) returns the algebraically largestelement in A if A is a vector.

Returns a row vector containing

the largest elements in each

column if A is a matrix

min(A) same as max(A) but returns

minimum values

diag(A) returns the main diagonal of A


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Practice Set:

1. Let A = [1, 10, 6, 4; 7, 2, 5, 11;12, 0, 0, 8; 1, 3, 5, 6].

Find: sum (A), max(A), min(A), length(A),

size(A), sort(A), diag(A)

2. Let B = magic(6). Verify that the sum of the

rows, columns and diagonals are equal.

3. Create an identity matrix with a size as

matrix A in problem 1.

ARRAY ADDRESSING/INDEXING


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ARRAY ADDRESSING/INDEXING

Array indices are the row and columnnumbers of an element in an array and are

used to keep track of the arrays elements.

Illustration:

Address Description

v(2) refers to 2nd element in the vector v

A(3,4) refers to the element in row 3,

column 4

v(:) represents all the row or column

elements of the vector v

v(1:3) represents the 1st through 3rd


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elements of v

A(:, 2) denotes all the elements in the 2nd

column of A

A(:, 1:4) denotes all the elements in the 1st

through 4th column of A

A(3, :) denotes all the elements in the 3 rd

row of A

A(2:4, :) denotes all the elements in the 2nd

through 4th row of AA(2:3,1:3) denotes all the elements in the 2nd

through 3rd row that are also in the

1st

Practice Set:


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1. Let v = [4, 2, 6, 0, 8] and A = [1, 7, 4, 0;

1, 5, 3, 2; 3, 1, 8, 5]. Determine theoutput of the following:

a. v(5)

b. v(2:4)

c. A(2,4)

d. A(3,2)

e. A(:, 3)

f. A(:, 2:4)g. A(2,:)

h. A(2:3, :)


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i. A(2:3, 3:4)

j. A(2,4) = 6

k. A(1, 5) = 2

l. B = A(:, 4: 1:2)

m. C = [1 1 1]; B(2, :) = C

ARRAY OPERATIONS


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(Element-by-element operations)

Symbols Operations

+ Scalar-array addition

Scalar-array subtraction

+ Array addition

Array subtraction

.* Array multiplication

./ Array right division

.\ Array left division

.^ Array exponentiation

Practice Set:


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1. Given:

A = [2, 3] B = [6, 7] b = 6Find: A + b, A b, A + B, A B, A .* B,

A ./ B, A .\ B, A .^ b, A .^ B

2. Given:

Find: A + B, C + D, A + C, B A, C B

3-21

06-2

7-53

A !

14-0

1050

2-31-

B !

2-14-

6-53C !

000000D !

1-2 43


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3. Given:

Find: 4A, 2B,

4. Given:

Find:AB, BA,CA, A

C

2-4

12A !

65-

43B !

B2

1A

151-

240

1-32

A !

213

02-2

1-5-1

B !

5-4-

6-4

20

C !

MATRIX OPERATIONS


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MATRIX OPERATIONS

Applies the same rule in ordinary matrices

operations in Mathematics.

Note:

1. + and apply the same rule as in element-by-element operations.

2. A/B = A*B-1 , where B-1 is the inverse of B.

3. A^2 = A*A4. AB is not defined .

Example:


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Let M = [1, 3, 6; 2, 4, 0; 5, 8, 9],

N = [4, 2; 6, 3; 1, 5] and O = [2, 2, 1;3, 4, 0; 1, 3, 7]. Find:

a. MN

b. M2

c. Show that MM-1=1

d. Show that M/O = MO-1

Use the command inv(A) to evaluate the

inverse of matrix A.

Determinants


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Use the command det(A) to evaluate the

determinant A.

Examples:

Evaluate the following determinants:

a. c.

b.129-

53-

143

211

1-32

05-2-3

7046-

4-503

4520

Solution of LinearSystems of

Al b i E ti


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Algebraic Equations

Consider a linear system of algebraicequations with n equations and n

unknowns:


Al b i E ti


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Algebraic Equations

In these equations, aij and bi areconstants, and the unknowns are

xi. We can rewrite the equations as:

where


Al b i E ti


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Algebraic Equations

We can find all the unknowns in the vectorx bydoing a few simple matrix manipulations. If we

premultiply both sides of the matrix equation by

the inverse of the A matrix we get:

which gives us our solution.

An Example


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p

Consider the system of 3 equations and 3

unknowns:

An Example


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p

Using MatLab:

>> A=[2 3 -1;-1 2 3;0 1 2]

A =

2 3 -1

-1 2 30 1 2

>> b=[-1 9 5]'

b =

-1

9

5

An Example


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p

Using MatLab:

>> x=inv(A)*b

x =

-11

2

Eigenvalues and Eigenvectors


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Many problems present themselves in terms ofan eigenvalue problem:

In this equation A is an n-by-n matrix, v is a

non-zero n-by-1 vector and is a scalar (which

may be either real or complex). Any value of

for which this equation has a solution is knownas an eigenvalue of the matrix A. It is

sometimes also called the characteristic value.



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The vector, v, which corresponds to this value iscalled an eigenvector. The eigenvalue problem

can be rewritten as

If v is non-zero, this equation will only have asolution if



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This equation is called the characteristic

equation ofA, and is an n

th

order polynomial in with n roots. These roots are called the

eigenvalues ofA. We will only deal with the

case of n distinct roots, though they may be

repeated. For each eigenvalue there will be aneigenvector for which the eigenvalue equation

is true.

Example


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Example

If

then the characteristic equation is

Example


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Example

and the two eigenvalues are

All that's left is to find the two eigenvectors. Let's find

the eigenvector, v1, associated with the eigenvector,1, first.

Example


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Example

so clearly

and the first eigenvector is any 2 element column vectorin which the two elements have equal magnitude and

opposite sign.

where k1 is an arbitrary constant. Note that we didn't

have to use +1 and -1, we could have used any two

quantities of equal magnitude and opposite sign.

Example


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Example

Going through the same procedure for the secondeigenvalue:

Using Matlab


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Using Matlab

A=[0 1;-2 -3]A =

0 1

-2 -3

[v,d]=eig(A)v =

0.7071 -0.4472

-0.7071 0.8944

d =

-1 0

0 -2

Using Matlab


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Using Matlab

The eigenvalues are the diagonal of the "d" matrix;1=-1, 2=-2. The eigenvectors are the columns of the

"v" matrix.

Note that MatLab chose different values for v1,1, etc...,but that the ratio of v1,1 to v1,2 and the ratio of v2,1 to v2,2are the same as our solution.

(MatLab chooses the values such that the sum of thesquares of the elements of the eigenvector equals

unity).


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Sequence & Series

Sequence & Series


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Sequence & Series

Arithmetic SeriesLet An = {1, 3, 5, 7, 9}. Find the value of

Matlab Code:

>> A=1:2:9;

>> s=sum(A)

s =

25

Sequence & Series


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Sequence & Series

Matlab Code:>> n=1:26

>> s=sum((5)./((n+5)+3))

s =

7.0018

Sequence & Series


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Sequence & Series

Matlab Code:

Sequence & Series


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Sequence & Series

Matlab Code:


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Eigenvalues and

Eigenvectors


Algebraic Equations


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Algebraic Equations

Consider a linear system of algebraicequations with n equations and n

unknowns:


Algebraic Equations


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Algebraic Equations

In these equations, aij and bi areconstants, and the unknowns are

xi. We can rewrite the equations as:

where


Algebraic Equations


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Algebraic Equations

We can find all the unknowns in the vectorx bydoing a few simple matrix manipulations. If we

premultiply both sides of the matrix equation by

the inverse of the A matrix we get:

which gives us our solution.

An Example


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Consider the system of 3 equations and 3

unknowns:

An Example


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Using MatLab:

>> A=[2 3 -1;-1 2 3;0 1 2]A =

2 3 -1

-1 2 30 1 2

>> b=[-1 9 5]'

b =

-1

9

5

An Example


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Using MatLab:

>> x=inv(A)*b

x =

-11

2



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g g

Many problems present themselves in terms ofan eigenvalue problem:

In this equation A is an n-by-n matrix, v is a

non-zero n-by-1 vector and is a scalar (which

may be either real or complex). Any value of

for which this equation has a solution is knownas an eigenvalue of the matrix A. It is

sometimes also called the characteristic value.



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g g

The vector, v, which corresponds to this value iscalled an eigenvector. The eigenvalue problem

can be rewritten as

If v is non-zero, this equation will only have asolution if



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g g

This equation is called the characteristic

equation ofA, and is an nth order polynomial in

with n roots. These roots are called the

eigenvalues ofA. We will only deal with the

case of n distinct roots, though they may be

repeated. For each eigenvalue there will be aneigenvector for which the eigenvalue equation

is true.

Example


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p

If

then the characteristic equation is


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Example


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p

so clearly

and the first eigenvector is any 2 element column vectorin which the two elements have equal magnitude and

opposite sign.

where k1 is an arbitrary constant. Note that we didn't

have to use +1 and -1, we could have used any two

quantities of equal magnitude and opposite sign.

Example


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p

Going through the same procedure for the secondeigenvalue:

Using Matlab


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g

A=[0 1;-2 -3]A =

0 1

-2 -3

[v,d]=eig(A)v =

0.7071 -0.4472

-0.7071 0.8944

d =

-1 0

0 -2

Using Matlab


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g

The eigenvalues are the diagonal of the "d" matrix;1=-1, 2=-2. The eigenvectors are the columns of the

"v" matrix.

Note that MatLab chose different values for v1,1, etc...,but that the ratio of v1,1 to v1,2 and the ratio of v2,1 to v2,2are the same as our solution.

(MatLab chooses the values such that the sum of thesquares of the elements of the eigenvector equals

unity).


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MATLAB

GRAPHICS2-D

FIGURE WINDOWS


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MATLAB directs graphics output toa window called figure that is

separate from the command

window. The figure function

creates figure windows.

Example: >>figure

2-D Plotting


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The plot function is used to produce two-dimensional curves, using x- and y-data

matrices specified by the user.

plot(xdata,ydata,clm)

You can plot multiple lines at once, using

pairs of x- and y-data, or triples of x, y.

plot(x1,y1,clm1,x2,y2,clm2)

Examples:


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Examples:

>> x = 0:10 ;>> y = 2*x + 3;

>> plot(x,y)

>> y1 = 4*x 2;>> y2 = x + 2;

>> plot(x,y1,x,y2)

Adding a Grid


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GRID ON creates a

grid on the current

figure

GRID OFF turns off

the grid from thecurrent figure

GRID toggles the

grid state

Color, Line Style and Marker


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The user can specify the color, linestyle and marker of a graph. If not, a bluesolid line without marker symbols, isproduced. Some of them are:

COLORS MARKERS LINE STYLES

y yellow . point -- dashed

m magenta x x-mark -. dashdot

c cyan * star : dotted

r red + plus

g green o circle ^ triangle (up)b blue s square p pentagram

w white d diamond h hexagram

k black v triangle (down)


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Graph a red sine wave

>> x = 0:0.1:2*pi;

>> y = sin(x);

>> plot(x,y,r)

>> grid on


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Try to graph the following using the different line styles,

markers and colors.

1. Graph y = 2cos3x

2. Graph the exponential function, logarithmic function,

inverse trigonometric function, hyperbolic function withappropriate domain.

Adding Additional Plots to a Figure


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By default, plot deletes existing lines and resets all axisproperties when a new line is drawn.

HOLD ON holds the current plot

HOLD OFF releases hold on the current plot

HOLD toggles the hold state


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>> x = 0:0.1:2*pi;

>> y = sin(x);

>> plot(x,y,r)

>> grid on>> hold on

>> plot(x, exp(-x),b:*)

Controlling Viewing Area


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LEFT mouse button zoom in

(x2)

RIGHT mouse button zoomout (x )

Double-click LEFT zoom out

completely

ZOOM ON allows user toselect viewing area

ZOOM OFF prevents

zooming operations

ZOOM toggles the zoomstate

ZOOM using the figure tool bar

GRAPH ANNOTATIONS


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ExampleType the following commands in the command window.

>>x=0:pi/20:2*pi;

>>y=sin(x);

>>plot(x,y,bs-,linewidth,2)

>>hold on

>>y1=cos(x);

>>plot(x,y1,r>:,linewidth,2)

ADDING A TITLE TO A GRAPH


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1. Using the Title Option on

the Insert Menu.

(i) Click the Insert menu inthe Figure window menu bar

and choose Title.

(ii) Enter the text of the label

and click anywhere in thefigure background to close

the text entry box.

There are several ways to add title to a graph:



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2. Using the Property Editor to

Add a Title.

(i) Double click on the axes on

the graph to open the

Property Editor.

(ii) Select the Style panel and

type in the text of your title in

the Title entry box.

(iii) ClickApply.



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3. Using the Title Function.

To add a title to a graph at the MATLAB command prompt,

use the title function.

Example:

>> title(Graph of Sine and Cosine Functions)


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ADDING A LEGEND TO A GRAPH


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There are two ways to add legend to a graph:

1. Using the Legend Option on the Insert Menu.

- Click on the Insert menu and choose Legend.

2. Using the Legend Function.To add a legend to a graph at the MATLAB command

prompt, use the legend function.

Example:>>legend( Sine Function , Cosine Function )

ADDING AXES LABELS TO A GRAPH

There are three ways to add labels to a graph:


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There are three ways to add labels to a graph:

1. Using the Label Options on the Insert Menu.(i) Click on the Insert menu and choose label

option that corresponds to the axes you want

to label.

(ii) Enter the text of the label, or edit the text of

an existing label.

2. Using the Property Editor.

(i) Start plot editing mode.

(ii) Double click on the axes on the graph to

open the Property Editor.(iii) Select the Labels panel and enter the text of

the label in the appropriate text entry box.

(iv) ClickApply.

3 C

ADDING AXES LABELS TO A GRAPH


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3. Using the Label Commands.

To add x, y and z axis labels to a graph use xlabel,ylabel and zlabel functions.

Example:

>>xlabel( x-axis,FontSize,16)

ADDING TEXT ANNOTATIONS TO A

GRAPH


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1. Creating Text Annotations in Plot Editing Mode.

(i) Click on the Insert menu and choose the Text

option or click the text button in the figure

window toolbar.(ii) Position the cursor where you want to add a

text annotation in the graph and click.

(iii) Enter a text.

(iv) Click anywhere in the figure background to

close the text entry box.

ADDING TEXT ANNOTATIONS TO A

GRAPH


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2. Adding Text Annotations with the text or gtextCommand.

To create annotations using text function, the text

and its location must be specified using the x and y

coordinates.Example:

a. >>str1(1) = {Sine Function:}

>>str1(2) = {y=sin(x)}

>>text(3,0.5,str1)

b. >>str2(1) = {Cosine Function: }

>>str2(2) = {y1=cos(x)}

>>text(0.3,-0.6,str2)

ADDING TEXT ANNOTATIONS TO A GRAPH


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SAVING FIGURES


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You can savefigures with SaveorSave Asthrough the Filemenu on theFigure Window.

This will create a.fig file.

Displaying Multiple Plots per

Figure


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g

Format: subplot(m,n,i)

This function breaks the figure window into m-by-n

matrix of small subplots and selects the ith subplot

for the current plot.

Examples:

1. >> subplot(2,2,1) 2. >> subplot(2,3,1)

>> subplot(2,2,2) >> subplot(2,3,2)

>> subplot(2,2,3) >> subplot(2,3,3)>> subplot(2,2,4) >> subplot(2,3,4)

>> subplot(2,3,5)

>> subplot(2,3,6)

MULTIPLE PLOTSPER FIGURE


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subplot(2,2,i)

where i = 1 to 4subplot(2,3,i)

where i = 1 to 6

1 2

3 4

1 2 3

4 5 6

Example

Type the following commandsin the command window.


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>>x=0:pi/20:2*pi;>>y=sin(x);

>>subplot(1,2,1)

>>plot(x,y,bs-,linewidth,2)

>>y1=cos(x);>>subplot(1,2,2)

>>plot(x,y1,r>:,linewidth,2)


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BASIC PLOTTING

COMMANDS

ezplot

l t i t f ti l tt f


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ezplot is an easy to use function plotter for

algebraic and transcendental functions,

parametric equations, implicit and explicit

functions.

ezplot(f)

plots the expression f = f(x) over the

default domain -2T < x < T

ezplot(f,[a,b])

plots f = f(x) over a < x < b

Examples:

>> subplot(2,3,1)


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>> ezplot(cos(x))

>> subplot(2,3,2)

>> ezplot(cos(x),[0, pi])

>> subplot(2,3,3)

>> ezplot(1/y-log(y)+log(-1+y)+x-1)

>> subplot(2,3,4)

>> ezplot(x^2+y^2-4)

>> subplot(2,3,5)

>> ezplot(x^3+y^3-5*x*y+1/5,[-3,3])>> subplot(2,3,6)

>> ezplot(sin(t),cos(t))


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POLAR CURVESPolar in polar coordinates can be created using the


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p g

polar(t,r,S) function, where t is the angle vector in radians, ris the radius vector, and S is an optional character string

describing the color, marker symbol, and/or line style.

Example

>>t=linspace(0,2*pi);

>>r=sin(2*t).*cos(2*t);

>>polar(t,r)

>>title(Polar Plot)

HISTOGRAMHistogram illustrates the distribution of values in a vector.

hist(y) draws a 10 bin histogram for the data in vector y


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Example>>x=-2.9:0.2:2.9; %specify the bins to use

>>y=randn(5000,1);%generate 5000 random points>>hist(y,x) %draw the histogram

>>title(Histogram of Gaussian Data)

hist(y) draws a 10-bin histogram for the data in vector y.

hist(y,n), where n is a scalar, draws a histogram with n bins.hist(y,x), where x is a vector, draws a histogram using the

bins specified in x.

PIE CHARTStandard pie charts can be created using the pie(a,b) function,


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where a is a vector of values and b is an optional logical

vectors describing a slice or slices to be pulled out of the pie

chart. The pie3 function renders the pie chart with a 3-D

appearance.

Example

>>a=[0.5 1 1.6 1.2 0.8 2.1];>>subplot(1,2,1)

>>pie(a,a==max(a)); %produces

chart a and pull out the biggest

slice.

>>subplot(1,2,2)>>explode=[1 0 0 0 0 0 ];

>>pie(a,explode) % Which part is

pulled out?

BAR GRAPHSBar graphs display vector or matrix data. By default, a bar

graph represents each element in matrix Bars in a 2-D graph


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graph represents each element in matrix. Bars in a 2-D graph,

created by barfunction, are distributed along the x-axis witheach element in a column drawn at a different location. All

elements in a row are clustered around the same location on

the x-axis.

Example

>> y =[5 2 1; 8 7 3; 9 8 6; 5 5 5;4 3 2];

>> subplot(1,2,1)

>> bar(y)

>> subplot(1,2,2)

>>bar3(y)

SPECIALIZED PLOT COMMANDS


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The following are some other specialized plotcommands:

area

pie3

rose

stairs

stem3

quiver

compass

feather


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MATLAB

GRAPHICS3-D

ezsurf

ezsurf(f) creates a graph of f(x,y), where f is a


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string that represents a mathematicalfunction of two variables, such as x and y.

Example:

>> subplot(1,2,1)

>> ezsurf('x^2+y^2')

>> subplot(1,2,2)

>> ezsurf('x^2-y^2')

ezmeshezmesh(f) creates a graph of f(x,y), where f is a

symbolic expression that represents a mathematical


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symbolic expression that represents a mathematical

function of two variables, such as x and y.

Example:

>> subplot(1,2,1)

>> ezmesh('x^2+y^2')

>> subplot(1,2,2)

>> ezmesh('x^2-y^2')

OTHER PLOTTING COMMANDS


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mesh

contour

contour3

waterfall

surf

plot

FIGURE WINDOW TOOLS


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ENJOY MATLAB GRAPHICS !


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Introduction to MATLAB


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Programming

Programs are contained in m-files

- Plain files not binary files produced by word


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Plain files not binary files produced by word

- Files must have an .m extension m-files must be in path

When you call an M-file function from either the command lineor

from within another M-file, MATLAB parses the function andstores it

in memory.

This prevents MATLAB from having to reparse a function eachtime you call it during a session.

The compiled function remains in memory until you clear itusing the

CLEAR command, or until you quit MATLAB.

TWO TYPES OF M-FILES1 S i t


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1. Scripts

- Automate long sequences of commands

2. Functions

- provide extensibility to MATLAB. They allow you to add newfunctions to the existing functions

Script M-files- Standard ASCII text files


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- Standard ASCII text files

- Contains series of MATLAB expressions stored together in apredefined sequence.

- Saved with an .m extension and are called by simply typingthe filename without the extension in the command window.

Downside ofusing Scriptfiles:

- All variables created are added to the workspace.

- Variables already existing in the workspace may be overwritten

Functions M-files- Functions are subprograms


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Functions are subprograms

- Use input and output parameters to communicate with otherfunction

Differences between Script & Function M-files:

Structural Syntax - must contain keyword FUNCTION at thebeginning of the first line

Function Workspaces, Inputs & Outputs- A function does not work with variables in the base MATLAB

workspace.

As a result, all information to be transferred between theMATLAB workspace and a function must bepassedas inputs

& outputs.

Structure of a Function M-file


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function y = mean(x)

% MEAN Average or mean value.

% For vectors, MEAN(x) returns the mean value.

% For matrices, MEAN(x) is a row vector

% containing the mean value of each column.

[m,n] = size(x);

if m == 1

m = n;end

y = sum(x)/m;

Keyword: function Function Name (same as file name .m)

Output Argument(s) Input Argument(s)

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MATLAB

Code

output_value = mean(input_value) Command Line Syntax

Inputs to Functions :

Text Input and Output


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Inputs to Functions :

input function can be used to prompt the user for numeric or

string input

Input parameters to functions are preferred

Text output from functions:

disp function for simple output

fprintffunction for formatted output.

Prompting for User InputTh i t f ti b d t t th f i


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The input function can be used to prompt the user for numeric

or string input.

Syntax: entry = input (TextDisplay)

entry = input (TextDisplay, s )

Examples:>> x = input (Enter a value for x)

Enter a value for x %serves as a prompt on the screen thatsignifies that MATLAB is awaitingfor user

response.

>>yourName = input (Enter your name,s);%like the previous

command but will return the entered string as a

text variable rather than as a variable name or

numerical value.

Text Output with disp- Output to the command window is achieved with the disp


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Output to the command window is achieved with the disp

functiondisp Simple to use. Provides limited control over appearance of

output.

Syntax: disp (variable)

disp (text string)

Examples:

>>poly_roots = [ 1.2400 2.5600 5.6400 ]

>> disp ('The roots of the cubic polynomial are')

The roots of the cubic polynomial are

>> disp (poly_roots)

1.2400 2.5600 5.6400

>> disp ('The roots of the cubic polynomial are') , disp (poly_roots)

The roots of the cubic polynomial are

1.2400 2.5600 5.6400

disp Simple to use. Provides limited control over appearance ofoutput.


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Syntax: disp (variable)disp (text string)

Examples:>>poly_roots = [ 1.2400 2.5600 5.6400 ]>> disp ('The roots of the cubic polynomial are')

The roots of the cubic polynomial are>> disp (poly_roots)

1.2400 2.5600 5.6400>> disp ('The roots of the cubic polynomial are') , disp (poly_roots)The roots of the cubic polynomial are

1.2400 2.5600 5.6400

Relational and Logical Operators

Operators Description


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p p

< Less than

> Greater than

= Greater than or equal to

~= Not equal to

== Equal to

& AND

| OR

~ NOT

xor Exclusive OR

-Will output 1 for true conditions and 0 for false conditions

- Can be used with scalars and arrays

Other Logical built-in functions

Operators Description


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all (A)

Returns true if all elements in a vectorA aretrue.

Returns false if one or more elements arefalse

any (A)

Returns 1 (true) if any element in a vectorA

is true (nonzero).Returns 0 (false) if all elements are false.

find (A)IfA is a vector, returns the indices of thenonzero elements

find (A>d)IfA is a vector, returns the address of theelements that are larger than d (anyrelational operator can be used)

SUMMARY

Relational operators involve comparison of two values.


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The result of a relational operation is a logical (True/False)value.

Logical operators combine (or negate) logical values to

produce another logical value.

Example : Analysis of temp dataThe following were the daily max temp in (F) in Washington during

the month of April,2002.


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the month of April,2002.

58 73 73 53 30 48 56 73 73 66 69 63 74 82 84 91 93 89 91 8059 69 56 64 63 66 64 74 63 69.

Use relational logical operators to determine the following:a. The number of days the temp was above 75b. the number of days the temp was between 65 and 80

Examples:

>> r = [ 8 12 9 4 23 19 10 ]


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[ ]

r =

8 12 9 4 23 19 10

>>s=r>t=r(s) % use s for addresses in vector r to create a vector t

t=

8 9 4 10

>> w=r(r


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Example


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function areacircle

r=input('Give radius of circle: ');

a=pi*r^2;

fprintf('The area of circle with radius %.2f is %.6f\n',r,a);

end

Flow control - selection


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The if-elseif-else construction

if

elseif

else

end

if height>170disp(tall)

elseif height


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teenager, adult and senior given as the input of the variableage.

function check_age

age=input('Enter your age: ')

if age

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