metotos numericos

8/17/2019 metotos numericos

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The purpose of the course is to provideengineers with plenty of mathemacial tools tosolve all di erent types of equations .

Issam Almustafa

Numerical analysis (Numerical methods


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35 !"ams35 #inal !"am$% wor&shops ( in class'

% assitence

)estirictions of the

course


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*ell phone

N+ ,-! +# *! /0+N!1,)IN2 * A--

+N ! A4- 1A +, 0A6! T+ !A6! +,) *!/0+N! +N 4 1!-7+T0!)8I-! +, 0A6! N+)I20T T+ 4A7! T0! ! A4


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They have start and end date9 if not su:mittedon time you lose the assignment points .

Assignments and

wor&shops


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!"ams!"am ; 8ritten*hapter only.

session <

!"am$ ; writtensession =*hapter $ ;

a' $% pts for writing a program for one of the rooting >ndingmethods ( done in class with e"planation '.

:' % pts for ma&ing :isection method in !"cel ( in class 'c' ?% pts written includes two pro:lems .

!"am 3 ;-ession 5

*hapter 3 9 < ;

-olving a set of pro:lems from *hapter 3 which should :edelivered printed .


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8henever you ma&e a measurement9 the num:er of meaningful

digits that you write down implies the error in the measurement.

#or e"ample if you say that the length of an o:@ect is %.cant >gures

In other words we are not sure of the was rouded up or down .

If we would round it again to two decimal places to :e %.


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)ules of signi>cat>gures


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c notation ascant.

8hy is that G

*Due to scientifcnotation


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%.%%5 C 5 quantity is 5 9 magnitude H$showing how small it is .

5%% C 5 the quantity is 5 the $ show how :igit is

!"amples


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Accuracy Notations• 3.14159 is accurate to the hundred-thousandths place

• 1000 is accurate to the thousands place

• 1000.0 is accurate to the tenths place

• 0.00035 is accurate to the hundred-thousandths place• 0.000350 is accurate to the millionths place (note the extra zero)• 1006 is accurate to the units place

• 560 is accurate to the tens place• 560 . is accurate to the units place (note the decimal point)• 560.0 is accurate to the tenths place


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as )eglas de )odondeoH4enor que 59 se de@a el dKgito precedente intacto

!@emplos; -upongamos que se desea redondear los siguientes nLmeros a Tres cifras signi>cativas;

5. $3 M 5. $

$H4ayor que 59 se aumenta una unidad el dKgito precedente.

. $? M . 3

3H ,n 5 seguido de cualquier dKgito diferente de cero9 se aumentaunaunidad el dKgito precedente.

=.


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CASE A:In rounding o numbers, the last fgurekept should be unchanged i the frstfgure dropped is less than !"

#or e$ample, i onl% one decimalis to be kept, then &"'(( becomes&"'"

CASE ):In rounding o numbers, the last fgure

kept should be increased b% i thefrst fgure dropped is greater than !"

#or e"ample9 if only two decimals areto :e &ept9 then .< ?$ :ecomes

.


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!"ample

./0)E1

.umber odecimalplacesdesired

numberbecomes to

desireddecimalplace

&"'(( .<

&"'23( $ .


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• Round 742,396 to four, three, and two significant digits:

742,400 (four significant digits) 742,000 (three significant digits) 740,000 (two significant digits)

• Round 0.07284 to four, three, and two significant digits:

0.07284 (four significant digits) 0.0728 (three significant digits)

0.073 (two significant digits) • Round 231.45 to four, three, and two significant digits

231.5 (four significant digits) 231 (three significant digits)

230 (two significant digits)

4ore !"amples ;


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How do ou round when the !i"e ou a #unch o$ num#ers to add% &ou would add (or

su#tract) the num#ers as usual' #ut then ou would round the answer to the same decimal

place as the least-accurate num#er.• Round to the appropriate number of significant digits:

13.214 + 234.6 + 7.0350 + 6.38

oo in! at the num#ers' * see that the second num#er' +34.6' is onl accurate to the tenths

place, all the other num#ers are accurate to a !reater num#er o$ decimal places. o m

answer will ha"e to #e rounded to the tenths place

13.+14 / +34.6 / .0350 / 6.3 2 +61.++90

oundin! to the tenths place' * !et op ri!ht © 2000-2011 All Rights Reserved

13.+14 / +34.6 / .0350 / 6.3 2 261.2

Rounding Addition


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Rounding Multiplication

How do ou round' when the !i"e ou num#ers to multipl (or di"ide)% &ou would

multipl (or di"ide) the num#ers as usual' #ut then ou would round the answer to the same

num#er o$ si!ni$icant di!its as the least-accurate num#er.• Simplify, and round to the appropriate number of i!nifi"ant di!it #

16.+35 0.+1 5

irst' * note that 5 has onl one si!ni$icant di!it' so * will ha"e to round m $inal answer to

one si!ni$icant di!it. 7he product is

16.+35 0.+1 5 2 17.614975

ince * can onl claim one accurate si!ni$icant di!it' * will need to round 1 .6149 5 to +0'

which is accurate to one si!ni$icant di!it.

16.+35 0.+1 5 2 20


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Another e"ample

• $ind the produ"t of 0.00435 and 4.6 to the appropriate number of di!it .

irst * multipl

0.00435 4.6 2 0.02001

oo in! at the ori!inal num#ers' * see that 4.6 has onl two si!ni$icant di!its' so * will

ha"e to round 0.0+001 to two si!ni$icant di!its. 7he + is the $irst si!ni$icant di!it' so the 0

$ollowin! it will ha"e to #e the second si!ni$icant di!its. *n other words' * must report the

answer as #ein!0.00435 4.6 2 0.020

7he answer should not #e 0.0+' #ecause 0.0+ has onl one si!ni$icant di!it, namel ' the 8+8. 7hetrailin! zero in 0.0+0 indicates that 8this is accurate to the thousandths place' or two si!ni$icantdi!its8' and is there$ore a necessar part o$ the answer.


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/ractica


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)eview last wee&/recision and accuracyIntro to !rror)elative !rror and a:solute errors!"ercises pages ( 9 = 9 9 3 9


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Precission and Accuracy


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8hat is an errorG

!rrors are the result of calculating appro"imation

or "arious reasons we ma not #e a#le to o#tain the exact "alue and we must #e content with an

approximation to the true "alue.

ome o$ these reasons are the $ollowin!.

• *mper$ect ph sical measurements.

•

ropa!ation o$ ound-o$$ errors

. OOO The value measured is called appro"imation relative to the valuecalculated according to a theory which is considered the true one.OOO


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!valuating the mean when

given a series of values

2


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6ength 7mm8 De iation rom

mean

$.

%.$

.5


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)elative and A:solute

error The a:solute error9 δ 9 is the magnitude of theerror

($' δ C R x H x a R.


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serie de Taylor de!"ample $ ;


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The more terms we add the more accurate our appro"imation is 9thus we can calculate two values which we will call present valueand previous value in order to estimate the error :etween them

/revious value

/resent value

1≈ x

e 15.0

≈e

:11

x xe +≈ 5.1

5.0≈e


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! C ( .5H 'S .5 %% C 33.3

Aplicando la formula assuming that.5 is the unrounded value and is

the rounded value (appro"imated asit drops o the other terms'.


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!"ercises on pages ( 9 = 9 9 3 9


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!cuaciones continua!cuaciones descontinua

#undamentos

matematicos


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H cuando la funcion se puede di:u@ar sinquitar la mano del papel .

$H la funcion es continua en un punto cuandoen este punto la funciUn es de>nida .

/asos para sa:er la

continuidad


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H que es una funcion continua$H e@emplo3H que es una funcion no continuaca raiD G

H que signi>ca numero imaginario?H raiD de ecuaciones con numero imaginario

#undamentos matematicos


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#unciUn continua

O 1i:@uar y mostrar


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!@emplo 3 ;

O a funcion es continua si en un punto " C c ellado iDuerdo es igual al lado derecho .

O !s continua cuando es de>nida en un puntodado


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#uncion descontinua

O -i limitamos " para sea siempre mayor de %entonces tendramos una funcion continua.

!@emplo ;


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!@emplo$ ;

No tiene raiD


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-emana <

Arreglar un error de continuidad de la semana pasada( disculpa ' en $<

*omentarios9 me avisan si vamos rapido o mandan suscomentarios a mi correo si lo pre>eren asi

ssam.almustafaWgmail.com4odelos de pro:lemas (cuales son ' y sus pasos

)epaso so:re el concepto del raiD

2enerar gra>cas usando !"cel para veri>car el raiD

IntroducciUn al metodo :isecciUn y su diagrama de Xu@o

taller


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4odelos de pro:lemas y pasos pararesolverlas

H Te da la ecuaciUn y un punto9 mas el valor Y" te pide ;

H valor real$H valor apro"imado

3H valor de error a:soluto y relativo

$H Te da dos puntos mas Y" y te pide ;

H valor apro"imado caundo " C c

3H Te da una gra>ca y la funciUn con dos valores de "te pide ;

H calcular el error a:solulto $H muestra en la grai>ca el error


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4odelos de pro:lemas y pasos pararesolverlas

ca y pide ;

marcar los puntos adonde la funcion tiene raiD .

5H Te da una ecuacion de serie de taylor y pide ;

H calcular el primer valor apro"imado hasta nterminos $H calcular el segundo valor apro"imado hasta nZmterminos 3H evaluar el error por cientoH Te da una ecuaciUn en forma

H calcular el raiD de la funcion si e"iste

?H Te da una ecuaciUn en forma

H calcular el raiD imaginario


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a funciUn tiene raiD cuando ;

H!n este punto es continua

$H cruDa el e@e "

!ncontrar el raiD

!@emplo ;

H $" Z C %$H

O2ra>ca

n ro ucc n a me o o e


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n ro ucc n a me o o e:isecciUn

Calculo 9umano :H /asa al otro lado el H< .

$H Tomar el raiD para los am:os lados.

El Calculo 9umano en la computadoraseria asi :

/orque tendramos que escri:ir mas codigo y para cadaecuacion otro coidgo .

H veri>que cuanto terminos hay ( mas terminos mascodigo )$H veri>que si es cuadrado ( si no GG[['3H veri>que si el signo de segundo termino es negativo opostivio


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H 8hat mathematical modeling G$H inear and polynomial (@ust quadratic in this course '

regression .3H *reate scatterplot in !"celts the data usingthe formulas to >nd the coe\cients Ja] and J:].5H #ind coe\cient for quadratic regression using anonline application .

H wor&shop ; solve one pro:lem of linear regressionand solve a quadratic regression using the !"cel.-olve one pro:lem of Newton s interpolation second

order

Tody


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There are many types of mathematicalmodeling 9 that one we are interested in is

called Linear model which is :asically a wayto descri:e a set of data in a form of a linearequation to :e a:le to evaluate a point that isnot in the data set .

4athematical

modeling


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!"ample ;

8rite equation ; solve it

/l i !" l


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/lot in !"cel

1ata plot

inear regression /olynomialregression of second


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Another e"ample

0ow does !"cel


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H '

0ow does ! celcalculate the

functionG


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H pro:lem ; -olve on paper

practice


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-olve in !"cel and calculate the answer

!"ample $


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-olve in !"cel

!"ample 3

Newton s 1ivided


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This method is used to connect data pointsthat lie e"actly on a graph of a function .

The function can :e linear ; a"Z : 9 orpolynomialaZ: or even higher degrees .

Newton s 1ivided1iference

Interpretation

!" l


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!"ample ;

E$ample


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E$ample The upward velocity of a roc&et is given as afunction of time in Ta:le (#igure 3'.

Ta:le 6elocity as a function of time.t (s' v(t' (mSs'% %

% $$?.%<5 3 $.?

$% 5 ?.35$$.5 %$.=?3% =% . ?

1etermine the value of the velocity at t C seconds using >rst order polyinterpolation :y Newton s divided di erence polynomial method.


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d


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Vuadratic

interpolation,sed when we want to get a more accurate value:etween two points or more .


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-ecant method


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