Week01_Introduction to Numerical (1)

8/19/2019 Week01_Introduction to Numerical (1)

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CHE 555NUMERICAL METHODSAND OPTIMIZATION


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WEEK 1

Introduction to Numerical Methods Mathematical modeling

Approximation and round off errors

Truncation errors and Taylor Series 2


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At the end of this topic, the students will be able:

• To describe numerical techniques as compared

to analytical methods

• To use Taylor series expansion to approximate

a function• To perform error analysis associated with

numerical methods

LESSON OUTCOMES


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4

Why Numerical Method?

• Could handle large systems of equations,

nonlinearity and complex geometries that is notcommon

• It pro!ide approximate solutions to many of theengineering pro"lems


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• #o$erful analysis tool in pro"lem sol!ing andunderstanding pro"lem in mathematical language

• Techniques "y $hich mathematical pro"lems areformulated, so that they can "e sol!ed $ith

arithmetic operations• The role of numerical method in sol!ing engineering

pro"lem%

5

What is Numerical Method&cont'

PROBLEMFORMULATION

Fundamental laws are usedto deelop mathematical

equations that can representthe specific problem

SOLUTION

!uitable numerical methodsare then selected to sole

the mathematical equations

INTERPRETATION

The results obtained canthen be used to

predict"analy#e"understandthe specific problem better


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$

%s the use of mathematics to• &escribe real world phenomena• %nesti'ate important questions about the obsered world• (xplain real world phenomena

• Test ideas• )a*e predictions about the real world

• The real world refers to• (n'ineerin' +hysics• +hysiolo'y (colo'y• ildlife mana'ement -hemistry• (conomics !ports• (tc

Mathematical modeling


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• A mathematical model is represented as a functionalrelationship of the form

.


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• (ependent !aria"le

)"ser!ed "eha!iour*state*phenomenon of a systemCharacteristic that reflects "eha!iour or state of the system

ie y, f(x), f(t)

• Independent !aria"le

(imension that determine a system ie time, t , x

• #arameter

+uantity that ser!es to relate to functions and !aria"les

eflecti!e of the system's properties or composition -ex% T, #, p./

• 0orcing functions

1xternal influence that acts on system ie acceleration gra!ity, g

/


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(xample

0

Apply 1ewtons second law, F 3 ma

And also can write as

(q relates a linear position x

dependent ariable6 to the appliedforce, F forcin' function6 and thetime, t independent ariable6 Themass, m is the only parameter inthe aboe model

F dt

xd m

or

F dt

dvm

=

=

2

2

dt

dxv =


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1xample of mathematical modeling

78


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77

(xample Assume that interested to predict the elocity of the fallin' parachutist with time

9se fundamental *nowled'e to find amathematical equation correlates theelocity to the arious forces actin' on theparachutist

Ne$ton's 2nd la$ of Motion

3the time rate change of momentum of a bodyis equal to the resulting force acting on it 4

The model is formulated as

F = ma

F5net force acting on the "ody -N/

m5mass of the o"6ect -7g/

a5its acceleration -m*s2/


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72

)odel relates acceleration of fallin'obect to the forces actin' on it,

differential equation6

m

F

dt

dv

dt

dv

=

= aonaccelerati

resistanceairof forceup$ard

gra!ityof forcedo$n$ard

=

=

+=

U F

D F

U F

D F F

cvU F

mg D F

−=

=

vm

c g

m

cvmg

dt

dv−=

−=


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7;

(xact or analytical solution: it exactly satisfies the ori'inal equation

dependent ariable

independent ariable

forcin' functions

par ameter

t,s ,m"s6

8 888

2 7$48

4 2...

$ ;5$4

/ 4778

78 44/.

72 4./.

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Analytical solution of the parachutist

∞


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74

9nfortunately, there are many mathematicalmodels that cannot be soled exactly

1umerical solution that approximates the exact solution


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75

The use of finite difference to approximate thefirst deriatie of with respect to t

Approximate ornumerical solution

/-8

/-/8

-

8

/-/8

-

it

ii

it

it

ii

it

it

vmc g

t t

vv

t t

vv

t

v

dt

dv

vm

c g

m

cvmg

dt

dv

−=

−

−

−

−

=

∆

∆≅

−=

−

=

+

+

+

+

/-8/-/-/- 8 iit t t

t t vm

c g vv

iii

−

−+= +

+


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7$

t,s ,m"s6

8 888

2 70$8

4 ;288

$ ;0/5

/ 44/2

78 4.0.

72 400$

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1umerical solution

-omparison between the exact andnumerical solution

∞

/- 8/-/-/- 8 iit t t t t vmc g vv

iii−

−+=

++


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Approximation and oundoff 1rrors

• Significant figures

0/

0/80

8880/

Num"ers to "e used in confidence

2 significant figures



7.


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7/

Important of signifian! fig"r!s in n"m!ria# m!t$o%s&

• 1umerical methods yield approximate results, therefore,need to deelop criteria to specify the confident inapproximate result

• Althou'h quantities such as π, e, or


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• Accuracy and Precision

70

Ina"rat! ' impr!is!

a"rat! ' pr!is!Ina"rat! ' pr!is!

a"rat! ' impr!is!

Inr!asing a"ra(

I n r ! a s i n g p

r !

i s i o n

>ow closely a computedor measured alue a'reewith the true alue

>ow closely indiidualcomputed or measuredalues a'ree with eachother


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• Error definitions

True !alue

Approximation

!alue

1rror

True !alue 5 1rror : Approximation !alue

28

Tr"nation !rrors ? resultwhen approximations areused to represent exactmathematical procedures

Ro"n%)off !rrors ? resultwhen numbers hain' alimited si'nificant fi'ures

are used to representexact numbers


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• εt designates true percent relati!e error

True !alue 5 error : approximation !alue

True error - E t /5 true !alue ; approximation

27

76

26

;6


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Calculation of errors

True !alue of length of a "ridge is 8


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.o$e!er , in actual situation, true !alue is rarely a!aila"leTherefore, need to estimate the true !alue approximation

In numerical method, iterative approach is used to

compute ans$er, in $hich error is estimated as the

difference "et$een pre!ious and current approximations

The signs of error can "e negati!e or positi!e,

A"solute !alue of error, εa need to "e lo$er than

prespecified percent tolerance, ε s

n is significant figures

2;

46

56


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1rror estimates for iterati!e method

Suppose that $e ha!e exponential function as,

Starting $ith the simplest !ersion, e x58, addterms to estimate e!" Compute true -εt/ and

approximate error -εa/ after each term is added

until εa falls "elo$ ε s , conforming to @significant figures Note that true !alue of e!"

is 89B28

24


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• Ans$er

25


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• Round-off errors

ln 2 5


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• )ther example of roundoff error

2.


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Truncation errors and Taylor series

2/

• Truncation errors

• Truncation error is the discrepancy introduced by the fact thatnumerical methods may employ approximations to represent

exact mathematical operations and quantities

• Truncation error are errors resulted from usin' an approximationin place of an exact mathematical procedure

• The difference between the calculated alue usin' exactmathematical equation and approximation mathematicalequation


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Zero order

First order

Second

order

nth order

20

+roides a means to predict a function alue at one point in terms of the

function alue and its deriatie at another point

• Taylor series


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;8

$6

.6

Taylor series by definin' a step si#e h = x i+1 - x i

8/8-

/F8-

/- ++

+=

nn

n hn

f #

ξ


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;7


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;2

Get say $e truncated the Taylor series expansion after HeroEorder term to

yield

emainder term, n for Hero order !ersion

Get truncate the remainder itself,

This result is still inexact "ecause neglected second and higher order terms

emainder term, n, accounts for all terms from -n:8/ to infinity

It also usually expressed as%

• Remainder for the Taylor series E!ansion

/-/- 8 ii x x f f ≅

+

F@F2

III @

/-@

2/-

/-< +++≅ h f

h f

h f # iii

x x

x

h f #i x /-<

I≅

/- 8+= nn h$ #

8

/8-

/F8-/- +

+

+= n

n

n hn

f # ξ


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Remainder for the Taylor series E!ansion

;;


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;4

Alternati!e simplification that tranforms the approximation into an

equi!alence "ased on graphical insight derivative mean%value theorem states that if a function f(x) and its

deri!ati!e are continous o!er inter!al from xi to xi&',

there exist at least one point on the function that has a slope, designated "y

f(), parallel to line 6oining f (xi ) and f(xi&' )

Thus,

So,

ero order ersion

First order ersion


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• "umericaldifferentiation

0or$ard finite di!ided difference

Jac7$ard finite di!ided difference

Centered finite di!ided difference

;5


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;$

0or$ard finite di!ided difference approximation of first deri!ati!e

Jac7$ard finite di!ided difference approximation of first deri!ati!e

Centered finite di!ided difference approximation of first deri!ati!e

Where,746

756

7$6

7.6

Where,

Where,

$(h)h

f ) f*(x

h #

h ) f(x ) f(x ) f*(x

ii

iii

+∇

=

+−

= −88

8−−= ii x xh

88 −+ −=−= iiii x x x xh


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;.

9se forward and bac*ward difference approximations of O(h) and a centereddifference approximation of O(h2 ) to estimate the first deriatie of

at x 3 85 usin' a step si#e h 3 85 @epeat usin' h 3 825 Also calculate the

true percent relatie error for each approximation

E*amp#! +E*amp#! ,-, +t!*t .oo/00

282D


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;/

E*!ris!

9se forward and bac*ward and a centered difference to estimate the first

deriatie of the function

at x 3 85 usin' a step si#e h385 @epeat usin' h 3 825 Also calculate the

true percent relatie error for each approximation

Ans&

h385F&):7525 470BC&):8/.5 7/$8B

-&):7288 77$;Bh3825F&):72$/.5 7/82BC&):804;.5 7227B-&):778$25 207B

C


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;0

• Second for$ard finite difference approximation of higher deri!ati!es

• Second "ac7$ard finite difference approximation of higher deri!ati!es

• Second centred finite difference approximation of higher deri!ati!es

• >i'her deriaties


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• Error !ro!agation

This section is to study how errors in numbers canpropa'ate throu'h mathematical functions %f we multiplytwo numbers that hae errors, we would li*e to estimatethe error in the product

%f a function f is dependent on

a6 a sin'le independent ariable x : fx6

b6 two independent ariables x and y : fx, y6

c6 seeral independent ariables x7, x2, x;, ,xn : fx7 , x2 ,, xn 6


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47

Function of a sin'le ariableDet x be the true alue and

xE be an approximate alue of x

Then, T!( for fx6 computed near fxE6 is 'ien by

Truncatin' after the first deriatie term and rearran'in' the remainin' terms

to 'ie

where

(q276 proides 2 capabilities:

7 to approximate the error in fx6 *nowin' its deriatie

2 to approximate the error in the independent ariable x

276

K/-2

K/-IIK/-K 2 +−+−+= x x

x f x x ) f*(x f(x+) f(x)

KK/I

K/K/-I

x(x f f(x+)

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

∆=∆

−≈−

x x x x

x f x f (x f

t !aria"leindependenof errortheof estimateanisKK

functiontheof errortheof estimateanisK/-/-K/I

−≡∆

−≡∆


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E*amp#!

ien a alue of xE 3 25 with an error of GxE 3 887, estimate the

resultin' error in the function, fx63x;

So#"tion

Hr the true alue lies between 754;.5 and 75/725 %n fact, if x ?240,fx6 could be 754;/2 and if x ? 257, it would be 75/7;2The first order error analysis proides a fairly close estimate of the true error

8BCD


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4;

E*!ris!

Inowin' a alue of xE 3 28 with an error of ΔxE 3 887, estimate the resultin'

error in the function

fx6 3 85x;J87x2K8/xJ8.

Ans: f286345 ± 88$4


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Function of a more than Hne ariable

226

2;6

Refer section #$%$%for eam!les


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• elati!e error

• Condition num"er

Refer section #$%$&

for eam!le


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• Condition no equals 8 indicates that function's

relati!e error is identical to the relati!e error inx

• Condition no greater than 8 indicates relati!e

error is amplified• Condition no less than 8 indicates relati!e

error is attenuated

• 0unction $ith !ery large !alues are said to "eill%conditioned

4$


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Total numerical errors 5 truncation error : round off error

4.

oundoff error L "y increase no of significant figures orreduce no of computation in analysis

Truncation error L "y decreasing step siHe -h/ or increase

no of computation in analysis

• Total "umerical Error


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Control numerical error

a!oid su"tract 2 nearly equal num"ers to

a!oid loss of significance

se Taylor series for truncation and roundoff

error analysis

#erform numerical experiments

E repeat computation $ith different step siHe or methodand compare results

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Week01_Introduction to Numerical (1)

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