lecture 3 polynomial interpolation for upload

7/24/2019 Lecture 3 Polynomial Interpolation for Upload

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CURVE FITTINGLECTURE 3

2/20/16 CS 131: NUMERICAL METHODS - PHILIPCHRISTIAN ZUNIGA



MOTIVATION

Data i !!a""# $i%&' a a (i)*&t& &t +, %a"!& +, a )+'ti'!+! (

!!a" *+"& tat +'& ,a)& i' ti it!ati+' i &' a&( t+ &4t*a)'+ %a"!& i $i%&' i' tat +i't.

2/20/16 CS 131: NUMERICAL METHODS - PHCHRISTIAN ZUNIGA



C!*%& Fitti'$15 G&'&*at& a i'$"& )!*%& tat i"" ,+""+ t& att&*' ta&' a a $*+i't i"" & +i't +' t& )!*%&.

25 G&'&*at& a i'$"& )!*%&/+* a &*i& +, )!*%& tat i"" )+'tai' &a)$*+!.




Li'&a* R&$*&i+' Li'&

Fit a "i'& tat i"" &t a*+4iat& t& (ata &t.




Li'&a* R&$*&i+' Li'&




E**+* +, t& Li'&a* R&$*&i+' LI'&




R&$*&i+' C+&8)i&'t

i at ti& )a""&( t& *&$*&i+' )+&8)i&'t. T& )"+&* * it i t+ 1 t& t& (ata i. * 91 i"i& tat t& *&$*&i+' "i'& t+ta""# )+'tai' a"" t& (




R&$*&i+' C+&8)i&'t i at ti& )a""&( t& *&$*&i+' )+&8)i&'t. T& )"+&* * it i t+ 1 t& +t& (ata i. * 91 i"i& tat t& *&$*&i+' "i'& t+ta""# )+'tai' a"" t& (a




Fai"!*& +, Li'&a* R&$*&i+'




P+"#'+ia" R&$*&i+'& )a' &4t&'( t& t&)'i;!& & !&( i' t& "i'&a* )a& t+ (&*i%& a ;!a(*ati) *&$*&i+' <a'( ,+* i$&* +*(&* *&$*&i+'5

Ti i'%+"%& +"%i'$ a #t& +, 3 &;!ati+' a'( 3 !''+'.




P+"#'+ia" R&$*&i+'Fi'( a ;!a(*ati) ,+*!"a tat )a' =t t& ,+""+i'$ (ata.




P+"#'+ia" R&$*&i+'




I't&*+"ati+'

Gi%&' '>1 +i't t&*& &4it +'"# +'& 't (&$*&& +"#'+ia" tt*+!$ +i't. I't&*+"ati+' i t& *+)& +, i(&'ti,#i'$ t& &)i=)

tat a& t*+!$ a"" +, t& '>1 +i't.




I't&*+"ati+'

Li'&a* I't&*+"ati+': Gi%&' t& ,+*!"a ,+* t& "i'& )+'tai'i'$ t&&i:

Fi'( t& "i'& )+''&)ti'$ t& ,+""+i'$ +i't: <125 <3 ?5




I't&*+"ati+'

@!a(*ati) I't&*+"ati+': T& ta'(a*( ,+* +, a ;!a(*ati) +"#'+ia

:

Ti )a' & *itt&' t+:

&*&:




I't&*+"ati+'




I't&*+"ati+' <&4a"&5Fi'( t& ;!a(*ati) ,!')ti+' tat a& t*+!$ t& ,+""+i'$ +i't:125 <2-5




I't&*+"ati+' <&4a"&5Fi'( t& ;!a(*ati) ,!')ti+' tat a& t*+!$ t& ,+""+i'$ +i't:125 <2-5

2/20/16CS 131: NUMERICAL METHODS - PH

CHRISTIAN ZUNIGA

@!a(*ati) I't&*+"ati+': T& ta'(a*( ,+* +, a ;!a(*ati) +"#'+ia:

Ti )a' & *itt&' t+:

&*&:



G&'&*a" N&t+' I't&*+"ati+'


CHRISTIAN ZUNIGA



G&'&*a" N&t+' I't&*+"ati+'

N&t+'B Di%i(&( Di&*&')& I't&*+"ati'$ P+"#'+ia"


CHRISTIAN ZUNIGA



La$*a'$& I't&*+"ati'$ P+"#'+ia"Si"# a *&,+*!"ati+' +, t& N&t+'B P+"#'+ia" tat a%+i( t& )+t& (i%i(&( (i&*&')&.


CHRISTIAN ZUNIGA

Lagrange Interpolating



Lagrange InterpolatingPolynomials

• Linear version (n = 1):

Used for 2 points of data: (xo,f(xo)) and (x1,f(x1)),

)()()( 1

01

00

10

11 x f

x x

x x x f

x x

x x x f

−

−+

−

−=

)( x Lo )(1 x L



Lagrange Interpolating Polynom

• Second order version (n = 2):

1,)(1 ≠ j x L

( ) ( )( )( )

( )( )

( ) ( )( )( )

( ) ( ) )(

)(

)()(

2

1202

10

1

2101

20

0

2010

212

x f x x x x

x x x x

x f

x x x x

x x x x

x f x x x x

x x x x x f

−−

−−+

−−

−−+

−−

−−=

2,)(2 ≠ j x L

0,)( ≠ j x Lo

l i



Lagrange InterpolatingPolynomials - Example

U& a La$*a'$& i't&*+"ati'$ +"#'+ia" +

t& =*t a'( &)+'( +*(&* t+ &%a"!at& "'<25+' t& ai +, t& (ata:

10 = x 0)1ln()( 0 == x f

41 = x62 = x

386294.1)4ln()( 1 == x f

791760.1)6ln()( 2 == x f

L I t l ti P l i l



Lagrange Interpolating PolynomialsExample (cont’d)

First order polynomial:

)()()( 1

01

0

0

10

1

1 x f x x

x x x f

x x

x x x f

−

−+

−

−=

462098.0386294.114120

4142)2(1 =⋅

−

−+⋅

−

−= f

L I t l ti P l i l



Lagrange Interpolating Polynomials –Example (cont’d)

Second order polynomial:

6 0

6 x

40

4 x

x x

x x

x x

x x x L

2o

2

1o

1o

−

−⋅

−

−=

−

−⋅

−

−=)(

6 4

6 x

04

0 x

x x

x x

x x

x x x L

21

2

o1

o1

−

−⋅

−

−=

−

−⋅

−

−=)(

46

4 x

06

0 x

x x

x x

x x

x x x L

12

1

o2

o2

−

−⋅

−

−=

−

−⋅

−

−=)(

L I t l ti P l i l




5658444.0791760.1)46)(16(

)42)(12(

386294.1)64)(14(

)62)(12(

0)61)(41(

)62)(42()2(2

=−−

−−+

⋅−−

−−+

⋅−−

−−= f

∑==

n

0iiin x f x L x f )()()( ()( j x x

x x

x L

n

0 j ji

j

i −

−=

∏=

La ran e Interpolatin Pol nomials



S&)*&t Sa*i'$

A' <' 5 &)*&t a*i'$ )&& i a )&& &*& ' &+"& *&)&i%& a&)*&t I, at"&at &*+' a$*&& t+ a*& t&i* a*t +, t& &)*&t t&'

+!"( & &ai"# *&)+%&*&(.

15 '+"&($& +, "& ta' a*& +'Bt a& it &ai&* t+ (&t&*i'& t&)*&t.

25 '+"&($& +, at "&at a*& +!"( a& it &a# t+ )+!t&/(&t+*i$i'a" &)*&t.

2/20/16

CS 131: NUMERICAL METHODS - PH

CHRISTIAN ZUNIGA



S&)*&t Sa*i'$

&+"&

2/20/16


CHRISTIAN ZUNIGA



S&)*&t Sa*i'$

6 a*&

2/20/16


CHRISTIAN ZUNIGA



S&)*&t Sa*i'$

6 a*&

- Ha%i'$ "& ta' 6 a*& +it &ai&* t+ (i)+%&* t& &)*&t

- Ha%i'$ a'# 6 +!"(

)+!tati+'a""# &a# t+ )+&)*&t.

2/20/16


CHRISTIAN ZUNIGA



I""!t*ati+'

- Gi%& &a) &*+' a i'$"& (i$it- A $*+! i) a 6 (i$it )a' '+ t&

&)*&t.

2/20/16


CHRISTIAN ZUNIGA



S&)*&t Sa*i'$

T& T*!t&( Ti*( Pa'+ t& &)*&t (5 i""

a <-15 (&$*&& +"#'+)+'ta't t&* i ( i.&.


CHRISTIAN ZUNIGA



S&)*&t Sa*i'$

T& T*!t&( Ti*( Pa'+ t& &)*&t (5 i""

a <-15 (&$*&& +"#'+)+'ta't t&* i ( i.&.

Ea) &&* +, t*&)&i%& i/&* a*&

&)*&t. T& TTP i"" &'&a) +, t& &&* <t& &&*B ID5


CHRISTIAN ZUNIGA



S&)*&t Sa*i'$

A'# 6 &+"& + a*&)a' *&)+'t*!)t t& &it&* !i'$ La$*a'$& P+* N&t+'B P+"#'+ia".


CHRISTIAN ZUNIGA



S&)*&t Sa*i'$

N& &&* )a' a((&( &)a!& t& t*!a*t# )a' !t $&'&a*& ,*+ t& +"#'+ia".


CHRISTIAN ZUNIGA



S"i'& I't&*+"ati+'

I' t& *&%i+! &)ti+' & =tt&( ' (ata &t

i'$"& '-1 (&$*&& +"#'+ia". S"i'& ,!')ti+' )+''&)ti'$ +"#'+ia"

)+'&)!ti%& !&t +, +i't.


CHRISTIAN ZUNIGA



S"i'& F!')ti+'

U!a""# & !& +"#'+ia" a "i'& ,!')ti+

Ea# t+ &%a"!at& Ea# t+ (i&*&'tiat&

Ea# t+ i't&$*at&


CHRISTIAN ZUNIGA



Li'&a* S"i'&


CHRISTIAN ZUNIGA

Given ( ) ( ) ( ) ( )nnnn

y x y x y x y x ,,,......,,,,111100 −−

, fit linear splines to the data. This sipl! inv

forin" the #onse#$tive data thro$"h strai"ht lines. %o if the a&ove data is "iven in an as#e

order, the linear splines are "iven &! ( ))( ii x f y =

Figure : Linear splines



Li'&a* S"i'&


CHRISTIAN ZUNIGA



Li'&a* S"i'&


CHRISTIAN ZUNIGA

T& !a*( %&"+)it# +, a *+)&t i $i%&' a a ,!')ti+' +, ti& i' Ta"& 1%&"+)it# at t916 &)+'( !i'$ "i'&a* "i'&.

tie (s) vel ('s)

0 0

10 227.04

15 362.78

20 517.35

22.5 602.97

30 901.67



@!a(*ati) S"i'&


CHRISTIAN ZUNIGA

Given ( ) ( ) ( ) ( )nnnn y x y x y x y x ,,,,......,,,,

111100 −−, fit $adrati# splines thro$"h the data. The

are "iven &!

,)(11

2

1 c xb xa x f ++=

10 x x x ≤≤

,22

2

2 c xb xa ++=

21 x x x ≤≤

.

.

.

,2

nnn c xb xa ++=

nn x x x ≤≤

−1



@!a(*ati) S"i'&


CHRISTIAN ZUNIGA

a#h $adrati# spline "oes thro$"h t*o #onse#$tive data points

)( 0101

2

01 x f c xb xa =++

)(1111

2

11 x f c xb xa =++ .

.

.

)( 11

2

1 −−− =++ iiiiii x f c xb xa

)(2

iiiiii x f c xb xa =++ .

.

.

)( 11

2

1 −−− =++ nnnnnn x f c xb xa

)(2

nnnnnn x f c xb xa =++

This #ondition "ives 2n e$ations



@!a(*ati) S"i'&


CHRISTIAN ZUNIGA

The first derivatives of t*o $adrati# splines are #ontin$o$s at the interior points.

+or eaple, the derivative of the first spline

11

2

1 c xb xa ++ is 112 b xa +

The derivative of the se#ond spline

22

2

2 c xb xa ++ is 222 b xa +

and the t*o are e$al at 1 x x = "ivin"

212111 22 b xab xa +=+

022 212111 =−−+ b xab xa



@!a(*ati) S"i'&


CHRISTIAN ZUNIGA

%iilarl! at the other interior points,

022 323222 =−−+ b xab xa

.

.

.

022 11 =−−+

++ iiiiii b xab xa

.

.

.

022 1111 =−−+

−−−− nnnnnn b xab xa

-e have (n1) s$#h e$ations. The total n$&er of e$ations is )13()1()2( −=−+ nnn .

-e #an ass$e that the first spline is linear, that is 01 =a

@!a(*ati) S"i'& E4a"&



THE UPARD VELOCIT OF AROCET IS GIVEN AS AFUNCTION OF TIME. USING@UADRATIC SPLINES

a5 FIND THE VELOCIT AT T916SECONDS

5 FIND THE ACCELERATION AT T916 SECONDS

)5 FIND THE DISTANCE COVERED

JETEEN T911 AND T916SECONDS

t %<t5

/

0 0

1022.0

K

1?362.

20 ?1.3?

22.?602.

3001.6

Data a'( P"+t



t %<t5

/

0 0

1022.0

K

1?362.

20

?1.3

?

22.?602.

3001.6

S " i



S+"!ti+'

L&t ! &t ! t&

,)( 11

2

1 ct bt at v ++= 100 ≤≤ t

,222

2 ct bt a ++= 1510 ≤≤ t

,33

2

3 ct bt a ++= 2015 ≤≤ t

,44

2

4

ct bt a ++= 2220 ≤≤ t

,55

2

5 ct bt a ++= 5.22 ≤≤ t

Ea) S"i'& G+& T*+!$ T+C i i



C+'&)!ti%& Data P+i't

,)(11

2

1

ct bt at v ++= 100 ≤≤ t

0)0()0( 11

2

1 =++ cba

04.227)10()10( 11

2

1 =++ cba

Ea) S"i'& G+& T*+!$T+ C+'&)!ti%& Data



t v(t)

s m/s

0 0

10 227.04

15 362.78

20 517.35

22.5 602.9730 901.67

T+ C+'&)!ti%& DataP+i't

227)10()10( 22

2

2 =++ cba

362)15()15( 222

2 =++ cba

362)15()15( 332

3 =++ cba

517)20()20( 33

2

3 =++ cba

901)30()30( 55

2

5 =++ cba

517)20()20(44

2

4

=++ cba

602)5.22()5.22( 44

2

4 =++ cba

602)5.22()5.22( 55

2

5 =++ cba

D&*i%ati%& a*& C+'ti'!+! at I't&*i+* DP i t



P+i't

,)( 11

2

1 ct bt at v ++= 100 ≤≤ t

,2222 ct bt a ++= 1510 ≤≤ t

( ) ( )22

2

2

10

11

2

1

=

++=++

tt

ct bt adt

d ct bt a

dt

d

( ) ( ) 10221011 22 == +=+ t t bt abt a

( ) ( ) 2211 102102 baba +=+

02020 2211 =−−+ baba

D&*i%ati%& a*& )+'ti'!+!t I t i D t P i t



at I't&*i+* Data P+i't

At t910

At t91?

At t920

At t922.?

0)10(2)10(2 2211 =−−+ baba

0)15(2)15(2 3322 =−−+ baba

0)20(2)20(2 4433 =−−+ baba

)5.22(2)5.22(2 5544 −−+ baba



Lat E;!ati+'

01 =a

Fi'a" S&t +, E;!ati+'



Fi'a" S&t +, E;!ati+'

C+&8)i&'t +, S"i'&

iaibic



C+&8)i&'t +, S"i'&

i ai bi ci

1 0 22.0K 02 0. K.2 .

3 -0.13?6

3?.66 -1K1.61

K 1.60K -33.?6

??K.??

? 0.20

2.6 -1?2.13

lecture 3 polynomial interpolation for upload

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