Linear Least Squares Approximation and Polynomial Interpolation: A Practical Example, Cheat Sheet of Mathematics

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Car 8.^1.^1

Given data : from above data (^) , we have

i Ki Yi i zi yi niyi ui

1 O 1 I 010. 00 ,^0

(^30) , 5 1 , 6487 3 0 , (^5 1) , 6487 0 , 82435 0 , 25 (^40) , (^75 2) , 117 4 0 , (^75 2) , 117 1 , 58775 0 , 5625 (^512) , (^7183 5 1 2) , 7183 2 , 7183 1 , 0

let -^5

P(r) =^ do +^ aix be^ the^ linear least^ square Zi= 12 , 58 , 7685 , 4514 1 , 875

do =^1 ,^875.^0 ,^768

• (^5) , 4514. (^2) , (^5) = 0 polynomial for the^ given data.^ We^ know^ that^ : ,^89968

5. 1 , 875 =^ (2, 5)

do Z.^ Zyi-Z^ Kiyi^.^ Exi^ =>^ P(u)^ =^0 , 09960 +^1 , 707842 m(z) - (Zxi) (^) an = 5.^5 ,^4541 -^2 ,^5.^8 ,^768 = (^1) , 70784

5. 1 , 875 -^ (2, 5)

Cu 8. 1. 2

yi = +a= 5 We^ have^ :^ =^1000 +^ 55a1^ +^38592 (^44) = 5500 + (^) 305a + (^) 3025a

S

• iyi = ai + a 4532 , 8 = (^) 385a0 + (^) 3025a1 + (^) 25333a = (^) Ag =^0 , 406667 an =^1 ,^15485

gi^ = a

S

ac =^0 ,^0348485 j (^) Ki ri u ut Yi nigi rigi => P(u) =^0 , 406667 +^1 , 15485x +^0 ,0348485x

110 10 1,0 1 , 01 , 3 1 , 3 1 , 3 1 , 3 j Ri gi^ P(ui)^ (Plui)^ -y: /

C 204 , 00 , 016 , 03 , 5 7 ,0 14, 0 2 , 8 I 1 , 0 1 , 31159636 0 , 0878

(^3 3) ,0 9, 027 , (^081) , (^04) , 2 12637, 8 13 , 4 2 2 , (^0 3) , 5 2 , (^855760) , 415

4 4 , 016 , 064 , 0256 , 05 ,^0 20 ,^000 , 0 320 , 03 3 ,^0 4 , 2 4 , 18485 0 , 0002

5 50 25, 0 125 , 0 625 , 07 ,0 35, 0 175, 0875 ,^84 4 , 05 , 0 5 , 58364 0 , 3406

Z (^7) , (^049) , 0343 , 0 2401, (^010) ,^1 70 , 7 494, 9 3464 , (^36 6) , 0 0. (^80) , (^59030) , 044 (^88064) , 0512 ,^04096 ,^012 ,^5 100 ,^0000 ,^06400 ,^0 Z^7 ,^0 10 ,^1 10 , 19818 0 , 0096 (^9 9). (^0 01) , (^0729) , (^06561) , (^013) ,0 117, 0 1053 , 09477 , (^0 8 88) & (^12) ,^511 , 87576 0 , 3897

10 10,0 100 , 0 1000,0 10000 , 0 15 , 6 156 , 0 1560 ,^0 15600 , 09 90 13 ,^0 13 , 62303 0 , 3882

Total (^55) , 0385 , (^03025) ,0 25333, (^081) , (^0572) , 44532 , 8 10 10 , (^015) , 6 15, (^440) , 0256 Total (^55) , (^0 385) , (^0 1) , 7035 => E =^1 , 7035

Cu 8. 1. 3

i (^1 2) S 4 5 6 = 1 Taco^ : Ni 1 , 0 1 , 1 1 , 3 1 ,^51 , 9 2 ,^18 ,^9 yinomia

1,0 1 , 331 2 ,^197 3 , 3756 , 859 9 ,^26124 ,^023 niyi = dia

u 1 , 01 , 46428565 , 06213 ,^83219 , 44842 , 863

it (^1) , (^01) , 6113 , (^7137) , (^59424) , (^76140) , 84179 , (^519) niyi = na

uP 1 , 0 1 , 772 4 , 82711 , 39147 ,^04685 , 766151 , 8016

Yi^1 ,84 1,^ 96 2,^212 ,^ 45 2,^943 ,^1814 ,^58 =niyi^ =^ na

yiRi 1 ,^842 ,^1562 ,^8733 ,^6755 ,^5866 ,^67822 ,^808

Yik?^1 , 84 2,^3723 ,^7355 ,^512 10 ,^61314 ,^02438 ,^096690 +^0.^ 9 an^ +^14 , 17dc^ +^24 ,^ 023a3^ =^14 ,^58

Yix?^1 , 84 2 ,^6094 ,^0550 ,^26920 ,^16529 ,^4567 ,^1888 ,9a0^ +^14 ,^1721 +^24 ,^ 023a2^ +^42 ,^06393 =^22 ,^008

I - E)

San

Taco! (^2) Yi (^) = do + a (^14) , 1700 +^24 , 023an +^42 ,863ae +^79 , 52as =^38 , 096

1 =^1

02390 +^42 , 862991 +^79 , 52a2 +^151 , 801az =^67 , 19

S

.Kiyi^ = adia ao = (^0) , 629019

an =^1 ,^18501

S

E)

E A2 = 0 , 0353325 /Gga20g

As =-^0 ,^0100472