Chapter 5. Curve fitting
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1 Chapter 5 Curve ttg
2 Assgmet please use ecell Gve the data elow use least squares regresso to t a a straght le a power equato c a saturato-growthrate equato ad d a paraola. Fd the r value ad justy whch s the est equato to represet the data y Curve ttg
3 Iterpolato pg 488 Polyomal Iterpolato s a commo method to determe termedate values etwee data pots. Geeral equato or th order polyomal s: a a a... a Polyomal terpolato cossts o determg the uque th -order polyomal that ts + data pot. For + data pots there s oly oe polyomal o order that passes through all the pots. For eample there s oly oe straght le rst-order polyomal that coects two pots Fg. 8.a ad oly oe paraola coects a set o three pots Fg.8.. Curve ttg
4 4 Two popular alteratve mathematcal ormats used to epress a terpolatg polyomal: a. Newto polyomal. Lagrage polyomal 8. Newto s Dvded-Derece Iterpolatg Polyomals The most popular ad useul polyomal orms. Cossts o the rst- ad secod-order versos. 8.. Lear Iterpolato The smplest orm o terpolato s to coect two data pots wth a straght le. Curve ttg
5 Fgure 8.: graphcal depcto o lear terpolato. 5 Curve ttg
6 6 Ths lear terpolato techque ca e depcted graphcally as show g 8. whch the smlar tragles ca e rearraged to yeld a learterpolato ormula; s reer to rst order terpolato polyomal -The term s a te-dvded-derece appromato o the rst dervatve. I geeral the smaller the terval etwee the data pots the etter the appromato. Curve ttg
7 7 Eample 7 Estmate the atural logarthm o usg lear terpolato. Frst perorm the computato y terpolatg etwee l = ad l 6 = The repeat the procedure ut use a smaller terval rom l to l Note that the true value o l s Soluto: By usg equato 8. a lear terpolato or l rom o = to = 6 to gve; = = = =? =6 = t * 48.% Curve ttg
8 8 The usg the smaller terval rom o = to = 4 yelds; = = = =? =4 = t.6947 *.% Thus usg the shorter terval reduces the percet relatve error to t =.%. Both terpolatos are show Fg.8. alog wth true ucto. Curve ttg
9 9 Fgure 8.: Comparso o two lear terpolatos wth deret tervals. Curve ttg
10 QUIZ 4 Estmate the logarthm o 5 to the ase log5 usg lear terpolato. a Iterpolate etwee log 4=.66 ad log6=.7785 Iterpolate etwee log4.5=.655 ad log 5.5=.7467 For each o the terpolatos compute the percet relatve error ased o the true value Curve ttg
11 8.. Quadratc Iterpolato The error eample 8. lear terpolato resulted rom appromato a curve wth a straght le. Wth data pots the estmato ca e mproved wth a secod-order Polyomal quadratc polyomal or paraola. Thus; Although equato 8. seem to der rom the geeral polyomal equato 8. the two equatos are equvalet y multplyg the terms equato 8. to yeld; = o + o + + o o Curve ttg
12 or collectg terms = a o + a + a where; a o = o o + o a = o a = Thus equatos 8. ad 8. are alteratve equvalet ormulatos o the uque secod-order polyomal jog three pots. To determe the values o coecet o ad rearrage ad use Eq 8. susttute Eq 8.4 to 8. ad susttute Eq 8.4 ad 8.5 to Eq 8. to yeld Eq 8.6. Curve ttg
13 Eample 8 Ft a secod-order polyomal to the three pots used lear terpolato eample o = = = 4 =.8694 = 6 = Use the polyomal to evaluate l Soluto : Curve ttg
14 4 Susttutg these values to equato 8. yelds the quadratc ormula: = o + o + o = whch ca e evaluated at = or; = whch represets a relatve error o t = 8.4%. Thus the curvature troduced y the quadratc ormula Fg.8.4 mproves the terpolato compared wth the result otaed usg straght les eample 8. Fg.8.. Curve ttg
15 5 Fgure 8.4: The use o quadratc ormula to estmate l mproves the terpolato. Curve ttg
16 Eample 9 Gve the data calculate.4 usg Newto s polyomals o order to. Soluto Curve ttg or.4
17 Use equatos to d ad. st order eed [ From eq 8.4; From eq 8.5; From equato 8. Curve ttg
18 d order eed [ From equato 8.6; From eq 8. or d order Curve ttg
19 9 QUIZ 5 Gve the data calculate 4 usg Newto s polyomal o order to Curve ttg
20 8.. Geeral Form o Newto s Iterpolatg Polyomals The th order polyomal s: = o + o + o o For th-order polyomal + data pots are requred: [ o o [.... [ The we used these data pots ad ollowg equato s to evaluate the coecets [ [ [ Curve ttg
21 Where the racketed [ ucto evaluatos are te dvded dereces; For eample the rst te dvded derece s represeted geerally as; Smlarly the th te dvded derece s Curve ttg k k j j k j j j j [ [ [ derece dvded Secod te [ [ [ [
22 These dereces ca e used to evaluate the coecets equatos 8.8 through 8. whch ca the susttuted to equato 8.7 to yeld the terpolatg polyomal called as Newto s dvded-derece terpolatg polyomal; Curve ttg
23 Eample From eample 8. data pots at = =4 ad =6 were used to estmate l wth a paraola. Now addg a ourth pot =5; =.6948 Estmate l wth a thrd order Newto s terpolatg Polyomal. = =4 =6 =5 = =.8694 = =.6948 Soluto The thrd-order polyomal wth = s = o + o + o + o - The rst dvded dereces are use eq 8.:.8694 [ Curve ttg
24 The secod dvded dereces use equato 8.: Curve ttg [ [ [ [ [ [ [ [
25 5 The thrd dvded dereces: [ [ [ Fally usg eqs : [ Isert all values to eqs 8.7; whch represets a relatve error o t = 9.%. Curve ttg
26 QUIZ 6 6 Calculate 4 wth a thrd ad ourth order Newto s terpolatg polyomal Curve ttg
27 Soluto st FDD; Curve ttg [ [ [
28 d FDD rd FDD Curve ttg [ [ [ [ [ [.5 [ 6.75 [ [ [
29 9 Fally = o + o + o + o Curve ttg
30 Chapter 5 Curve ttg Lear regresso epoetal model power equato ad saturato growth rate equato Polyomal Regresso Polyomal Iterpolato Lear terpolato Quadratc Iterpolato Newto DD Lagrage Iterpolato Sple Iterpolato
31 8. Lagrage Iterpolatg Polyomals The Lagrage terpolatg polyomal s smply a reormulato o the Newto s polyomal that avods the computato o dvded dereces: Where desgates the product o. For eample the lear verso = s j j j j L L where Curve ttg
32 Ad the secod-order verso = s For = Curve ttg
33 Eample Use a Lagrage terpolatg polyomal o the rst ad secod order to evaluate l ased o the data gve. = = = 4 =.8694 = 6 =.7976 Soluto: Frst-order polyomal at = use eq. 8.; Secod-order polyomal at = use eq 8.; Curve ttg
34 Quz Quz 7 Gve the data calculate 4 usg the Lagrage polyomals o order to
35 8.6 Sple Iterpolato 5 I the prevous sectos the th order polyomals were used to terpolate etwee + data pots. For eample we ca derve a perect seveth-order polyomal or eght pots Fgure 8.4 ac However there are cases where these uctos ca lead to erroeous results. A alteratve approach s to apply lowerorder polyomals to susets o data pots. Such coectg polyomals are called sple uctos. There are three types o sple terpolatos;. Lear sple/ st order sple. Quadratc sple. Cuc sple Curve ttg
36 6 Fgure 8.4: Illustrato shows the sples are superor to hgher order terpolatg polyomals. Curve ttg
37 8.6. Lear Sples The smplest coecto etwee two pots s a straght le. The rst order sples or a group o ordered data pots ca e deed as a set o lear uctos Ths equatos ca e used to evaluated the ucto at ay pot etwee ad y rst locatg the terval wth whch the pot les. The the approprate equato s used to determe the ucto value wth the terval. The method detcal to lear terpolato. m m m m where where where where 7 Curve ttg
38 Dsadvatages o st Order Sples see gure Not smooth. At the data pots where two sples meet called a kot the slope chages aruptly These dsadvatages ca e overcome y usg hgher-order polyomal sples that esure smoothess at the kot. Curve ttg
39 9 Eample a: Frst-order Sples Ft the data the tale elow wth rst-order sples. Evaluate the ucto at = 5. 5 =..5 = 4.5. = 7..5 = 9..5 Soluto: Determe the slopes etwee pots or every tervals. For terval = 4.5 to = 7 usg eq 8.7: m Curve ttg
40 4 m m Thereore at = 5; m Curve ttg
41 Quadratc sples Quadratc sples s use to derve a d order polyomal or each terval etwee data pots. Geeral polyomal or each terval ca e represeted as; a c For + data pots = there are tervals ad ukow costats the a s s ad c s hece equatos are requred. Fgure 8.7 shows the otato used to derve quadratc sples. Curve ttg
42 4 Fgure 8.7 : tervals wth 4 data pots. Curve ttg
43 The equatos/codtos are requred to evaluate the ukows. The ucto values o adjacet polyomals must e equal at the teror kots. or = to. Thereore the total s codtos.. The rst ad the last uctos must pass through ed pots. Ths adds two addtoal equatos ad total o + = codtos. Curve ttg 4 c a c a c a c a
44 44. The rst dervatves at the teror kots must e equal. ' a a a or = to. Ths provdes aother - or a total o + = codtos. 4. Assume the secod dervatve s zero at the rst pot. a = a Curve ttg
45 45 Eample : Quadratc sples Ft quadratc sples to the same data used eample a. Use the results to estmate the value at = data pots = tervals = 9 ukows.e. a c a c a c Curve ttg
46 Solutos a. = = 4 equatos Usg eqs 8.9 ad 8.; or = to ; Curve ttg 46 c a c a ' or or c a c a c a c a
47 47 Thereore; a a a a c 7 c 7 c 4.5 c Passg the rst ad last uctos + = codtos.e. =6 equatos. Usg eqs 8. ad 8. a a For = ad = = 9; a a 9 c c c 9 c.5.5 Curve ttg
48 Thereore; c. The rst dervatves at the teror kots must e equal. Usg eq. 8. Curve ttg c a c a For For ' a a a a a a a
49 49 Thereore; 9a 4a 9a 4a d. Secod dervatve s zero at the rst pot.e.. Usg eq. 8.4; a a = Because ths equato speces a we have 8 smultaeous equatos. Curve ttg
50 I matr orm: Curve ttg c a c a c
51 5 These equatos are solved wth the results: a = = - c = 5.5 a =.64 = c = 8.46 a = -.6 = 4.6 c = -9. Susttuted to the orgal quadratc equatos: a = < < 4.5 = < < 7. = < < 9. Thereore at = 5 s: 5 = =.66 c Curve ttg
52 Chapter 5 Curve ttg
53 5 Assgmet please use ecell Gve the data elow use least squares regresso to t a a straght le a power equato c a saturato-growthrate equato ad d a paraola. Fd the r value ad justy whch s the est equato to represet the data y Curve ttg
54 Soluto a The regresso o y versus to gve y y = R = The regress o o log y versus log to gve log log y Thereore = = ad =.8577 ad the power model s 5 4 y = R =.955 Curve 4 ttg 5 6
55 c The regresso o /y versus / to gve y Thereore = /.996 = 5.9 ad = = ad the saturato-growth-rate model s y y = R = Curve ttg
56 d The polyomal regresso to t a paraola y The model ad the data ca e plotted as 5 4 y = R = Comparso o ts: The lear t s ovously adequate. Although the power t ollows the geeral tred o the data t s also adequate ecause the resduals do ot appear to e radomly dstruted aroud the est t le ad t has a lower r tha the saturato ad paraolc models. The est ts are or the saturato-growth-rate ad the paraolc models. They oth have radomly dstruted resduals ad they have smlar hgh coecets o determato. The saturato model has a slghtly hgher r. Although the derece s proaly ot statstcally sgcat the asece o addtoal ormato we ca coclude that the saturato model represets the est t. Curve ttg
57 5 4 y = R = y = R = y = R = y = R =
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