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the_ basicapproach to fita curve or u ,"1, is of curves rhat passdirectly through each of the points such data usually orii,inatesfrom tables Examples are valuesForthe density of water or for the heaica_ pacity of gasesas a function of temperature The esti_ mation of valuesbetweenwell-known discretepoints is called inrerpolarion(Fig pT4 lb and e Curve Fittingond Engineering ond Science your ftrst exposure[o curve fitting may have been to deter_ mtne lntermediatevalues from mbulated data_for instance, from interest tables for *";;;; ;;: or from steam fables for thermodynamics Trir the remainder of your career,you will ,rnroughout irequentoccasion estimate to intermediate values fave trom such tabJes
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tJi,i';,"):,::!::1'j-"H',narureis"uudt,o,t' squares ,rgrriiion (FigpT4 tar)rluuru! wucrs theoaul ls Known to very Dre_ fc1nd, whererue datais knownro be very pre_ cise
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(c) P FIGURE T4I (b) regression, lineor through doto points: leosl-squores five curve fo Three ottempts fit o "best" io)
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havebeentab' and scientificproperties Although many of the widely usedengineering availablein this convenientform Special ulated,there are a greatmany more that are not your own dataand develop and new problemcontextsoften requirethat you measure cases Two types of applicationsare generally encountered your own predictive relationships testing data:trend analysisand hypothesis when fitting experimental pattern of the data to make predicTrend analysis representsthe processof using the with high precision,you might utilize interpotions For caseswherethe datais measured regression with least-squares lating polynomialsImprecisedatais often analyzed variable values of the dependent Trend analyslsmay be used to predict or forecast observeddata or interpolation This can involve extrapolationbeyond the limits of the of within the range of the data All fields of engineeringand scienceinvolve problems this type A second application of experimentalcurve fitting is hypothesistesting Here,an are dataIf the model coefficients with measured model is compared existingmathematical
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to valuesthat bestflt the observed dataOn the ulrknown,it nray be necessary determine are of other hand,if estimates the model coetTicients alreadyavailable,it may be appropripledictecl valuesof the nrodelwith observed valuesto testthe adequacy of ate to compare on the model Ofien alternativenodels are comparedand the "best" one is selected the basisof empiricalobservations curvefitting is irnIn additionto the foregoing engineering scientific and applications, portantin other nurnerical methodssuchas integrationand the approxirnate solutionof dil'Finally,culve-fitting ferential equations techniqtres be usedto derivesimplefunctions can cornplicated functioDs to approximate
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that Chap l3 focuses linear regression; is hol'r'to deon After a brief review of statistics discussing terminethe "best" straightline througha setof uncertain datapointsBesides quantitativc line we also present how to calculate slopeand intercept this straight the of fbr In we ser'the and visualmethods evaluating validity of the lesults acldition, describe fbr of equations eral approaches the linearization nonlinear andmultiplelinearreglessiou of /4 begins with brief discussions polynomial fit regressirtn cubics,or higher-order Pctlynomial dealswith developinga bc'st clf parabolas which is depolynomialsThis is followed by a clescription multiple linear regression, of variable), is a linear functionof two or more signedtbr the casewhere the dependent exvariables 12, This approach specialutility for evaluating has r1, irrdependent , xtn is on of perimental of datawherethe variable interest dependent a nunlber differentfactors are we how polynomialand multiple regression After multiple regression, illustrate modelAmong otherthings,this will allow of both subsets a generullinear least-squares its statisand discuss general matrir representation ofregression a us to introduce concise Finallythe last sections Chap l4 ale del'otedtononlinearregression of tical properties to fit This approachis designedto computea least-squares of a nonlinearequaticln data interpolutionis describecl In Clnp /5, the alternativeclrrve-tittillg techniquecaTled valuesbetween previously, intermediate interpolation usedfbr estimating is As discussed We the for precise ln are datapclints Chap15,polynomials derived this purpose introduce to linesand parabolas connect of by basicconcept polynomialinterpolation usingstraight polynomillTwo procedure fitting an rrth-ordel ior points Thenwe developa generalized fbrm The first, called in fbr thesepolynomials equation fbrmatsare presentcd expressing whenthe appropriate orderof the polynopolt'nontial, preferable is Newton's interpolotirrg mial is unknown The second,called the Lagrange interpolotingpoltrutmial, has advanrvhenthe properolder is known befbrehand tages This for an Finally, Chup l6 presents alternativetechnique fitting precisedatapoints fashion to lits callecl splineirrterytolation, polynomials databut in a piecewise techniclrre, well suitedfbr fitting datathat is genelallysmoothbut exhibits As suchit is particularly is interpolation with an overviewof how piecewise The encls abruptlocalclranges chapter imolemented MATLAB in