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Althoughits graphical basisis appealing, naturalsplineis only one of several the end conditionsthat can be specified splines for Two of the mostpopularare at Clunlted End ConditionThis option involvesspecifyingthe first derivatives tlrefirst it and lastnodes This is sometimes calleda "clamped"splinebecause is what occurs when you clamp the end of a draftingsplineso that it hasa desiredslopeFor example, if zerolirst derivatives specilied, splinewill level off or becomehorizontalat the the are ends " Not-a-Knot" End ConditionA third alternative to fbrce continuity of the third deis knotsSincethe splinealready specifies that rivativeat the second andthe next-to-last the tunction value and its first and secondderivativesare equal at theseknots specimeansthat the samecubic functionswill apply to fying continuous third derivalives segments Sincethe first internal knotsno longer eachofthe first and iasttwo adjacent represent thejunction of two different cubic functions,they are no longer true knots Hencethis caseis refenedto as the "not-a-knot" contlitionIt hasthe additionalpropusingan ordinarycubic erty thatfor firur points,it yieldsthe sameresultasis obtained polynomialof the sortdescribed Chap 15 interpolating in

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These conditionscan be readily applied by using Eq tt625t fbr the interior knots i:2,3,,n2 , a n d u s i n g f i r s t ( l ) a n d l a s t e q u a t(in -n s a s w r i t t e n i n T a b1e 2 o 1) l6 Figure 165showsa comparison the threeendconditions appliedfo tit the datafrom of as rble 16l The clampedcaseis setup so that the derivatives the endsareequalto zero at As expected, spline fit for the clarnpedcaselevels off at the endsIn contrast,the the natul'aland not-a-knotcasesfollow the trend of the data points nrore closelyNotice how the the natural spline tendsto straighten as would be expected out because secondderivatives go to zero at the endsBecause hasnonzerosecondderivativesat the ends,the notit a-knotexhibitsmore curvature

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SSIS AND ETL DBAs often move data in and out of databases, and at the same time, perform transfor-

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mations on the data as it is moved. This is often done for data warehouses and application integration. This process is often referred to as extract, transform, and load (ETL), and in SQL Server this is commonly performed using SQL Server Integration Services (SSIS).

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The spline MATLAB hasseveral built-in functionsto implementpiecewise interpolation in The pchip funcfunction performscubic splineinterpolationas described this chapter tion implementspiecewisecubic Hermite interpolationThe interpl function canalso implement spline and Hermite interpolation, can also perform a numberof othertypes but of piecewise interpolation 1651 MATTAB Function: spline Cubic splinescan be easily computedwith the built-in MATLAB function, spli-ne Ithas the generalsyntax,

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