53these polynomials to interpolate bothfand its first derivative:sj(xj-1) =f(xj-1),j=1, . . . ,n;sj(xj) =f(xj),j=1, . . . ,n;s0j(xj-1) =f0(xj-1),j=1, . . . ,n;s0j(xj) =f0(xj),j=1, . . . ,n.To satisfy these conditions, takesjto be the Hermite interpolant tothe data(xj-1,f(xj-1),f0(xj-1))and(xj,f(xj),f0(xj)). The resultingfunction,S(x):=sj(x)forx∈[xj-1,xj], will always have a contin-uous derivative,S∈C1[x0,xn], but generallyS6∈C2[x0,xn]due todiscontinuities in the second derivative at the interpolation points.In many applications, we lack specific values forS0(xj) =f0(xj);we simply want the functionS(x)to be assmoothas possible. Thatmotivates our next topic: splines.

5400.20.40.60.81-202468101214xf(x)S(x)00.20.40.60.81-20-1001020304050xS0(x)f0(x)00.20.40.60.81-600-400-2000200400600xS00(x)f00(x)Figure1.19:Piecewise cubic Hermiteinterpolant tof(x) =sin(20x) +e5x/2atn=5 uniformly spaced points (top),and the derivative of this interpolant(middle). Now both the interpolant andits derivative are continuous, and thederivative interpolatesf0. However,the second derivative of the interpolantnow has jump discontinuities (bottom).

55lecture 9:Introduction to Splines1.11SplinesSpline fitting, our next topic in interpolation theory, is an essentialtool for engineering design. As in the last lecture, we strive to inter-polate data using low-degree polynomials between consecutive gridpoints. The piecewise linear functions of Section1.10were simple,but suffered from unsightly kinks at each interpolation point, reflect-ing a discontinuity in the first derivative. By increasing the degree ofthe polynomial used to modelfon each subinterval, we can obtainsmoother functions.Long before numerical analysts gottheir hands on them, ‘splines’ wereused in the woodworking, shipbuild-ing, and aircraft industries. In suchwork ‘splines’ refer to thin pieces ofwood that are bent between physicalconstraints calledducks(apparentlythese were also calleddogsandratsinsome settings; modern versions aresometimes calledwhalesbecause of theirshape). The spline, a thin beam, bendsgracefully between the ducks to givea graceful curve. For some discussionof this history, see the brief ‘History ofSplines’ note by James Epperson in the19July1998NA Digest, linked from theclass website. For a beautiful derivationof cubic splines from Euler’s beamequation—that is, from the originalphysical situation, see Gilbert Strang’sIntroduction to Applied Mathematics,Wellesley Cambridge Press,1986.1.11.1Cubic splines: first approachRather than settingS0(xj)to a particular value, suppose we simplyrequireS0to be continuous throughout[x0,xn]. This added freedomallows us to impose a further condition: requireS00to be continuouson[x0,xn], too. The polynomials we construct are calledcubic splines.In spline parlance, the interpolation points{xj}nj=0are calledknots.These cubic spine requirements can be written as:sj(xj-1) =f(xj-1),j=1, . . . ,n;sj(xj) =f(xj),j=1, . . . ,n;s0j(xj) =s0j+1(xj),j=1, . . . ,n-1;s00j(xj) =s00j+1(xj),j=1, . . . ,n-1.

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