Cubic Spline Questions

Transcription

1 Cubic Spline Questions. Find natural cubic splines which interpolate the following dataset of, points:.0,.,.,.0, 7.0,.,.0,0.; estimate the value for. Solution: Step : Use the n- cubic spline equations to find the second derivatives slide 0: : : Natural splines have 0 so the sstem simplifies to: Step : ppl these solutions to obtain the cubic spline equations from formulae for,, C, slides 7 & 77 for the three subintervals i.e., with,,. For :...0 C [ ]. [ ].

2 The spline equation, which applied for the first subinterval.0 < <., is then given b: C [ ].7 S For : C [ ]7.0. [ ]7.0. The spline equation, which applied for the second subinterval. < < 7.0, is then given b:..0. [. [ ] C 0. 7 ] S For : C [ ] [ 7 ].0 7.0

3 The spline equation, which applied for the third subinterval 7.0 < <.0, is then given b: C [ ]. 0 S The three equations S, S and S constitute the natural cubic spline equations for over the interval.0 < < To estimate the value at, use the spline equation for the second sub-interval S:

4 . Find the interpolating cubic splines for the five logarithmic breakpoints,ln,,ln,,ln,,ln, and,ln. Use [i.e. three eercises to do here]: a natural endpoint conditions; b specified endslopes of at and / at ; and c not-a-knot conditions. Compare the graphs corresponding to the three different endpoint conditions. For whichever of these endpoint conditions can be used with the built-in Matlab function, compare our solution to that found with Matlab. Solution: Step : Use the n- cubic spline equations to find the second derivatives. We end up with the following sstem of equations denoted Sstem : ln ln ln ln ln : ln ln ln ln ln : ln ln ln ln ln : Step : Find,, C, slides 7 & 77 for the four subintervals. For : ] [ C ] [

5 For : ] [ C ] [ For : ] [ C ] [ For : ] [ C ] [

6 a Using natural endpoint conditions Natural splines have 0 so Sstem simplifies to: ln ln 0 ln The spline equation, which applied for the first subinterval < <, is then given b: C ln ln 0 [ ] 0. S The spline equation, which applied for the second subinterval < <, is then given b: C ln ln [ ] 0. [ ] 0.00 S The spline equation, which applied for the third subinterval < <, is then given b: C ln ln [ ] 0.00 [ ] S The spline equation, which applied for the fourth subinterval < <, is then given b: C ln ln [ ] S

7 b Using specified endslopes of at and / at NOTE: There was a tpo on the Web, the endslope at is not zero! We need to set the values of & to values calculated from equation [] from slide 7 in the notes so that & have the desired values. For the left endpoint, use. For the right endpoint, use. d d where and d t, : d t, d d : ln ln 0 ln ln ln 0 ln With these conditions, Sstem simplifies to: ln ln ln ln ln The spline equation, which applied for the first subinterval < <, is then given b: C ln ln [ ] 0. [ ] 0.7 S

8 The spline equation, which applied for the second subinterval < <, is then given b: C ln ln [ ] 0.7 [ ] 0. S The spline equation, which applied for the third subinterval < <, is then given b: C ln ln [ ] 0. [ ] S The spline equation, which applied for the fourth subinterval < <, is then given b: ln [ C ln [ ] ] S c Using not-a-knot conditions We wish to enforce the continuit of the third derivative at the first and last internal knots i.e., and. To begin, we take the derivative of the second derivative found on slide 7: d d d d Continuit at these internal knots mean: t : 0 t : 0

9 With these conditions, Sstem simplifies to: 0 ln ln ln The spline equation, which applied for the first subinterval < <, is then given b: C ln ln [ ] 0. [ ] 0.77 S The spline equation, which applied for the second subinterval < <, is then given b: C ln ln [ ] 0.77 [ ] 0.0 S The spline equation, which applied for the third subinterval < <, is then given b: C ln ln [ ] 0.0 [ ] 0.0 S The spline equation, which applied for the fourth subinterval < <, is then given b: ln [ 0.00 C ln [ ] ] S

10 Please note that with this last case, since we are using the not-a-knot conditions, S and S are in fact the same cubic function, as is the case with S and S. Therefore, if ou are confident in our results, ou actuall onl need to find two spline equations for this question when using the not-a-knot conditions! In each case, the four equations S, S, S and S constitute the natural cubic spline equations for over the interval < <. The following are the plots of these splines in Matlab. The built-in spline function in Matlab is capable of finding splines using the fied endslopes condition and the not-aknot condition.. Cublic spline using natural endpoints

11 . Cublic spline using fied endslopes hand dotted Matlab spline function solid Cublic spline using not-a-knot conditions hand dotted Matlab spline function solid