1. Chapter 15 Curve Fitting : Splines
Gab Byung Chae

3. Oscillation in a higher order interpolation

4. The alternative : spline

5. Notation used to derive splines
n-1 intervals in n data points

6. Splines in 1st, 2nd, 3rd order
Linear spline
Quadratic spline
Cubic spline

7. 15.2 Linear Splines linear splines passing thru
Example : (3,2.5),(4.5,1),(7,2.5),(9,.5) -> (5,?)

8. 15.3 Quadratic Splines
The objective in quadratic splines is to derive a second-order polynomial for each interval between data points
The polynomial :
si(x) =ai + bi(x-xi)+ ci(x-xi)2
For n data points, there are n-1 intervals and, consequently, 3(n-1) unknown constants to evaluate.  3(n-1) equations or conditions are required to evaluate the unknowns.

10. Let for all i=1,,..,n-1
1. Continuity condition : the function must pass through all the points.
--- (15.6)
for all i=1,,..,n-1 : (n-1) equations
2. The function values of adjacent polynomials must be equal at the knots.
--- (15.8)
for all i=1,,..,n-1 : (n-1) equations

11. Let for all i=1,,..,n-1
3. The first derivatives at the interior nodes must be equal.
(n-2) eq.s
--- (15.9)
4. Assume that the second derivative is zero at the first point.
1 equation
Example : (3,2.5),(4.5,1),(7,2.5),(9,.5) -> (5,?)

12. Example 15.2
Problem] Fit quadratic splines to the data in Table Use the results to estimate the value at x=5.

13. Equation (15.9) creates 2 conditions with
Solution] Equation (15.8) is written for i=1 through 3 (with c1=0) to give
f1+ b1h1 = f2
f2+ b2h2 + c2h22 = f3
f3+ b3h3 + c3h32 = f4
Equation (15.9) creates 2 conditions with
c1=0 :
b1 = b2
b2+ 2c2h2 = b3

14. f1 = h1 = = 1.5
f2 = h2 = = 2.5
f3 = h3 = = 2.0
f4 = 0.5
b = 1.0
b c = 2.5
b c3 = 0.5
b1 - b = 0
b2 - b c = 0

15. b2 = -1 c2 = 0.64 b3= 2.2 c3 = -1.6 s1(x) = 2.5 –(x-3)
s2(x) = 1.0 –(x-4.5)+0.64(x-4.5)2
s3(x) = (x-7.0)-1.6(5-7.0)2
x=5 이므로
s2(5) = 1.0 –(5-4.5)+0.64(5-4.5)2 =0.66

16. 15.4 Cubic Splines a. (n-1) eq.s b. (n-1) eq.s c. (n-2) eq.s
d. (n-2) eq.s
e. 2 eq.s

17. Solving Cubic Splines

18. Example 15.3
Fit cubic splines to the data . Utilize the results to estimate the value at x =5

19. Solution : employ Eq.(15.27)
f1 = h1 = = 1.5
f2 = h2 = = 2.5
f3 = h3 = = 2.0
f4 = 0.5

20. So we have
Using MATLAB the results are
c1 = c2 =
c3 = c4 = 0

21. Using Eq. (15.21) and (15.18),
b1 = d1 =
b2 = d2 =
b3 = d3 =

22. End Conditions
Clamped End Condition : Specifying the first derivatives at the first and last nodes.
Not-a-Knot End Condition : Force continuity of the third derivative at the second and the next-to-last knots. – same cubic functions will apply to each of the first and last two adjacent segments,

23. 15.5 Piecewise interpolation in Matlab
MATLAB function : spline
Syntax :
yy = spline(x, y, xx)

24. Example 15.4 Splines in MATLAB
Use MATLAB to fit 9 equally spaced data points sampled from this function in the interval [-1, 1].
a) a not-a-knot spline b) a clamped spline with end slopes of f1’=1 and f’n-1 = -4

25. Solution : a)
>> x = linspace(-1,1,9);
>> y = 1./(1+25*x.^2);
>> xx = linspace(-1,1);
>> yy = spline(x,y,xx);
>> yr = 1./(1+25*xx.^2);
>> plot(x,y,’o’,xx,yy,xx,yr,’—’)

26. Cubic spline of the Runge function

27. b)
>> yc =[1 y -4];
>> yyc =spline(x,yx,xx);
>> plot(x,y,’o’,xx,yyc,xx,yr,’—’)

28. 15.5.2 MATLAB function : interp1
Implement a number of different types of piecewise one-dimensional interpolation.
Syntax :
yi = interp1(x,y,xi, ‘method’)
yi = a vector containing the results of the interpolation as evaluated at the points in the vector xi , and ‘method’ = the desired method.

29. method Nearest – nearest neighbor interpolation.
Linear – linear interpolation
Spline – piecewise cubic spline interpolation(identical to the spline function)
pchip and cubic – piecewise cubic Hermite interpolation.
The default is linear.

30. MATLAB’s methods >> yy=spline(x,y,xx); cubic spilne
>> yy=interp1(x,y,xx): (piecewise) linear spline
>> yy=interp1(x,y,xx,’spline’): cubic spline
>> yy=interp1(x,y,xx,’nearest’) : nearest neighbor method
>> yy=interp1(x,y,xx,’pchip’): piecewise cubic Hermite interpolation
>> zz=interp2(x,y,z,xx,yy): 2 dimensional interpolation

31. Example 15.5 Trade-offs Using interpl
Linear, nearest, cubic spline, piecewise cubic Hermite interpolation

32. >> t = [ ];
>> v = [ ];
>> tt = linspace(0,110);
>> vl=interp1(t,v,tt); %(piecewise) linear spline
>> plot(t,v,’o’,tt,vl)
>> vn=interp1(t,v,tt,’nearest’); % nearest
>> plot(t,v,’o’,tt,vn)
>> vs=interp1(t,v,tt,’spline’); % cubic spline
>> plot(t,v,’o’,tt,vs)
>> vh=interp1(t,v,tt,’pchip’); % piecewise cubic Hermite interpolation
>> plot(t,v,’o’,tt,vh)