1. Curve Fitting: Splines
Chapter 16
Curve Fitting: Splines

2. Spline Interpolation
There are cases where polynomial interpolation is bad
Noisy data
Sharp Corners (slope discontinuity)
Humped or Flat data
Overshoot
Oscillations

3. Example: f(x) = sqrt(abs(x))
Interpolation at 4, 3, 2, 1, 0, 1, 2, 3, 4

4. Spline Interpolation Cubic interpolation 7th-order 5th-order
Linear spline

5. Spline Interpolation Idea behind splines
Use lower order polynomials to connect subsets of data points
Make connections between adjacent splines smooth
Lower order polynomials avoid oscillations and overshoots

6. Spline Interpolation
Use “piecewise” polynomials instead of a single polynomial
Spline -- a thin, flexible metal or wooden lath
Bent the lath around pegs at the required points
Spline curves -- curves of minimum strain energy
Piecewise Linear Interpolation
Piecewise Quadratic Interpolation
Piecewise Cubic Interpolation (cubic splines)

7. Zero curvatures at end points
Drafting Spline
Continuous function and derivatives
Zero curvatures at end points

8. Splines There are n-1 intervals and n data points
si(x) is a piecewise low-order polynomial

9. Spline fits of a set of 4 points

11. Example: Ship Lines
waterline

12. Bow
Stern

13. Spline interpolation for submerged hull (below waterline)
Original database
Spline interpolation for submerged hull (below waterline)

14. Linear Splines b is the slope between points
Connect each two points with straight line functions connecting each pair of points
b is the slope between points

15. Linear Splines
si (x) = ai + bi (x  xi)
x x x x4

16. Identical to Lagrange interpolating polynomials
Linear Splines
Identical to Lagrange interpolating polynomials

17. Linear splines Connect each two points with straight line
Functions connecting each pair of points are
slope

18. Linear splines are exactly the same as linear interpolation!
Example:

19. Linear Splines
Problem with linear splines -- not smooth at data points (or knots)
First derivative (slope) is not continuous
Use higher-order splines to get continuous derivatives
Equate derivatives at neighboring splines
Continuous functional values and derivatives

20. Quadratic Splines Quadratic splines - continuous first derivatives
May have discontinuous second and higher derivatives
Derive second order polynomial between each pair of points
For n points (i=1,…,n): (n-1) intervals & 3(n-1) unknown parameters (a’s, b’s, and c’s)
Need 3(n-1) equations

21. Quadratic Splines 3(n-1) unknowns si(x) = ai + bi (xxi) + ci(xxi)2
f(x5)
s2 (x)
f(x2)
s4 (x)
f(x3)
s1 (x)
s3 (x)
f(x4)
f(x1)
Interval Interval Interval 3 Interval 4
x x x x x5
3(n-1) unknowns
si(x) = ai + bi (xxi) + ci(xxi)2

22. Piecewise Quadratic Splines
Example with n = 5
The functional must pass through all the points xi (continuity condition)
The function values of adjacent polynomials must be equal at all nodes (identical to condition 1.) 2(n-2) + 2 = 2(n-1)
The 1st derivatives are continuous at interior nodes xi, (n-2)
Assume that the second derivatives is zero at the first points
3(n1) equations for 3(n1) unknowns
2(n1) + (n2) + (1) = 3(n1)

23. Piecewise Quadratic Splines
Function must pass through all the points x = xi (2n  2 eqns)
This also ensure the same function values of adjacent polynomials at the interior knots (hi = xi+1 – xi)
Apply to every interval (4 intervals, 8 equations)

24. Piecewise Quadratic Splines
Continuous slope at interior knots x = xi (n2 equations)
Apply to interior knots x2 , x3 and x4 (3 equations)
Zero curvatures at x = x1 (1 equation)

25. Example: Quadratic Spline

26. Quadratic Spline Interpolation
function [A, b] = quadratic(x, f)
% exact solution f(x)=x^3-5x^2+3x+4
% x=[ ]; f=[ ];
h1=x(2)-x(1); h2=x(3)-x(2); h3=x(4)-x(3); h4=x(5)-x(4);
f1=f(1); f2=f(2); f3=f(3); f4=f(4); f5=f(5);
A=[
0 h1 h1^
h2 h2^
h3 h3^
h4 h4^2 *h *h *h ];
b=[f1; f2-f1; f2; f3-f2; f3; f4-f3; f4; f5-f4; 0; 0; 0; 0];

27. Quadratic Spline Interpolation
>> x=[ ]; f=[ ];
>> [A,b]=quadratic(x,f)
p = A\b
9.0000
4.0000
7.3333
dx = 0.1;
z1=x(1):dx:x(2); s1=p(1)+p(2)*(z1-x(1))+p(3)*(z1-x(1)).^2;
z2=x(2):dx:x(3); s2=p(4)+p(5)*(z2-x(2))+p(6)*(z2-x(2)).^2;
z3=x(3):dx:x(4); s3=p(7)+p(8)*(z3-x(3))+p(9)*(z3-x(3)).^2;
z4=x(4):dx:x(5); s4=p(10)+p(11)*(z4-x(4))+p(12)*(z4-x(4)).^2;
H=plot(z1,s1,'r',z2,s2,'b',z3,s3,'m',z4,s4,'g');
xx=-1:0.1:6; fexact=xx.^3-5*xx.^2+3*xx+4;
hold on; G=plot(xx,fexact,'c',x,f,'ro');
set(H,'LineWidth',3); set(G,'LineWidth',3,'MarkerSize',8);
H1=xlabel('x'); set(H1,'FontSize',15);
H2=ylabel('f(x)'); set(H2,'FontSize',15);
H3=title('f(x)=x^3-5x^2+3x+4'); set(H3,'FontSize',15);

28. Quadratic Spline
s4(x)
Exact function
s2(x)
s1(x)
s3(x)

29. Cubic Splines Cubic splines avoid the straight line and the over-swing
Can develop method like we did for quadratic – 4(n–1) unknowns – 4(n–1) equations
interior knot equality
end point fixed
interior knot first derivative equality
assume derivative value if needed

30. Piecewise Cubic Splines
s2 (x)
sn-1 (x)
s1 (x)
Continuous slopes and curvatures x1 x2 x3
xn-1 xn
4(n1) unknowns

31. Piecewise Cubic Splines
s2”(x)
Sn-1”(x)
s1”(x)
s3”(x) x0 x1 x2
xn-1 xn
si (x) - piecewise cubic polynomials
si’(x) - piecewise quadratic polynomials (slope)
si”(x) - piecewise linear polynomials (curvatures)
Reduce to (n-1) unknowns and (n-1) equations for si’’

32. Cubic Splines
Piecewise cubic polynomial with continuous derivatives up to order 2
The function must pass through all the data points
gives 2(n-1) equations
i = 1,2,…, n-1

33. Cubic Splines
2. First derivatives at the interior nodes must be equal:
(n-2) equations
3. Second derivatives at the interior nodes must be equal:

34. Cubic Splines The last two equations
4. Two additional conditions are needed (arbitrary)
The last two equations
Total equations: 2(n-1) + (n-2) + (n-2) +2 = 4(n-1)

35. Cubic Splines Solve for (ai, bi, ci, di) – see textbook for details
Tridiagonal system with boundary conditions c1 = cn= 0

36. Cubic Splines
Tridiagonal matrix

37. Hand Calculations

38. Hand Calculations
can be further simplified since c1 = c4 = 0 (natural spline)

39. Cubic Spline Interpolation

40. Cubic Splines
Piecewise cubic splines (cubic polynomials)

41. Cubic Spline Interpolation
The exact solution is a cubic function
Why cubic spline interpolation does not give the exact solution for a cubic polynomial?
Because the conditions on the end knots are different!
In general, f (x0)  0 and f (xn)  0 !!

42. Cubic Spline Interpolation

43. >> xx=[-1 0 2 5 6]; f = [-5 4 -2 19 58];
>> Cubic_spline(xx,f);
Resulting piecewise function:
s1 =
( )+( )*(x-( ))
s2 =
( )*(x-( )).^2+( )*(x-( )).^3
( )+( )*(x-( ))
( )*(x-( )).^2+( )*(x-( )).^3
( )+( )*(x-( ))
( )*(x-( )).^2+( )*(x-( )).^3
( )+( )*(x-( ))
( )*(x-( )).^2+( )*(x-( )).^3
(i = 1)
(i = 2)
(i = 3)
(i = 4)

44. Cubic Spline Interpolation
Piecewise cubic Continuous slope Continuous curvature
zero curvatures at end knots
exact solution

45. End Conditions of Splines
Natural spline : zero second derivative (curvature) at end points
Clamped spline: prescribed first derivative (clamped) at end points
Not-a-Knot spline: continuous third derivative at x2 and xn-1

46. Continuous third derivatives at x2 and xn-1
>> x = linspace(-1,1,9); y = 1./(1+25*x.^2); >> xx = linspace(-1,1,100); yy = spline(x,y,xx); >> yr=1./(1+25*xx.^2); >> H=plot(x,y,'o',xx,yy,xx,yr,'--');
Continuous third derivatives at x2 and xn-1
Comparison of Runge’s function (dashed red line) with a 9-point not-a-knot spline fit generated with MATLAB (solid green line)

47. Clamped End Spline - Use 11 values including slopes at end points
>> yc = [1 y –4]; % 1 and -4 are the 1st-order derivatives (or slopes)at 1st & last point, respectively. >> yyc = spline(x,yc,xx); >> >> H=plot(x,y,'o',xx,yyc,xx,yr,'--');
Note that first derivatives of 1 and 4 are specified at the left and right boundaries, respectively.

48. MATLAB Functions One-dimensional interpolations yy = spline (x, y, xx)
yi = interp1 (x, y, xi, ‘method’)
yi = interp1 (x, y, xi, ‘linear’)
yi = interp1 (x, y, xi, ‘spline’) – not-a-knot spline
yi = interp1 (x, y,, xi, ‘pchip’) – cubic Hermite
yi = interp1 (x, y,, xi, ‘cubic’) – cubic Hermite
yi = interp1 (x, y,, xi, ‘nearest’) – nearest neighbor

49. MATLAB Function: interp1
Piecewise polynomial interpolation on velocity time series for an automobile

50. Spline Interpolations
>> x=[ ]
x =
>> y=[ ]
y =
>> xi=1:0.1:10;
>> y1=interp1(x,y,xi,'linear');
>> y2=interp1(x,y,xi,'spline');
>> y3=interp1(x,y,xi,'cubic');
>> plot(x,y,'ro',xi,y1,xi,y2,xi,y3,'LineWidth',2,'MarkerSize',12)
>> print -djpeg spline00.jpg

51. Spline Interpolations
Linear
Not-a-knot
Cubic

52. MATLAB’s Functions Two-dimensional interpolations
zi = interp2 (x, y, z, xi, yi)

53. MATLAB Function: interp2
>> [x,y,z]=peaks(100); [xi,yi]=meshgrid(-3:0.1:3,-3:0.1:3);
>> zi = interp2(x,y,z,xi,yi); surf(xi,yi,zi)
>> print -djpeg075 peaks1.jpg
>> [x,y,z]=peaks(10); [xi,yi]=meshgrid(-3:0.1:3,-3:0.1:3);
>> print -djpeg075 peaks2.jpg
10  10 data base
100 100 data base

54. CVEN 302-501 Homework No. 11 Chapter 16 Chapter 19
Problem 16.1(a) (30) – Hand Calculation
Problem 16.1(b) (20)– MATLAB program not-a-knot spline: yy = spline(x, y, xx)
Problem 16.1(c)(20) – MATLAB program
Chapter 19
Problem 19.2 (20)– Hand Calculation
Problem 19.4 (20)– Hand Calculation
Due on Wed. 11/05/08 at the beginning of the period