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Chapter 16

Chapter 16. Curve Fitting: Splines. Spline Interpolation. There are cases where polynomial interpolation is bad Noisy data Sharp Corners (slope discontinuity) Humped or Flat data Overshoot Oscillations. Example : f ( x ) = sqrt ( abs ( x )).

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Chapter 16

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  1. 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. 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

  10. Example: Ship Lines waterline

  11. Bow Stern

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

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

  14. Linear Splines si (x) = ai + bi (x xi) x1 x2 x3 x4

  15. Linear Splines Identical to Lagrange interpolating polynomials

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

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

  18. 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

  19. 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

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

  21. Piecewise Quadratic Splines • 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 Example with n = 5 3(n1) equations for 3(n1) unknowns 2(n1) + (n2) + (1) = 3(n1)

  22. 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)

  23. 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)

  24. Example: Quadratic Spline

  25. Quadratic Spline Interpolation function [A, b] = quadratic(x, f) % exact solution f(x)=x^3-5x^2+3x+4 % x=[-1 0 2 5 6]; f=[-5 4 -2 19 58]; 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=[1 0 0 0 0 0 0 0 0 0 0 0 0 h1 h1^2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 h2 h2^2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 h3 h3^2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 h4 h4^2 0 1 2*h1 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 2*h2 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 2*h3 0 -1 0 0 0 1 0 0 0 0 0 0 0 0 0]; b=[f1; f2-f1; f2; f3-f2; f3; f4-f3; f4; f5-f4; 0; 0; 0; 0];

  26. Quadratic Spline Interpolation >> x=[-1 0 2 5 6]; f=[-5 4 -2 19 58]; >> [A,b]=quadratic(x,f) p = A\b -5.0000 9.0000 0 4.0000 9.0000 -6.0000 -2.0000 -15.0000 7.3333 19.0000 29.0000 10.0000 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);

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

  28. 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

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

  30. 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’’

  31. 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

  32. 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: (n-2) equations

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

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

  35. Cubic Splines Tridiagonal matrix

  36. Hand Calculations

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

  38. Cubic Spline Interpolation

  39. Cubic Splines • Piecewise cubic splines (cubic polynomials)

  40. 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 !!

  41. Cubic Spline Interpolation

  42. >> xx=[-1 0 2 5 6]; f = [-5 4 -2 19 58]; >> Cubic_spline(xx,f); Resulting piecewise function: s1 = (-5.000000)+(11.081218)*(x-(-1.000000)) s2 = (0.000000)*(x-(-1.000000)).^2+(-2.081218)*(x-(-1.000000)).^3 Resulting piecewise function: s1 = (4.000000)+(4.837563)*(x-(0.000000)) s2 = (-6.243655)*(x-(0.000000)).^2+(1.162437)*(x-(0.000000)).^3 Resulting piecewise function: s1 = (-2.000000)+(-6.187817)*(x-(2.000000)) s2 = (0.730964)*(x-(2.000000)).^2+(1.221658)*(x-(2.000000)).^3 Resulting piecewise function: s1 = (19.000000)+(31.182741)*(x-(5.000000)) s2 = (11.725888)*(x-(5.000000)).^2+(-3.908629)*(x-(5.000000)).^3 (i = 1) (i = 2) (i = 3) (i = 4)

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

  44. 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

  45. >> 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)

  46. 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.

  47. 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

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

  49. Spline Interpolations >> x=[1 2 4 7 9 10] x = 1 2 4 7 9 10 >> y=[3 -2 5 4 9 2] y = 3 -2 5 4 9 2 >> 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

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