Splines and Piecewise Interpolation. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

Transcription

1 Splines and Piecewise Interpolation Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

2 Splines n 1 intervals and n data points 2

3 Splines (cont.) Go through the points Match first derivatives at the interior point Match first and second derivatives 3

4 Linear spline 4

5 Example: first-order spline Q: Evaluate the function at x = 5 A: Choose the second interval from x = 4.5 to x = 7 5

6 Table lookup using sequential search function yi = TableLook(x, y, xx) n = length(x); if xx < x(1) xx > x(n) error('interpolation outside range') end % sequential search i = 1; while(1) if xx <= x(i + 1), break, end i = i + 1; end % linear interpolation yi = y(i) + (y(i+1)-y(i))/(x(i+1)-x(i))*(xx-x(i)); 6

7 Table lookup using binary search function yi = TableLookBin(x, y, xx) n = length(x); if xx < x(1) xx > x(n) error('interpolation outside range') end % binary search il = 1; iu = n; while (1) if iu - il <= 1, break, end im = fix((il + iu) / 2); if x(im) < xx il = im; T = [ ]; else density = [ iu = im; ]; end val=tablelookbin(t,density,350); end % linear interpolation yi = y(il) + (y(il+1)-y(il))/(x(il+1)-x(il))*(xx - x(il)); 7

8 Quadratic spline Go through the points Let Match first derivatives at the interior point 8

9 Example: Quadratic spline f 1 f 3 Go through the points (Assume that the 2 nd derivative is zero at the first point, c 1 =0) f 2 f 4 Match first derivatives at the interior points 9

10 Cubic spline Go through the points Let Match first derivatives at the interior point Match second derivatives at the interior point 10

11 Cubic spline (cont.) substitute d i (Go through the node) substitute b i b i-1 (Match first derivatives at the node) Let Equations 2 n-1 11

12 Cubic spline (cont.) Assume that the 2 nd derivative is zero at the first node c 1 =0 Match second derivatives at node (Equation 1) Assume that the 2 nd derivative is zero at the last node (Equation n) 12

13 Cubic spline (cont.) Tridiagonal matrix! 13

14 Example: Natural cubic spline 14

15 Example: Natural cubic spline (cont.) substitute 15

16 End conditions (The 2 nd derivatives =0 at the ends) Not-a-Knot end condition: Force continuity of the 3 rd derivative at the second and the next-to-last knots. 16

17 MATLAB built-in function to implement piecewise interpolation Example: Runge s function 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,'--') 17

18 MATLAB built-in function to implement piecewise interpolation (Clamped condition) Create a new vector yc that has the desired first derivatives as its first and last elements x = linspace(-1,1,9); y = 1./(1+25*x.^2); xx = linspace(-1,1); yc = [1 y -4]; yyc = spline(x,yc,xx); yr = 1./(1+25*xx.^2); plot(x,y,'o',xx,yyc,xx,yr,'--') 18

19 MATLAB built-in function to implement piecewise interpolation yi = interp1(x, y, xi, 'method'); 'nearest nearest neighbor interpolation. (zero-order polynomials) 'linear linear interpolation 'spline piecewise cubic spline interpolation (identical to the spline function) 'cubic piecewise cubic interpolation 'pchip' piecewise cubic Hermite interpolation 19

20 Example using MATLAB s interp1 function t = [ ]; v = [ ]; tt = linspace(0,110); vl = interp1(t,v,tt); % default: linear interpolation plot(t,v,'o',tt,vl) 20

21 interp1(t,v,tt) interp1(t,v,tt,'nearest') interp1(t,v,tt,'spline') interp1(t,v,tt,'pchip') 21

22 Multidimensional interpolation Bilinear interpolation 22

23 Bilinear interpolation Using the Lagrange interpolation 23

24 Example of bilinear interpolation The measured temperatures at a number of coordinates on the surface of a rectangular heated plate: Use bilinear interpolation to estimate the temperature at x i = 5.25 and y i =

25 Multidimensional interpolation in MATLAB 2- and 3-dimensional piecewise interpolation by interp2 and interp3 zi = interp2(x, y, z, xi, yi, 'method'); The methods can be linear, nearest, spline, or cubic x=[2 9]; y=[1 6]; z=[ ;55 70]; zi=interp2(x,y,z,5.25,4.8); 25

26 Example of 2-D interpolation The temperature distribution on a rectangular plate for the range 2 x 0 and 0 y 3 clear, clc, clf x=linspace(-2,0,100); y=linspace(0,3,100); 8 7 [X,Y] = meshgrid(x,y); 6 5 Z=2+X-Y+2*X.^2+2*X.*Y+Y.^2; cs=surfc(x,y,z); f(x 1,x 2 ) xlabel('x_1'); ylabel('x_2'); zlabel('f(x_1,x_2)'); x x

27 Example of 2-D interpolation (cont.) x=linspace(-2,0,9); y=linspace(0,3,9); [X,Y] = meshgrid(x,y); Z=2+X-Y+2*X.^2+2*X.*Y+Y.^2; xunk=-1.63; yunk=1.627; ztrue=2+xunk-yunk+2*xunk.^2+2*xunk.*yunk+yunk.^2; zlinear=interp2(x,y,z,xunk,yunk); et_linear=abs((ztrue-zlinear)/ztrue)*100; zspline=interp2(x,y,z,xunk,yunk,'spline'); et_spline=abs((ztrue-zspline)/ztrue)*100; 27

28 Reference Steven C. Chapra "Applied Numerical Methods with MATLA B", 3rd ed., McGraw Hill,