第12讲内插和外推方法.pptVIP

Chapter 3, Interpolation and Extrapolation;Interpolation Extrapolation;低级多项式;Polynomial Interpolation;Vandermonde Matrix Equation;Condition Number;Norms;Commonly Used Norms;Lagrange’s Formula;Joseph-Louis Lagrange (1736-1813);Neville’s Algorithm;Determine P12 from P1 P2;Determine P123 from P12 P23;Recursion Relation for P;Use Small Difference C D;Deriving the Relation among C D;Evaluate P(3) given 4-points (0,1), (1,2), (2,3),(4,0).;polint( ) Program;polint(), continued; Subroutine polint(xa,ya,n,x,y,dy) Integer n, nmax Parameter (nmax=10) Integer i,m,ns Real den,dif,dift,ho,hp,w,c(nmax),d(nmax) Ns=1 Dif=abs(x-xa(1)) Do 11 i=1,n Dift=abs(x-xa(i)) If (dift.lt.dif) then Ns=i Idf=dift Endif ;; D(i)=hp*den C(i)=ho*den Enddo 12 If(2*ns.lt.n-m) then Dy=c(ns+1) Else Dy=d(ns) Ns=ns-1 Endif Y=y+dy Enddo 13 Return end ;Piecewise Linear Interpolation;Piecewise Polynomial Interpolation;Cubic Spline;;2. We now discuss Interpolation Extrapolation The function passes through all (or at least most) points. ;;;;Lagrange Interpolating Polynomials (CC 18.2) Given n+1 data points (xi,yi), find the nth order polynomial: y = pn(x) = a0 + a1x + a2x2 + … + anxn that passes through all of the points. The Lagrangian polynomials approach employs a set of nth order polynomials, Li(x), such that:;;i;;;;;Cubic Splines;;;;;Natural Cubic Splines;;;;;;;x = 0:10; y = sin(x); xx = 0:0.25:10; yy = spline(x,y,xx); plot(x,y,o,xx,yy) ;npts = 10; xy = [randn(1,npts); randn(1,npts)]; plot(xy(1,:),xy(2,:),ro,LineWidth,2); for n = 1:npts, text(xy(1,n),xy(2,n),[ num2str(n)]) end set(gca,XTick,[],YTick,[]) cv = cscvn(xy); fnplt(c