三次样条插值函数的构造与Matlab实现

自动测量与控制O.I.Automation20062511AutomaticMeasurementandControl2006,Vol.25,No.11·76·10061576200611007603Matlab11,21.6500312.4427001MatlabMatlabO242.1AConstructionandRealizationofCubicSplineInterpolationFunctionXUXiao-yong1,ZHONGTai-yong1,2(1.SchoolofMathematics&ComputerScience,YunnanNationalitiesUniversity,Kunming,650031,China;2.Dept.ofMathematics,YunyangTeacherCollege,Danjiangkou442700,China)Abstract:Theboundaryconditionsofcubicsplineweregivenbythepracticalstateattheextremepointsofdiscretefunction.Takingthefirstboundaryconditionasanexample,itusedthesecondorderderivativeofthejunctionstorepresentthecubicsplineinterpolationfunction,andsolvedtherelativeequationswithchasingmethod.TheuniversalprocedureofcubicsplinewasrealizedundertheenvironmentofMatlab,whichcouldcomputetheinterpolationpointanddirectlydisplaytheexactexpressionoffunctionandsegmentalcurveineachinterval.Keywords:Cubicspline;Interpolationfunction;Matlabprogram0Matlab1[a,b]ax1x2xnby1,y2,ynS(x)S(xj)yjj1,2,,nS(x)[xj,xj+1]j1,2,,n1j1Matlab1S(x)[a,b]S(x)S(x)[xj,xj+1]1Sj(x)ajx3bjx2cjxdj,(j1,2,,n1)(1)aj,bj,cj,djS(xj)yj,S(xj0)=S(xj0),(j=2,n1)(2)S'(xj0)=S'(xj0),S(xj0)=S(xj0),(j=2,n1)(3)(2)(3)n3(n2)4n64(n1)231S'(x1)y1'S'(xn)yn'2S(x1)y1S(xn)yny1yn03yf(x)baS(x)S'(x10)S'(xn0)S(x0)S(xn0)12S(x)[xj,xj+1]j=1,2,,n1S(x)[xj,xj+1]S(x)[xj,xj+1]j=1,2,,n1S(xj)=MjS(xj+1)=Mj+1S(x)1jjjjj1j''MhxxMhxx)x(S++−+−=hjxj+1xjS(x)1220060721200609091983自动测量与控制O.I.Automation20062511AutomaticMeasurementandControl2006,Vol.25,No.11·77·AjBjS(x)S(x)jj1jj3jjj31jBxAMh6)xx(Mh6)xx()x(S++−+−=++hjxj+1xjj=1,2,,n1S(xj)yjS(xj+1)yj+1jjjjjj3j1jyBxAMh6)xx(=++−+(4)1jj1j1j1jj3j1jyBxAMh6)xx(+++++=++−(5)(5)(4)j1jjj2jj1jyyhAh6MM−=+−++jj1jjj1jjh6MMhyyA−−−=++Aj(5)1jjj1jjj1j2j1j1jjxh6MMhyyh6MyB+++++⎟⎟⎠⎞⎜⎜⎝⎛−−−−−=jj1j2j1jj1j2jjj1jjj1jjj1j2j1j1jj1jjj1jjjhxxyh6Mhxxh6Myxh6MMhyyh6Myxh6MMhyyBxA−⎟⎟⎠⎞⎜⎜⎝⎛−+−⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛−−−−−+⎟⎟⎠⎞⎜⎜⎝⎛−−−=+++++++++++(6)jj2j1j1jj1j2jjj1jj3jjj31jhxxh6Myhxxh6MyMh6)xx(Mh6)xx()x(S−⎟⎟⎠⎞⎜⎜⎝⎛−+−⎟⎟⎠⎞⎜⎜⎝⎛−+−+−=+++++(7)S(x)jj1jjj1j1jj2jjj21j'h6MMhyyMh2)xx(Mh2)xx()x(S−−−+−+−−=++++(8)S(x)[xj-1,xj][xj,xj+1]j2,,n1)0x(S)0x(Sj'j'+=−xj)0x(S)0x(Sj'1jj'j+=−+j1j1j1jj1j1jj'jM3hhyyM6h)0x(S−−−−−+−+=−1jjjj1jjjj'1jM6hhyyM3h)0x(S+++−−+−=+j1j1j1jj1j1j1jjjj1jjjM3hhyyM6hM6hhyyM3h−−−−−+++−+=−−+−j1j1j1jjjj1j1jj1jjj1jj1j1jhh1hyyhyy6MhhhM2Mhhh+⎟⎟⎠⎞⎜⎜⎝⎛−−−=++++−−−++−−−−(9)j1j1jjhhh+=µ−−j1j1j1jjjj1jjhh1hyyhyy6d+⎟⎟⎠⎞⎜⎜⎝⎛−−−=−−−+jj1µ−=γµjMj2Mj(1µj)Mj+1djj2,3,,n1S'(x1)y'1,S'(xn)y'nS'(x)nn1nn1211M2MMM2β=+γβ=α+−(10)⎟⎟⎠⎞⎜⎜⎝⎛−−=β=γ=α'111211n1yhyyh611⎟⎟⎠⎞⎜⎜⎝⎛−−=β−−−1n1nn'n1nnhyyyh6⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡ββ=⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡γαγαγ−−−−n1n21n1n21n1n1n11ddMMMM22212MMOOO(11)(11)(7)[a,b]3nax1x2xnby1,y2,,yny'1,y'2x0hjxj+1xjj1,2,,n1j1j1j1jjjj1jjj1j1jjhh1hyyhyy6dhhh+⎟⎟⎠⎞⎜⎜⎝⎛−−−=+=µ−−−+−−(j=2,,n1)⎟⎟⎠⎞⎜⎜⎝⎛−−=β⎟⎟⎠⎞⎜⎜⎝⎛−−=β−−−1n1nn'n1nn'111211hyyyh6yhyyh6(11)sj(x)x0]x,x[1jj+y04MatlabMatlab[3]function[]=spline3(X,Y,dY,x0,m)N=size(X,2);s0=dY(1);sN=dY(2);interval=0.025;disp('x0')x0h=zeros(1,N-1);fori=1:N-1h(1,i)=X(i+1)-X(i);endd(1,1)=6*((Y(1,2)-Y(1,1))/h(1,1)-s0)/h(1,1);d(N,1)=6*(sN-(Y(1,N)-Y(1,N-1))/h(1,N-1))/h(1,N-1);fori=2:N-1d(i,1)=6*((Y(1,i+1)-Y(1,i))/h(1,i)-(Y(1,i)-Y(1,i-1))/h(1,i-1))/(h(1,i)+h(1,i-1));end自动测量与控制O.I.Automation20062511AutomaticMeasurementandControl2006,Vol.25,No.11·78·mu=zeros(1,N-1);md=zeros(1,N-1);md(1,N-1)=1;mu(1,1)=1;fori=1:N-2u=h(1,i+1)/(h(1,i)+h(1,i+1));mu(1,i+1)=u;md(1,i)=1-u;endp(1,1)=2;q(1,1)=mu(1,1)/2;fori=2:N-1p(1,i)=2-md(1,i-1)*q(1,i-1);q(1,i)=mu(1,i)/p(1,i);endp(1,N)=2-md(1,N-1)*q(1,N-1);y=zeros(1,N);y(1,1)=d(1)/2;fori=2:Ny(1,i)=(d(i)-md(1,i-1)*y(1,i-1))/p(1,i);endx=zeros(1,N);x(1,N)=y(1,N);fori=N-1:-1:1x(1,i)=y(1,i)-q(1,i)*x(1,i+1);endfprintf('M\n');M=x;fprintf('\n');symst;digits(m);fori=1:N-1pp(i)=M(i)*(X(i+1)-t)^3/(6*h(i))+M(i+1)*(t-X(i))^3/(6*h(i))+(Y(i)-M(i)*h(i)^2/6)*(X(i+1)-t)/h(i)+(Y(i+1)-M(i+1)*h(i)^2/6)*(t-X(i))/h(i);pp(i)=simplify(pp(i));coeff=sym2poly(pp(i));iflength(coeff)~=4tt=coeff(1:3);coeff(1:4)=0;coeff(2:4)=tt;endifx0X(i)&x0X(i+1)L=i;y0=coeff(1)*x0^3+coeff(2)*x0^2+coeff(3)*x0+coeff(4);endval=X(i):interval:X(i+1);fork=1:length(val)fval(k)=coeff(1)*val(k)^3+coeff(2)*val(k)^2+coeff(3)*val(k)+coeff(4);endifmod(i,2)==1plot(val,fval,'r+')elseplot(val,fval,'b.')endholdonclearvalfvalans=sym(coeff,'d');ans=poly2sym(ans,'t');fprintf('[%f,%f]\n',X(i),X(i+1));fprintf('S(%d)=',i);pretty(ans);endfprintf('x0[%f,%f]\n',X(L),X(L+1));fprintf('x0=%f\n',x0);y0X,YdYx0m[1]yf(x)11yf(x)x-1.5012y0.125-119[1.5,2]S(x)S'(1.5)0.75S'(2)14m5x01.5MatlabX[1.5012]Y[0.125119]dY[0.7514]x01.5m5spline3(X,Y,dY,x0,m)x0x01.5MM5.00004.00004.000016.0000[-1.5,0.0]1t2t)1(S23−+=[0.0,1.0]1t2)2(S2−=[1.0,2.0]3t6t4t2)3(S23−+−=x0[1.0,2.0]x01.5y03.75001-2-10121086420-215Matlab[1],,.[M].:.2002,(6):5563.[2].[J].,2000,(6).[3],,,.MATLAB[M].:,2003.10061576200611032006052211,21.3410002.1000832MachineEquipmentFaultDetectionBasedonRealNumberReverseChoosingAlgorithmAbstract:Thechoosingalgorithmaddedtwokindsofchoosingalgorithmintothereversechoosingmechanismofrealnumbercoding.Atfirst,thedetectionsetwaskeptfarfromtheself-space;atsecond,thedetectionw