升洛火掌數(shù)值分析上機(jī)實(shí)驗(yàn)報(bào)告題目：插值法學(xué)生姓名學(xué)院名稱計(jì)算機(jī)學(xué)院專業(yè)計(jì)算機(jī)科學(xué)與技術(shù)時(shí)間實(shí)驗(yàn)?zāi)康?、掌握三種插值方法：牛頓多項(xiàng)式插值，三次樣條插值，拉格朗日插值2、學(xué)會(huì)matlab提供的插值函數(shù)的使用方法實(shí)驗(yàn)內(nèi)容1、已知函數(shù)在下列各點(diǎn)的值為xi0.20.40.60.81.0f(x)i0.980.920.810.640.38試用4次牛頓插值多項(xiàng)式七（x）及三次樣條函數(shù)，（x）（自然邊界條件）對(duì)數(shù)據(jù)進(jìn)行插值。用圖給出{（xi,yi）,X|=0.2+0.08i,i=0,1,11,10},P4（x）及S（x）。2、在區(qū)間［-1,1］上分別取n=10,20用兩組等距節(jié)點(diǎn)對(duì)龍格函數(shù)f（x）=1/（1+25x2）作多項(xiàng)式插值及三次樣條插值，對(duì)每個(gè)n值，分別畫出插值函數(shù)及f（x）的圖形。3、下列數(shù)據(jù)點(diǎn)的插值x01491625364964y012345678可以得到平方根函數(shù)的近似，在區(qū)間［0,64］上作圖。用這9個(gè)點(diǎn)作8次多項(xiàng)式插值L8(x)用三次樣條(第一邊界條件)程序求，(x)從得到結(jié)果看在[0,64]上，哪個(gè)插值更精確，在區(qū)間[0,1]上。兩種插值哪個(gè)更精確？實(shí)現(xiàn)方法進(jìn)入matlab開發(fā)環(huán)境依據(jù)算法編寫代碼調(diào)試程序運(yùn)行程序(1)牛頓插值多項(xiàng)式：P=f(x)+f[x,x](x-x)+f[x,x,x](x-x)(x-x)+???+f[x,x，…，x]

n00100120101n(x-x0)(x-x])三次樣條插值：用三次樣條插值函數(shù)由題目分析知，要求各點(diǎn)的沖值：一20000100.50020.50000—3.75000D.50020網(wǎng)0——T姍000,50020.500-a750000002\\m4\0實(shí)驗(yàn)代碼如下：(1)牛頓插值多項(xiàng)式程序：functionvarargout=newton(varargin)clear,clcx=[0.20.40.60.81.0];fx=[0.980.920.810.640.38];newtonchzh(x,fx);functionnewtonchzh(x,fx)n=length(x);FF=ones(n,n);FF(:,1)=fx';fori=2:nforj=i:nFF(j,i)=(FF(j,i-1)-FF(j-1,i-1))/(x(j)-x(j-i+1));endendfori=1:nfprintf('%4.2f',x(i));forj=1:ifprintf('%10.5f',FF(i,j));endfprintf('\n');end三次樣條插值程序：functionsanciyangtiao(n,s,t)x=[0.20.40.60.81.0];y=[0.980.920.810.640.38];n=5forj=1:1:n-1h(j)=x(j+1)-x(j);endforj=2:1:n-1r(j)=h(j)/(h(j)+h(j-1));endforj=1:1:n-1u(j)=1-r(j);endforj=1:1:n-1f(j)=(y(j+1)-y(j))/h(j);endforj=2:1:n-1d(j)=6*(f(j)-f(j-1))/(h(j-1)+h(j));endd(1)=0d(n)=0a=zeros(n,n);forj=1:1:na(j,j)=2;endr(1)=0;u(n)=0;forj=1:1:n-1a(j+1,j)=u(j+1);a(j,j+1)=r(j);endb=inv(a)m=b*d'p=zeros(n-1,4);forj=1:1:n-1p(j,1)=m(j)/(6*h(j));p(j,2)=m(j+1)/(6*h(j));p(j,3)=(y(j)-m(j)*(h(j)八2/6))/h(j);p(j,4)=(y(j+1)-m(j+1)*(h(j)八2/6))/h(j);endend圖程序：x=[0.20.40.60.81.0];y=[0.980.920.810.640.38];plot(x,y)holdonfori=1:1:5y(i)=0.98-0.3*(x(i)-0.2)-0.62500*(x(i)-0.2)*(x(i)-0.4)-0.20833*(x(i)-0.2)*(x(i)-0.4)*(x(i)-0.6)-0.52083*(x(i)-0.2)*(x(i)-0.4)*(x(i)-0.6)*(x(i)-0.8)endk=[011011]x0=0.2+0.08火kfori=1:1:4y0(i)=0.98-0.3*(x(i)-0.2)-0.62500*(x(i)-0.2)*(x(i)-0.4)-0.20833*(x(i)-0.2)*(x(i)-0.4)*(x(i)-0.6)-0.52083*(x(i)-0.2)*(x(i)-0.4)*(x(i)-0.6)*(x(i)-0.8)endplot(x0,y0,'o',x0,y0)holdony1=spline(x,y,x0)plot(x0,y1,'o')holdons=csape(x,y,'variational')fnplt(s,'r')holdongtext('EyzINui6xdE?±^^Q')gtext('O-I^In')gtext('4zIA￡iU2aOp')(2)多項(xiàng)式插值程序：function[C,D]=longge(X,Y)n=length(X);D=zeros(n,n)D(:,1)=Y'forj=2:nfork=j:nD(k,j)=(D(k,j-1)-D(k-1,j-1))/(X(k)-X(k-j+1));endendC=D(n,n);fork=(n-1):-1:1C=conv(C,poly(X(k)))m=length(C);C(m)=C(m)+D(k,k);endend三次樣條插值程序：functionS=longgesanci(X,Y,dx0,dxn)N=length(X)-1;H=diff(X);D=diff(Y)./H;A=H(2:N-1);B=2*(H(1:N-1)+H(2:N));C=H(2:N);U=6*diff(D);B(1)=B(1)-H(1)/2;U(1)=U(1)-3*(D(1));B(N-1)=B(N-1)-H(N)/2;U(N-1)=U(N-1)-3*(-D(N));fork=2:N-1temp=A(k-1)/B(k-1);B(k)=B(k)-temp*C(k-1);U(k)=U(k)-temp*U(k-1);endM(N)=U(N-1)/B(N-1);fork=N-2:-1:1M(k+1)=(U(k)-C(k)*M(k+2))/B(k);endM(1)=3*(D(1)-dx0)/H(1)-M(2)/2;M(N+1)=3*(dxn-D(N))/H(N)-M(N)/2;fork=0:N-1S(k+1,1)=(M(k+2)-M(k+1))/(6火H(k+1));S(k+1,2)=M(k+1)/2;S(k+1,3)=D(k+1)-H(k+1)*(2火M(k+1)+M(k+2))/6;S(k+1,4)=Y(k+1);endend三次樣條函數(shù)程序代碼：functionsanci3(n,s,t)y=[012345678];x=[01491625364964];n=9forj=1:1:n-1h(j)=x(j+1)-x(j);endforj=2:1:n-1r(j)=h(j)/(h(j)+h(j-1));endforj=1:1:n-1u(j)=1-r(j);endforj=1:1:n-1f(j)=(y(j+1)-y(j))/h(j);endforj=2:1:n-1d(j)=6*(f(j)-f(j-1))/(h(j-1)+h(j));endd(1)=0d(n)=0a=zeros(n,n);forj=1:1:na(j,j)=2;endr(1)=0;u(n)=0;forj=1:1:n-1a(j+1,j)=u(j+1);a(j,j+1)=r(j);endb=inv(a)m=b*d't=ap=zeros(n-1,4);

p(j,1)=m(j)/(6*h(j));p(j,2)=m(j+1)/(6*h(j));p(j,3)=(y(j)-m(j)*(h(j)八2/6))/h(j);p(j,4)=(y(j+1)-m(j+1)*(h(j)八2/6))/h(j);end拉格朗日插值程序：functiony=lagrange(x0,y0,x)n=length(x0);m=length(x);fori=1:mz=x(i);s=0.0;fork=1:np=1.0;forj=1:nifj~=kp=p*(z-x0(j))/(x0(k)-x0(j));endends=p*y0(k)+s;endy(i)=s;endend實(shí)驗(yàn)結(jié)果1.牛頓插值多項(xiàng)式結(jié)果:0.200.900000.400.0.200.900000.400.92000-0.300000.600.81000-0.55000-0.62500O.SO0.64000-0.85000-0.75000-0.208331.000.3S000-1.30000-1.12500-0.62500.52083所以有四次插值牛頓多項(xiàng)式為:P4（x）=0.98-0.3(x-0.2)-0.62500(x-0.2)(x-0.4)-0.20833(x-0.2)(x-0.4)(x-0.6)-0.52083(x-0.2)(x-0.4)(x-0.6)(x-0.8)三次樣條插值結(jié)果:>>sanciyangtiao-3.750000000000005-4.499999999999997-&750000000000005-3.750000000000005-4.-3.750000000000005-4.499999999999997-&750000000000005-3.750000000000005-4.499999999999997-6.750000000000005Coluims1through40.500000000000000-0,1339285714285710.535714285714286-0.1428571428571430.0357142857142860.03571428571428S-0.1428571428571430.571428571428571-0.142857142857143-0.0089285714285710.035714285714286-0.-0,1339285714285710.535714285714286-0.1428571428571430.0357142857142860.03571428571428S-0.1428571428571430.571428571428571-0.142857142857143-0.0089285714285710.035714285714286-0.1428671428571430.6357L42857142B6斯國(guó)口Co1'Jims1through40.500000000000000-0.13392S57112S5710.0357142857142860.500000000000000-0.13392S57112S5710.035714285714286-0.00892857U2S571000.535714285714236-0.1428571428571430.03571423571428600-0.U2857112H5714.30.5^1428571428571-0.142857142S57143000.0(3571428&7142B6-0.1428571428571430.5357L42357142S60CciliumE0-0.0089285711285710.035714285714286-0.13392857U285710.5000000000000000-1.6071^2857142361-1.071428571128568-3.1071^28571428600A得到m=(0-1.6071-1.0714-3.10710),則可得:5們.」-t)3-1.3393{r-f'.2)+4.9OO(f).4-.r)+4fi53fi(.r-C.2)if.^(1.2.(1.4':_—L339306-r)3-0.8S29(r-04、+4.6瑚@6—時(shí)+4.08SFM—0.4)if藁限遍(叫=[_o^mfO.8-r)3-2.58930?一商，+4.08ST(0.8-r)+3.3036(.—0.6)if*060.8][-258fi3|L0-r)3-0^-O.S):J4-3.30361]J.i-.r)+L9(r-D.S)ifh&LO]圖形為：2.多項(xiàng)式插值，n=10時(shí):n=20時(shí):三次樣條插值，n=10時(shí):n=20時(shí):三次樣條插值程序結(jié)果:命令行蜀口L-oimniis.inrougny0-0.0020253699400200.0-0.0020253699400200.005400986506719-0.0160679348574890.051232214863735-0.1696617005152340.575034382439513-0.13349012449491000.000548557692089-0.0014627671783030.004351732357237-0.0138753915255950.0a5350044702a43-0.15573847S5773950.5361535753840380-0.0001469237389520.000391812637206-0.a011fi56425956S80.003716622730070-0.0123080476880470.041715663304659-0.1436125648350100.5000000000000000-0.5209350826236720.055326鴕69964焚-0.02608^988814462a.000578307672522-0.00293190158Sd06-0.OOOS3038424B052-0.0009059940413180解得:M0=0;M1=-0.5209;M2=0.0558;M3=-0.0261;M4=0.0006;M5=-0.0029;M6=-0.0008;M「--0.0009;M8=0；則三次樣條函數(shù):‘0：1一上尸--O.D86a(.v-