数值分析算法设计.doc

一、题目 在飞机制造业中，机翼的加工是一项关键技术。由于机翼的尺寸很大，通常在图纸中只能标出某些关键点的数据。下表给出的是某型号飞机的机翼上缘轮廓线的部分数据。x0.004.749.5019.0038.0057.0076.00y0.005.328.1011.9716.1517.1016.34x95.00114.00133.00152.00171.00190.00y14.6312.169.697.033.990.00但是，在使用数控机床加工机翼时，由于机床走刀只能沿x方向和y方向走非常小的步长，因此机床编程时需要计算出轮廓线上x坐标每改变1个单位时y的相应坐标。请根据加工要求分别用分段线性插值法、分段二次多项式插值法、分段三次多项式插值法和三次样条插值法，对上表中的数据进行细化。二、算法设计程序中有三个函数实现对数据的处理，分段线性插值，分段二次多项式插值，分段三次多项式插值都在main函数中实现，三中处理方法均采用Lagrange插值多项式的方法。三次样条插值在yt函数中实现。具体设计如下：在次数不高于n的多项式集合，中寻求多项式使其满足条件 i=(0,1,n)满足此种条件的多项式成为n次插值多项式。令取插值基函数为 (k=0,1,n)显然，都是n次多项式，且具有下列性质因此，函数组必在点集上线性无关，并且就是满足插值条件的n次插值多项式。(1) 分段线性插值法将区间按照x的变化分成几个区间，在每段区间上运用线性定义计算区间内部的步长值。即在Lagrange插值公式中，令n=1(2) 分段二次多项式插值在x所在区间上相邻三点之间做出一条曲线，在中间一点与其余两点中点之间的步长值则取曲线上的值，依次计算下去，在两个端点附近的值则按照相邻的曲线上的取值选取。即在Lagrange插值公式中，令n=2(3) 分段三次多项式插值取在x上相邻的四点做曲线，只取其前三点区间内的值作为函数值。依次计算即得函数值。即在Lagrange插值公式中，令n=3（4）三次样条插值对于区间a,b上的一个分划:如果函数s(x)满足条件(1) s(x)在每个子区间(i=0,1,n-1)上是次数不高于k 的多项式；(2) s(x)在区间(a,b)上有k-1阶连续导数。则称s(x)是定义在a,b对应于分划的k次多项式插值。 对应于k次插值共需要确定n+k个系数，在题目中共需要n+3个系数，依据题目所给定条件，程序中选择第三种边界条件。在选择第三种边界条件时，将数组进行了周期性的扩充。在负半周将程序驻足完整。形成两个新的函数组dx,dy.在新的数组基础上应用第三种边界条件既将代入n元k次多项式函数，得到方程组求解方程组，将所的M值代入，即可求得相应的函数值。在求解M值时，程序中采用的是追赶法。三、全部源程序# include # include # define N 13# define M 191double xN=0.00,4.74,9.50,19.00,38.00,57.00,76.00,95.00,114.00,133.00,152.00,171.00,190.00;double yN=0.00,5.32,8.10,11.97,16.15,17.10,16.34,14.63,12.16,9.69,7.03,3.99,0.00;void yt();double div2(int i,int j)double m;m=(i-xj+1)*(i-xj+2)/(xj-xj+1)/(xj-xj+2)*yj;m+=(i-xj)*(i-xj+2)/(xj+1-xj)/(xj+1-xj+2)*yj+1;m+=(i-xj)*(i-xj+1)/(xj+2-xj)/(xj+2-xj+1)*yj+2;return(m);void main()int i=0,j=0;double solu1M=0,solu2M=0,solu3M=0;/*分段线性插值*/ for(i=0;i=xj)&(ixj+1)solu1i=(i-xj+1)/(xj-xj+1)*yj+(i-xj)/(xj+1-xj)*yj+1;elsei-;j+;solu1M=yN-1;printf(分段线性插值后结果为:n);for(i=0;iM;i+)if(i%10=0&i!=0)printf(n);printf(%.2ft,solu1i);printf(n);/*分段二次多项式差值*/j=0;for(i=0;i(x1+x2)/2;i+)solu2i=div2(i,j);j=j+1;for(i=int(xj+xj+1)/2)+1;i=(xj+xj+1)/2)&i(xj+2+xj+1)/2)solu2i=div2(i,j);elsei-;j+;for(i=int(xN-2+xN-3)/2)+1;iM-1;i+) solu2i=div2(i,j); solu2M=yN-1;printf(分段二次插值后结果为:n);for(i=0;iM;i+)if(i%10=0&i!=0)printf(n);printf(%.2ft,solu2i);printf(n);/*三次多项式插值*/j=0;for(i=0;i=xj&ixj+2)solu3i=(i-xj+1)*(i-xj+2)*(i-xj+3)/(xj-xj+1)/(xj-xj+2)/(xj-xj+3)*yj;solu3i+=(i-xj)*(i-xj+2)*(i-xj+3)/(xj+1-xj)/(xj+1-xj+2)/(xj+1-xj+3)*yj+1;solu3i+=(i-xj)*(i-xj+1)*(i-xj+3)/(xj+2-xj)/(xj+2-xj+1)/(xj+2-xj+3)*yj+2;solu3i+=(i-xj)*(i-xj+1)*(i-xj+2)/(xj+3-xj)/(xj+3-xj+1)/(xj+3-xj+2)*yj+3;elsei-;j+;solu3M-1=yN;printf(分段三次插值后结果为:n);for(i=0;iM;i+)if(i%10=0&i!=0)printf(n);printf(%.2ft,solu3i);printf(n);yt();/* 求三次样条插值*/void yt()double h24,u25,p25,d25,solu4380,dx25,dy25;double g25,e25,r25,s25,c25,b25,t25;double z=0;int i,j;for(i=0;iN;i+) /*构建新矩阵,形成周期函数*/dxi=xi;for(i=N;i25;i+)dxi=380.00-x24-i;for(i=0;iN;i+)dyi=yi;for(i=N;i25;i+)dyi=-1*y24-i;for(i=0;i24;i+)hi=dxi+1-dxi; for(i=1;i24;i+) /*计算矩阵中的各阶系数*/ui=hi-1/(hi-1+hi); pi=1-ui; di=6*(dyi+1-dyi)/hi-(dyi-dyi-1)/hi-1)/(hi-1+hi); u24=h0/(h0+h23); p24=1-u24; d24=6*(dy1-dy0)/h0)-(dy24-dy23)/h23)/(h0+h23); for(i=1;i25;i+) /*用追赶法求解矩阵c(M)阵的值*、 bi=2.00; g1=b1; for(i=1;i23;i+)ei=pi/gi; gi+1=bi+1-ui+1*ei; s1=u1/g1; for(i=2;i23;i+)si=-ui*si-1/gi; s23=(p23-u23*s22)/g23; r1=p24; for(i=2;i23;i+) ri=-ri-1*ei-1; r23=u24-r22*e22; for(i=1;i24;i+)z+=ri*si; r24=b24-z; t1=d1/g1; for(i=2;i24;i+)ti=(di-ui*ti-1)/gi; z=0; for(i=1;i0;i-)ci=ti-ei*ci+1-si*c24; c0=c24;j=0; /*回代求结果*/for(i=0;i=dxj&idxj+1)solu4i=cj*(dxj+1-i)*(dxj+1-i)*(dxj+1-i)/(6*hj); solu4i+=cj+1*(i-dxj)*(i-dxj)*(i-dxj)/(6*hj); solu4i+=(dyj-cj*hj*hj/6)*(dxj+1-i)/hj; solu4i+=(dyj+1-cj+1*hj*hj/6)*(i-dxj)/hj;elsei-;j+;printf(三次样条插值后结果为:n);for(i=0;iM;i+)if(i%10=0&i!=0)printf(n);printf(%.2ft,solu4i);printf(n);四、插值后的数据表分段线性插值后结果为:0.001.122.243.374.495.476.066.647.227.818.308.719.129.539.9310.3410.7511.1611.5611.9712.1912.4112.6312.8513.0713.2913.5113.7313.9514.1714.3914.6114.8315.0515.2715.4915.7115.9316.1516.2016.2516.3016.3516.4016.4516.5016.5516.6016.6516.7016.7516.8016.8516.9016.9517.0017.0517.1017.0617.0216.9816.9416.9016.8616.8216.7816.7416.7016.6616.6216.5816.5416.5016.4616.4216.3816.3416.2516.1616.0715.9815.8915.8015.7115.6215.5315.4415.3515.2615.1715.0814.9914.9014.8114.7214.6314.5014.3714.2414.1113.9813.8513.7213.5913.4613.3313.2013.0712.9412.8112.6812.5512.4212.2912.1612.0311.9011.7711.6411.5111.3811.2511.1210.9910.8610.7310.6010.4710.3410.2110.089.959.829.699.559.419.279.138.998.858.718.578.438.298.158.017.877.737.597.457.317.177.036.876.716.556.396.236.075.915.755.595.435.275.114.954.794.634.474.314.153.993.783.573.363.152.942.732.522.312.101.891.681.471.261.050.840.630.420.210.00分段二次插值后结果为:0.001.332.563.664.665.546.316.967.287.838.368.869.349.7910.2110.4910.8811.2511.6211.9712.3112.6312.9513.2413.5313.8014.0614.3114.5414.5714.7814.9915.1815.3615.5415.7015.8616.0116.1516.2816.4016.5116.6216.7116.8016.8816.9417.0016.8616.9116.9516.9817.0217.0417.0617.0817.0917.1017.1017.1017.0917.0817.0717.0417.0216.9916.9516.8216.7816.7316.6816.6316.5816.5216.4616.4016.3416.2716.2016.1316.0615.9815.9015.8215.7415.6515.5315.4415.3515.2515.1515.0514.9514.8514.7414.6314.5214.4114.2914.1714.0513.9313.8113.6813.5513.3313.2013.0712.9412.8112.6812.5512.4212.2912.1612.0311.9011.7711.6411.5111.3811.2511.1210.9910.8810.7510.6210.4910.3610.2310.099.969.829.699.559.429.289.159.018.878.738.598.458.348.208.057.917.777.627.487.337.187.036.886.736.586.426.276.115.955.805.645.485.325.154.994.834.664.504.334.163.993.823.653.483.303.132.952.772.602.422.242.061.871.691.511.321.140.950.760.00分段三次插值后结果为:0.001.412.653.744.705.526.246.877.417.888.368.859.319.7610.1810.5710.9511.3111.6511.9712.3012.6212.9213.2113.4913.7514.0014.2414.4714.6814.8915.0815.2715.4415.6015.7515.8916.0316.1516.2716.3816.4816.5716.6516.7316.8016.8616.9116.9617.0017.0317.0617.0817.0917.1017.1117.1117.1017.1017.0917.0717.0617.0317.0116.9816.9416.9116.8616.8216.7716.7216.6616.6016.5416.4816.4116.3416.2716.2016.1316.0515.9715.8915.8115.7215.6415.5515.4515.3615.2615.1615.0614.9614.8514.7414.6314.5114.3914.2714.1514.0213.9013.7713.6413.5113.3813.2413.1112.9812.8412.7112.5712.4312.3012.1612.0311.9011.7711.6511.5211.3911.2611.1311.0010.8710.7410.6110.4810.3510.2210.099.969.829.699.569.429.299.159.028.888.748.608.478.338.198.047.907.767.627.477.327.187.036.886.746.596.446.296.145.985.835.675.515.355.195.034.864.694.524.344.173.993.803.623.423.233.032.832.632.432.222.011.801.581.361.140.920.690.470.230.00三次样条插值后结果为:-0.001.252.463.614.645.546.276.897.417.888.328.769.209.6210.0410.4510.8511.2411.6111.9712.3112.6412.9513.2413.5213.7914.0414.2814.5114.7214.9215.1115.2915.4615.6215.7615.9016.0316.1516.2616.3716.4616.5516.6316.7116.7716.8316.8916.9316.9817.0117.0417.0617.0817.0917.1017.1017.1017.0917.0817.0717.0517.0217.0016.9616.9316.8916.8516.8016.7516