数值分析：2.5分段低次插值法.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,第二章 函数近似计算的插值问题,2.5,分段低次插值法,Numerical Value Analysis,1,2.5,分段低次插值法,一、高次插值的龙格,(,Runge,),现象,(,插值过程的收敛性问题,),问题,:,所构造的插值多项式 作为,近似函数,是否 的次数愈高,逼近 的效果,愈好,即,利用高次插值多项式的危险性,在,20,世纪初被,Runge,发现,.,2,例子,.,并作图比较,.,解,:,3,不同次数的,Lagrange,插值多项式的比较图,Runge,现象,-5,-4,-3,-2,-1,0,1,2,3,4,5,-1.5,-1,-0.5,0,0.5,1,1.5,2,n=2,n=4,n=6,n=8,n=10,f(x,)=1/(1+x,2,),4,在,-2,2,上,L,10,(x),对,f(x,),逼近较,好,但在端点附近很差,.,可以证明,即随着,n,的增长,L,n,(x,),在两,端点附近的振荡会越来越大,.,高次代数,插值所发生的这种现象称为,Runge,现象,.,在上个世纪初由,Runge,发现,.,这表明,:,并不是插值多项式的次数越高,插值效果越好,精度也不一定是随次数的提高而升高,.,结论,:,不适宜在大范围使用高次代数插值,.,5,解决办法,:,分段,低次,插值,;分段,光滑,插值,;,若从舍入误差分析,知当,n7,时,舍入误差亦会增大,.,可知,Runge,现象是由,f(x,),的高阶导数无界所致,.,6,分段,低次,插值,7,二、分段线性,Lagrange,插值,构造,Lagrange,线性插值,1.,分段线性插值的构造,8,-(1),-(2),显然,当 时,或者通过分段插值基函数 的线性组合来,表示,:,9,其中,且,10,也称折线插值,如右图,曲线的光滑性较差,在节点处有尖点,但如果增加节点的数量,减小步长,会改善插值效果,因此,则,11,由第二节定理,1,可知,n,次,Lagrange,插值多项式的余项为,2.,分段线性插值的误差估计,12,定理,13,三、分段三次,Hermite,插值,可构造两点三次,Hermite,插值多项式,14,其中,我们称,为分段三次,Hermite,插值多项式，其余项为,15,例2.,比较几种插值,.,我们分别用分段二次、三次,Lagrange,插值和,分段两点三次,Hermite,插值作比较,解,:,即,16,f(x),0.80000,0.30769,0.13793,0.07547,0.04160,H,3,(x),0.81250,0.30750,0.13750,0.07537,0.04159,x,0.5,1.5,2.5,3.5,4.8,R,3,(x)=f(x)-H,3,(x),-0.01250000000000,0.00019230769231,0.00043103448276,0.00009972579487,0.00001047427455,L,2,(x),0.87500,0.32500,0.12500,0.07206,0.04087,L,3,(x),0.80000,0.32500,0.13382,0.07443,0.04269,17,分段低次插值的特点,:,计算较容易,可以解决,Runge,现象,可保证收敛性,但插值多项式分段,插值曲线在节点处会出现尖点,不可导,优点,:,缺点,:,18,See you later!,19,