1(x?xk)2(xk?1?x)2maxf(4)(x)a?x?b4!1x?xk?xk?1?x22?[()]maxf(4)(x)a?x?b4!211??4h4maxf(4)(x)a?x?b4!2h4?maxf(4)(x)384a?x?b?
16．求一个次数不高于4次的多项式P（x），使它满足
P(0)?P?(0)?0,P(1)?P?(1)?0,P(2)?0
解：利用埃米尔特插值可得到次数不高于4的多项式
x0?0,x1?1y0?0,y1?1m0?0,m1?11
1H3(x)??yj?j(x)??mj?j(x)j?0j?0?0(x)?(1?2x?x0x?x12)()x0?x1x0?x1x?x1x?x02)()x1?x0x1?x0
?(1?2x)(x?1)2?1(x)?(1?2?(3?2x)x2?0(x)?x(x?1)2?1(x)?(x?1)x2232?H(x)?(3?2x)x?(x?1)x??x?2x 32设P(x)?H3(x)?A(x?x0)2(x?x1)2其中，A为待定常数
?P(2)?1?P(x)??x3?2x2?Ax2(x?1)?A?211从而P(x)?x2(x?3)2 4417．设f(x)?1/(1?x2)，在?5?x?5上取n?10，按等距节点求分段线性插值函数Ih(x)，计算各节点间中点处的Ih(x)与f(x)值，并估计误差。
18．解：若x0??5,x10?5则步长h?1,xi?x0?ih,i?0,1,?,10在小区间
Ih(x)?[xix,i?1f(x)?11?x2
]，分段线性插值函数为上
x?xi?1x?xif(xi)?f(xi?1) xi?xi?1xi?1?xi11?(x?x)i1?xi21?xi?12
?(xi?1?x)各节点间中点处的Ih(x)与
f(x)的值为
当x??4.5时，f(x)?0.0471,Ih(x)?0.0486 当
f(x??3.5x?)时，
f(x?)0.h0I?7x55当
x??2.5时，
0.h1I?3x7 9当
f(x??1.5x?)时，
f(x?)0.h3I?0x77当
x??0.5时，
0.h8I?0x0 0h2误差xmaxf(x)?Ih(x)?maxf??(?)i?x?xi?18?5?x?5又?f(x)?11?x2
?f?(x)??2x,(1?x2)26x2?2f??(x)?(1?x2)324x?24x3f???(x)?(1?x2)4令f???(x)?0得f??(x)的驻点为x1,2??1和x3?0
1f??(x1,2)?,f??(x3)??22
1?maxf(x)?Ih(x)??5?x?5418．求f(x)?x2在[a,b]上分段线性插值函数Ih(x)，并估计误差。 解：在区间[a,b]上，x0?a,xn?b,hi?xi?1?xi,i?0,1,?,n?1,
h?maxhi0?i?n?1?f(x)?x2?函数f(x)在小区间[xi,xi?1]上分段线性插值函数为
Ih(x)??x?xi?1x?xif(xi)?f(xi?1)xi?xi?1xi?1?xi12[xi(xi?1?x)?xi?12(x?xi)]hi误差为
1maxf(x)?Ih(x)?maxf??(?)?hi2xi?x?xi?18a???b?f(x)?x2
?f?(x)?2x,f??(x)?2h2?maxf(x)?Ih(x)?a?x?b419．求f(x)?x4在[a,b]上分段埃尔米特插值，并估计误差。
hi 解：在[a,b]区间上，x0?a,xn?b,hi?xi?1?xi,i?0,1,?,n?1,令h?0max?i?n?1?f(x)?x4,f?(x)?4x3?函数f(x)在区间[xi,xi?1]上的分段埃尔米特插
值函数为
Ih(x)?(?(?(?(x?xi?12x?xi)(1?2)f(xi)xi?xi?1xi?1?xix?xi2x?xi?1)(1?2)f(xi?1)xi?1?xixi?xi?1x?xi?12)(x?xi)f?(xi)xi?xi?1x?xi2)(x?xi?1)f?(xi?1)xi?1?xi
xi4?3(x?xi?1)2(hi?2x?2xi)hixi?14?3(x?xi)2(hi?2x?2xi?1)hi?4xi2(x?x)i?1(x?xi)2hi3
4xi?13?2(x?xi)2(x?xi?1)hi误差为
f(x)?Ih(x)?1(4)44 f(?)(x?xi)2(x?xi?1)2又?f(x)?x4hh4!?maxf(x)?Ih(x)?maxi?a?x?b0?i?n?116161hi4(4)?maxf(?)()24a?x?b2?f(4)(x)?4!?2420．给定数据表如下： Xj Yj
试求三次样条插值，并满足条件：
(1)S?(0.25)?1.0000,S?(0.53)?0.6868;
(2)S??(0.25)?S??(0.53)?0.h0?x1?x0?0.050.25 0.30 0.39 0.6245 0.45 0.53 0.5000 0.5477 0.6708 0.7280 解：
h1?x2?x1?0.09h2?x3?x2?0.06h3?x4?x3?0.08
??j???1?hj?1hj?1?hj,?j?hjhj?1?hj533,?2?,?3?,?4?11457
?1?924,?2?,?3?,?0?11457f(x1)?f(x0)f?x0,x1???0.9540x1?x0f?x1,x2??0.8533f?x2,x3??0.7717f?x3,x4??0.7150
(1)S?(x0)?1.0000,S?(x4)?0.6868d0?6(f?x1,x2??f0?)??5.5200h0f?x1,x2??f?x0,x1???4.3157h0?h1d1?6d2?6d3?6d4?由此得矩阵形式的方程组为 f?x2,x3??f?x1,x2???3.2640h1?h2f?x3,x4??f?x2,x3???2.4300h2?h36(f4??f?x3,x4?)??2.1150h3 2 1 M0 ?5.5200
5 14 2
3 59 14 M1 ?4.3157
2 5 2
3 7 M2 ? ?3.264 04 7 2
M3 ?2.4300
1 2 M4 ?2.1150
求解此方程组得
M0??2.0278,M1??1.4643M2??1.0313,M3??0.8070,M4??0.6539?三次样条表达式为
S(x)?Mj?(yj?(xj?1?x)36hj2?Mj?1(x?xj)36hjMj?1hj62Mjhj6)xj?1?xhj?(yj?1?)x?xjhj
(j?0,1,?,n?1)?将M0,M1,M2,M3,M4代入得