數(shù)值分析第四版習(xí)題及答案

1、第四版數(shù)值分析習(xí)題第一章緒論1.設(shè)x0x的相對(duì)誤差為δ求lnx的誤差.2.設(shè)x的相對(duì)誤差為2％求nx的相對(duì)誤差.3.下列各數(shù)都是經(jīng)過(guò)四舍五入得到的近似數(shù)即誤差限不超過(guò)最后一位的半個(gè)單位試指出它們是幾位有效數(shù)字:123451.10210.031385.656.43071.0.xxxxx??????4.利用公式(3.3)求下列各近似值的誤差限:12412324()()()ixxxiixxxiiixx??其中1234xxxx均為第3題所給的數(shù)

2、.5.計(jì)算球體積要使相對(duì)誤差限為1％問(wèn)度量半徑R時(shí)允許的相對(duì)誤差限是多少6.設(shè)028Y?按遞推公式11783100nnYY???(n=12…)計(jì)算到100Y.若取783≈27.982(五位有效數(shù)字)試問(wèn)計(jì)算100Y將有多大誤差7.求方程25610xx???的兩個(gè)根使它至少具有四位有效數(shù)字(783≈27.982).8.當(dāng)N充分大時(shí)怎樣求211Ndxx????9.正方形的邊長(zhǎng)大約為100㎝應(yīng)怎樣測(cè)量才能使其面積誤差不超過(guò)1㎝210.設(shè)212

3、Sgt?假定g是準(zhǔn)確的而對(duì)t的測(cè)量有0.1秒的誤差證明當(dāng)t增加時(shí)S的絕對(duì)誤差增加而相對(duì)誤差卻減小.11.序列ny滿足遞推關(guān)系1101nnyy???(n=12…)若021.41y??(三位有效數(shù)字)計(jì)算到10y時(shí)誤差有多大這個(gè)計(jì)算過(guò)程穩(wěn)定嗎12.計(jì)算6(21)f??取21.4?利用下列等式計(jì)算哪一個(gè)得到的結(jié)果最好36311(322)99702.(21)(322)????13.2()ln(1)fxxx???求f(30)的值.若開平方用六位函

4、數(shù)表問(wèn)求對(duì)數(shù)時(shí)誤差有多大若改用另一等價(jià)公式22ln(1)ln(1)xxxx??????計(jì)算求對(duì)數(shù)時(shí)誤差有多大14.試用消元法解方程組?1010121210102.xxxx????假定只用三位數(shù)計(jì)算問(wèn)結(jié)果是否可靠13.證明1200.njnjyyy????????14.若1011()nnnnfxaaxaxax????????有n個(gè)不同實(shí)根12nxxx?證明?10021.1()nknjknaknjjxfx??????????15.證明n階均差

5、有下列性質(zhì):i)若()()Fxcfx?則????0101nnFxxxcfxxx???ii)若()()()Fxfxgx??則??????010101nnnFxxxfxxxgxxx?????.16.74()31fxxxx????求017222f?????及018222f?????.17.證明兩點(diǎn)三次埃爾米特插值余項(xiàng)是(4)22311()()()()4!()kkkkRxfxxxxxx????????并由此求出分段三次埃爾米特插值的誤差限.18

6、.求一個(gè)次數(shù)不高于4次的多項(xiàng)式()Px使它滿足(0)(1)PPk???并由此求出分段三次埃爾米特插值的誤差限.19.試求出一個(gè)最高次數(shù)不高于4次的函數(shù)多項(xiàng)式()Px以便使它能夠滿足以下邊界條件(0)(0)0PP???(1)(1)1PP???(2)1P?.20.設(shè)??()fxCab?把??ab分為n等分試構(gòu)造一個(gè)臺(tái)階形的零次分段插值函數(shù)()nx?并證明當(dāng)n??時(shí)()nx?在??ab上一致收斂到()fx.21.設(shè)2()1(1)fxx??在5

7、5x???上取10n?按等距節(jié)點(diǎn)求分段線性插值函數(shù)()hIx計(jì)算各節(jié)點(diǎn)間中點(diǎn)處的()hIx與()fx的值并估計(jì)誤差.22.求2()fxx?在??ab上的分段線性插值函數(shù)()hIx并估計(jì)誤差.23.求4()fxx?在??ab上的分段埃爾米特插值并估計(jì)誤差.24.給定數(shù)據(jù)表如下:jx0.250.300.390.450.53jy0.50000.54770.62450.67080.7280試求三次樣條插值()Sx并滿足條件i)(0.25)1.0

8、000(0.53)0.6868SS????ii)(0.25)(0.53)0.SS????25.若??2()fxCab?()Sx是三次樣條函數(shù)證明i)????????222()()()()2()()()bbbbaaaafxdxSxdxfxSxdxSxfxSxdx????????????????ii)若()()(01)iifxSxin???式中ix為插值節(jié)點(diǎn)且01naxxxb??????則??????()()()()()()()()()ba