对函数[tex=2.214x2.357]Hqxa/UCqq6/+StWVpW6nUr/ywv3F3oCfiNclBLHvryo=[/tex] , 在区间[tex=2.0x1.357]AUoDsQBgen8/+sL3yGoyYA==[/tex]上用等距线性插值、等距[tex=3.857x1.0]aSjk4o7nmJkfQs7mJKmLIA==[/tex]3 次插值、等距样条插值，问步长[tex=0.643x1.0]uPu/UBwxTDghY6MHYDLmcA==[/tex]应取多少才能保证各自的截断误差小于[tex=2.0x1.214]FpeOfmuZawZqwM2eXSPGDw==[/tex] ?

2022-06-09

对函数[tex=2.214x2.357]Hqxa/UCqq6/+StWVpW6nUr/ywv3F3oCfiNclBLHvryo=[/tex] , 在区间[tex=2.0x1.357]AUoDsQBgen8/+sL3yGoyYA==[/tex]上用等距线性插值、等距[tex=3.857x1.0]aSjk4o7nmJkfQs7mJKmLIA==[/tex]3 次插值、等距样条插值，问步长[tex=0.643x1.0]uPu/UBwxTDghY6MHYDLmcA==[/tex]应取多少才能保证各自的截断误差小于[tex=2.0x1.214]FpeOfmuZawZqwM2eXSPGDw==[/tex] ?

答案：

解: 因为[tex=15.786x2.929]cUpWZ5FLuDIWKEpLMrDIGMvy5ilnME81SRLw6Vimo4ed7RmDKmfduMp9Ov7JAzyvhQyeuogEyhRKfNR42m8ZzyTUdpjwolhM8afKwEJWTO81v1BRMNgyt6Ti09JYm5lE[/tex], 所以[tex=19.0x2.857]t641V3I3SyDUiJyh8KjtjEvhaxf5lrA7p+QIgcgk4wsWpNydVcTJj6UBRYEGoapasTUU8HLB0sDmyZ7ESyXPRIHwiawsz11e0qcTHyYV6GoNXdSWaDRcDJnq2TqIPYmBT3IJzM45GQyEzjFgm1dq47hw+Pg+HMU0gkqT93EPpYc=[/tex], 因此[tex=11.071x2.786]58uX+ZvA4DEFmqa/P4Lo6HLZ5RZ95HOcpkM+OyZBITOUKlEvKtKXhYWqwYkDa0CYxA2WtMwk6LepD0sBiN12CuuwnCWWu5YBOJ0BohFaJGc=[/tex]若在区间[tex=2.0x1.357]AUoDsQBgen8/+sL3yGoyYA==[/tex]上用等距线性插值，则wu差为:[tex=12.214x2.429]1NX6vCmwSgugGP6peKdXsiOf9630RWhMnYT4VooRkpienX6Xv/Wf1Xmm8n54fiYhXZDlauYBb7o7K6X64dy9+A==[/tex], 对任[tex=3.071x1.357]ZplrpbYpqJE5cGwN4oUp3w==[/tex]欲使[tex=7.929x1.5]7Djk0XKlR/2dCteK2EjJD16OQvJDVVh47bTnLp1EpCg=[/tex], 只须[tex=14.643x1.571]BuX+E3lUbxadf8wzzw1ZluFtRllKn039j00d38mxa61RtaYl7aaP0rQZZkuAmLfS[/tex]若在区间[tex=2.0x1.357]AUoDsQBgen8/+sL3yGoyYA==[/tex]上用等距[tex=3.857x1.0]aSjk4o7nmJkfQs7mJKmLIA==[/tex]3 次插值，则误差为:[tex=24.214x2.857]dSJPonYB/jOfKRSV82CzkzWYuM4uRbXlU9t22A7ER3emKM7dLUxZun9v8qX7KL8qr4XRBVyOU/X2pzbELyyJXEEBbiXmPf50S9q9UsI8xkMuhAAUam2h7hLuSSyEuAQeTESJax/kTzNOPjeiEKcMsvDCeAIj3ba+twvEWF03liQHfrORg0lMr2FakNd8dsuPwTsU5ojkx58RdW0rFlmySeTuHwrqLdM5ebwwuFhQUNI=[/tex], 对任[tex=3.071x1.357]ZplrpbYpqJE5cGwN4oUp3w==[/tex]欲使[tex=8.643x1.5]dSJPonYB/jOfKRSV82CzkzccRUJxdrjOgGESKenSH3Z1d8oKjIuM4Hr4wKSC/huG[/tex],只须[tex=16.857x1.857]nZ1D1i/qswvp6ACv548a5HsVyj4gyMX6UxddqcZs2fLreVKvyqZ3IFVoYA41RVCAx119Ovt+8c2hsfJdKcjRKUw3KxPt58COvNvDrRtejLv/kS8Erimu0uwYfZFLU430[/tex]若在区间[tex=2.0x1.357]AUoDsQBgen8/+sL3yGoyYA==[/tex]上用等距样条插值，则山定理 5，有:[tex=18.5x2.357]X7TSsPAx7qZ/BQdzKUXAozuFbL+OzTWV5EbuUt8AhaAmjWpIc745/pfu0MeYQG754Xmu8QuwQq1GYxuU3KqEyM0HgpuGD8GAu1Z7yex27wlBJhU5WCUf8EiF23DBLSJEu1bbGvzJ6BaFEKLQGQ+cEuogjg/htVRXZSAGaK49vY4=[/tex]欲使[tex=7.714x1.5]X7TSsPAx7qZ/BQdzKUXAoyCWRQg5VdmTIdOzbuHoDro=[/tex]|,只须[tex=16.857x1.857]KhuWJN3Ku564+cthcX1bRc8e3CkGVO3TMzhzb+jz8eO4NS0P63TCNc17h7q3+h+ZZin5I6iR46/jiDKyH9+RTSoaXXdePAg7raNRpy9+oRc=[/tex]

举一反三

内容

0
给定[tex=3.571x1.357]0jgNZNb5KE0SpRQgBt7oQg==[/tex]，设x=0是4重插值节点，x=1是单重插值节点试求相应的Hermite插值公式，并估计误差[tex=4.071x1.357]ZHsKcW72rLaSaexOsDovRw==[/tex]
1
函数[tex=4.143x1.357]UtO6tkZzi2ddaLRNBsQlRA==[/tex]在区间[tex=2.0x1.357]AUoDsQBgen8/+sL3yGoyYA==[/tex]上的最小值是[input=type:blank,size:6][/input]
2
给定函数[tex=4.143x1.357]xe0pQFG03hsSf3z3JfzIEA==[/tex]的一个数表[img=660x189]178fcd27d6cd946.png[/img]试分别采用: (1) 双一次插值 (2) 对[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]二次、对[tex=0.5x1.0]yBR4oiFoTexGaFalQ7m8kg==[/tex] 一次的二元插值; (3) 双二元插值,计算[tex=4.786x1.357]o1tzBI5tDPKClK7CmPWmfQ==[/tex]的近似值。
3
给定数据表如下[tex=16.571x3.357]OOdTrLGt+hva56tTPivt0/r95dokJkdKTk98EGDhBNUwwvPSMqQ9+aeTQ2HsKPWOMIJr4R70TDuHuiTv3S1DPYIPY/mUZJEpF6rOX0bRjPCH63WJeZvfe7ZF1QmFGDncn7MUuSV83DZa00IIMCd1Ja1NlRLKWMLWZCYXBRlqB+N2xKlpGU5wpqwdGr4l6XVT[/tex]（1）用三次插值多项式计算[tex=2.571x1.286]WXl7aTmBZwDKTODr4AwoZA==[/tex]的近似值；（2）用二次插值多项式计算[tex=3.071x1.286]deYvyFCZtS5temkeHqdoNA==[/tex]的近似值；（3）用分段二次插值计算[tex=1.857x1.286]G6WxJ307HB2e1l7Qz3uNbQ==[/tex]（[tex=5.643x1.286]lusWrymqw2MApyoeZ9LVlaG7RaNinVoGHvhuWZKCNLdwng78wI5DjlIpIwT/lExY[/tex]）的近似值能保证有几位有效数字（不计舍入误差）？其中已知[tex=9.714x1.714]a4Eg/bvBEa+RCCx/mnsgtyYoIQK6IBNiDvTBn/Riyu8+ZlHYT+JWvLAydi5Cak5hpQzKmCFCR1NnV40o4yUpRQ==[/tex]。
4
找出函数[tex=3.571x0.929]1KQep/cJ4kpzQWTxXtALGw==[/tex]在区间[tex=1.857x1.357]J8sjPoB94GBNczqzS8+7VA==[/tex]的 11 个等距点上插值的 10 次多项式，打印出这个多项式的牛顿形式中的系数. 计算并打印这个多项式与[tex=3.571x0.929]1KQep/cJ4kpzQWTxXtALGw==[/tex]之差在区间[tex=1.857x1.357]zBn3HDCI1zB5XPvs1Jo9LQ==[/tex]的 33 个等距点上的值. 由此能得出什么结论?