23牛顿插值

§2.3均差与牛顿插值多项式拉格朗日插值多项式结构对称，使用方便。但由于是用基函数构成的插值，这样要增加一个节点时，所有的基函数必须全部重新计算，不具备承袭性，还造成计算量的浪费。这就启发我们去构造一种具有承袭性的插值多项式来克服这个缺点，也就是说，每增加一个节点时，只需增加相应的一项即可。这就是牛顿插值多项式。3.1差商（均差）及性质1差商（均差）已知y=)(xf函数表)()()()(1010nnxfxfxfxfxxxx),(jixxji当)(xf则在nnxxxxxx,,,,,,12110上平均变化率分别为：,)()(,010110xxxfxfxxf,)()(,121221xxxfxfxxf.)()(,111nnnnnnxxxfxfxxf，即有定义：定义为f(x)的差商3.1（）§2.3差商与牛顿插值多项式商）阶均差（均差也称为差的为kxf)(111021010],...,[],,...,[],...,,[kkkkkkxxxxxfxxxxfxxxf121020210],[],[],,[:xxxxfxxfxxxf二阶均差.,)()()(],[20000的一阶（均差）关于点为函数称定义kkkkxxxfxxxfxfxxf--=.,...,1,0),(][nixfxfii的零阶差商。关于称为iixxfxf)(][特别地均差及其性质2基本性质kjijkjiijxxxf00)()(kjjkjxxf01)()(（2）k阶差商kxxxf,,,10关于节点kxxx,,,10是对称的，或说均差与节点顺序无关，即例如：共6个ijkxxxf,,jikxxxf,,kjixxxf,,,,,jkixxxfkijxxxf,,ikjxxxf,,kxxxf,,,10kjkjjjjjjjjxxxxxxxxxxxf01110)())(())(()(的线性组合，即)(xf的k阶差商kxxxf,,,10是函数值)(,),(),(10kxfxfxf（1）kxxxf,,,10kxxxf,,,0101,,,xxxfkkkxxxf,,,10kjkjjjjjjjjxxxxxxxxxxxf01110)())(())(()(分析：当k=1时,01110010)()(][xxxfxxxfxxf，(1)可用归纳法证明。(2)利用(1)很容易得到。只证(1)010110)()(,xxxfxfxxf证明：（1）当k=1时,010110)()(,xxxfxfxxf011100)()(xxxfxxxf时成立，即有假设当nk111111121)())(()()(][njnjjjjjjjnxxxxxxxxxfxxxf，，njnjjjjjjjnxxxxxxxxxfxxxf011010)())(()()(][，，111111121)())(()()(][njnjjjjjjjnxxxxxxxxxfxxxf，，],,,[110nnxxxxf，则由定义0110121,,,,,xxxxxfxxxfnnn011xxnnjnjjjjjjjnjjnjxxxxxxxxxxxxxxxxxf1111101001)())(())(()()()(#)())(())(()(1011110njnjjjjjjjjxxxxxxxxxxxf时成立，即有假设当nknjnjjjjjjjnxxxxxxxxxfxxxf011010)())(()()(][，，))())(()(020100nxxxxxxxf)())(()((121111nnnnnxxxxxxxf(0阶差商)一阶差商二阶差商三阶差商k阶差商ix0x1x2x3x4xkx)(ixf)(0xf)(1xf)(2xf)(3xf)(4xf)(kxf表2.1],[10xxf],[21xxf],[32xxf],[43xxf],,[321xxxf],,[210xxxf],,[432xxxf],,,[3210xxxxf],,,[4321xxxxf],,,[10kxxxf],,[12kkkxxxf],[1kkxxf3差商表计算顺序:每次用前一列同行的差商与前一列上一行的差商再作差商。3.2牛顿插值多项式已知)(xfy函数表（3.1）,由差商定义及对称性，得000)()(,xxxfxfxxf)()(,)()(000axxxxfxfxf110010],[],[,,xxxxfxxfxxxf)()(,,],[],[110100bxxxxxfxxfxxf221010210,,,,],,,[xxxxxfxxxfxxxxf)()](,,,[,,,,221021010cxxxxxxfxxxfxxxfnnnnxxxxxfxxxfxxxf,,,,,,],,,[10100)()](,,,[,,,,,,01010dxxxxxfxxxfxxxfnnnn1牛顿插值多项式的推导，)()()()(1010nnxfxfxfxfxxxx3.1（）),(jixxji当将(b)式两边同乘以,)(0xx)()(,)()(000axxxxfxfxf)()(,,],[],[110100bxxxxxfxxfxxf)()](,,,[,,,,221021010cxxxxxxfxxxfxxxf)()](,,,[,,,,,,01010dxxxxxfxxxfxxxfnnnn)())((,,,11010nnxxxxxxxxxf))(())(](,,,[1100nnnxxxxxxxxxxxf))(](,,[)](,[)()(102100100xxxxxxxfxxxxfxfxf)(0xx],[0xxf)(,00xxxxf抵消))((10xxxx10,,xxxf)(,,110xxxxxf抵消)())((110nxxxxxx)](,,,[2210xxxxxxf10,,,nxxxf抵消)()(0xfxf],[10xxf)(0xx))((10xxxx210,,xxxf)(0xx)](,,,[,,,010nnnxxxxxfxxxf)())((110nxxxxxx)())((110nxxxxxx(d)式两边同乘以,把所有式子相加,得，))((10xxxx,(c)式两边同乘以))(())(](,,,[)())((,,,))(](,,[)](,[)()(110011010102100100nnnnnxxxxxxxxxxxfxxxxxxxxxfxxxxxxxfxxxxfxfxf记)())((,,,))(](,,[)](,[)()(11010102100100nnnxxxxxxxxxfxxxxxxxfxxxxfxfxP))(())(](,,,[)(1100nnnnxxxxxxxxxxxfxR))(())(](,,,[1100nnnxxxxxxxxxxxf)())((,,,))(](,,[)](,[)(11010102100100nnxxxxxxxxxfxxxxxxxfxxxxfxf---牛顿插值多项式---牛顿插值余项)(],,,[10jnjnxxxxxf可以验证),,1,0)(()(nixPxfini，即满足插值条件,因此)(xPn可得以下结论。)(xf)(xPn)(xRn定理),();,,1,0))((,(jixxnixfxjiii当则满足插值条件),,1,0(),()(nixPxfini的插值多项式为：)()()(xRxPxfnn（牛顿插值多项式）(3.2)其中，)](,[)()(0100xxxxfxfxPn)())(](,,,[11010nnxxxxxxxxxf---牛顿插值多项式)(],,,,[)(010ininnxxxxxxfxR---牛顿插值余项2n+1阶差商函数与导数的关系由n次插值多项式的唯一性，则有)()(xLxPnn,牛顿插值多项式)(xPn)(xLn与拉格朗日插值多项式都是次数小于或等于n的多项式,只是表达方式不同.?因为nnnHxLxP)()(，而的基函数可为:nHnxx,,,1)1()())((,,,1)2(1100nxxxxxxxx)(,),(),()3(10xlxlxln已知函数表)(xfy],[10xxf],,,[10nxxxf)(0xf牛顿插值多项式系数牛顿插值多项式系数牛顿插值多项式系数1)(nxf的若阶导数存在时，由插值多项式的唯一性有余项公式)()()(xPxfxRnn)(],,,,[010ininxxxxxxf)()!1()(0)1(ininxxnf！)1(],,,,[)1(10nfxxxxfnnn+1阶差商函数导数其中ba,且bax,为包含),,1,0(nixi区间.依赖于则n阶差商与导数,],[)()1(阶导数存在在区间设nbaxf的关系为！nfxxxfnn)(10],,,[其中，ba,的区间。，，，为包含nxxxba10,n+1阶差商函数与导数的关系定理则次多项式是一个若,)2(0nxaxfinii],,,[10kxxxf时当时当nkankn,,0。Newton插值的优点是：每增加一个节点，插值多项式只增加一项，即：],,,[)())(()()(110101nnnnxxxfxxxxxxxNxN因此便于递推运算。而且Newton插值的计算量小于Lagrange插值。的近似值。算插值多项式，并由此计次牛顿求的函数表给出)596.0(4,)(fxf.出均差表首先根据给定函数表造例1解析,8.065.055.04.003134.065.055.04.019733.055.0)4.0(28.04.0116.141075.04xxxxxxxxxxxNixif一阶均差二阶均差三阶均差四阶均差五阶均差1.116001.186000.280001.275730.358930.197331.384100.433480.213000.031341.515330.524920.228630.03126-0.00012一阶均差二阶均差三阶均差四阶均差五阶均差0.400.410750.550.578150.650.696750.800.888110.901.026521.050.25382,8.065.055.04.003134.065.055.04.019733.055.0)4.0(28.04.0116.141075.04xxxxxxxxxxxN于是63192.0596.0596.04Nf..6,8,7,4,1)(,5,4,3,2,1插值多项式求四次牛顿时设当iixfx练习kxkf(xk)一阶差商二阶差商三阶差商四阶差商012