差商及其性质

4.1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）（1.4§4差商与牛顿插值多项式定义4为函数在jixx,的一阶差商（一阶均差）；)(xf,],[],[)(,,,,ikjikjkjikjixxxxfxxfxfxxxxxxf称为y=在点kjixxx,,的二阶差商（二阶均差）；)(xf（3）一般由函数y=的n－1阶差商表可定义函数的n阶差商。)(xf)(xf称为函数y=在nxxx,,,10点的n阶差商（n阶均差）。],[jixx，称,)()()(,,ijijjijixxxfxfxfxxxxf（1）对于的一阶差商表，再作一次差商，即)(xf（2）由函数y=）（xfxxxxxxfnn],,,[],,,[1010011021,,,,,xxxxxfxxxfnnn即nxxxf,,,21110,,,nxxxfn－1阶差商2基本性质定理5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2.4],[10xxf],[21xxf],[32xxf],[43xxf],,[321xxxf],,[210xxxf],,[432xxxf],,,[3210xxxxf],,,[4321xxxxf],,,[10kxxxf],,[12kkkxxxf],[1kkxxf3差商表计算顺序:同列维尔法，即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。4.2牛顿插值多项式已知)(xfy函数表（4.1）,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）（1.4),(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6),();,,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）)2.4(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7则次多项式是一个若,)2(0nxaxfinii],,,[10kxxxf时当时当nkankn,,0计算步骤:(2)用秦九韶算法或着说用嵌套乘法计算.)(xpn3牛顿插值多项式计算次数(当k=n时)(1)计算差商表(计算的系数))(xpn(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],[10xxf],[21xxf],[32xxf],[43xxf],,[321xxxf],,[210xxxf],,[432xxxf],,,[3210xxxxf],,,[4321xxxxf],,,[10kxxxf],,[12kkkxxxf],[1kkxxf],[10xxfn1n],,[210xxxf2n],,,[3210xxxxf1],,,[10kxxxf除法次数(k=n):)(213212nnn(2)用秦九韶算法或着说用嵌套乘法计算.)(xpn)())((,,,))(](,,[)](,[)()(11010102100100nnnxxxxxxxxxfxxxxxxxfxxxxfxfxP乘法次数:n优点:(1)计算量小,较L-插值法减少了3-4倍.(2)当需要增加一个插值节点时,只需再计算一项,即)()(1xpxpkk)())((,,,10110kkxxxxxxxxxf---递推公式(适合计算机计算).],[{10xxf乘除法次数大约为:nnn)(212)(0xx)(0xx)(0xx)(0xx)(1xx)(1xx)(1xx)(0xf],,[{210xxxf}},,,10nxxxf)(1nxxnn232124两函数相乘的差商定理8（两函数相乘的差