数值计算方法第5讲牛顿插值总结课件

,Numerical Computational Method,数值计算方法,插 值 法,第 二,章,数值计算方法,国家精品课程,主讲教师：杨爱民,http:/210.31.198.78/eol/jpk/course/welcome.jsp?courseId=1220,问题的引入,思考,1,问题的由来,问题的实质,思考,2,提法的抽象,新概念的诞生,新概念的初识,学习计算方法的建议,思考,3,思考,4,准确理解概念,特性（独有的性质）,新算法研究,算法的警示,思考,5,思考,6,算法的应用,算法的进一步研究,思考,7,思考,8,算法原理,警示：,A,！,B!C!,能解决的专业问题,联想与展望,学习建议,第二章 插值法,插值法的一般理论,Newton,插值,Lagrange,插值,分段低次插值,Hermite,插值、样条插值,1,3,4,2,5,插,值,法,Lagrange,插值,Newton,插值,样条插值,误差估计,分段插值,两点式,点斜式,等距节点,算,法,比,较,推广方法,均差,差分,知 识 结 构 图,Chapter 2 Interpolation,一般理论,插值多项式,Newton,前插后插公式,差商及其性质,Newdon,插值,多项式的构造,Newdon,插值多项式余项,差分及其应用,Newdon,插值,法的基本思路,三、牛顿插值法,.,0,的一阶差商（亦称均差）,.,关于点,k,x,x,),(,),(,),(,0,0,0,为函数,称,k,k,k,x,f,x,x,x,f,x,f,x,x,f,-,-,=,牛顿,插值,法,差商及其性质,差商定义,二阶差商,:,K,阶差商,:,特别地,差商记号,牛顿,插值,法,差商及其性质,差商具有线性,牛顿,插值,法,差商及其性质,差商可表示为函数值的线性组合,差商与函数值的关系,牛顿,插值,法,差商及其性质,差商与函数值的关系,观察与思考,牛顿,插值,法,差商及其性质,差商与所含节点的顺序无关,建议记忆,差商与节点的关系,牛顿,插值,法,差商及其性质,差商与导数的关系,则,N,阶差商和,N,阶导数密切相关！,证明见后,牛顿,插值,法,差商性质总结,牛顿,插值,法,Newdon,插值,法的基本思路,建立,Newdon,插值,公式的理由,具有规律性,又有承袭性,不具有承袭性,每当增加一个节点时,不仅要增加求和的项数,而且以前的各项也必须重新计算,.,具有严格的规律性,便于记忆,优点,缺点,使其满足：,牛顿,插值,法,Newdon,插值,法的基本思路,通过节点,思考题,需要引入新的概念,牛顿,插值,法,Newdon,插值,多项式的构造,Newdon,插值,多项式的构造,归纳,构造多项式：,Newdon,插值,多项式,展开即为,由插值多项式存在唯一性可知：,Newdon,插值,多项式的余项,从而：,牛顿插值余项,由此可得差商的另一性质,差商与导数的关系,说明每增加一个结点，,Newton,插值多项式只增加一项，具有承袭性！,观察与思考,插,值,法,N_L,插值,多项式的比较,说明每增加一个结点，,Newton,插值多项式只增加一项，具有承袭性！,Newdon,Lagrange,基函数简便,承袭性,计算简便,基函数规律,可分析性,计算量较大,误差可估,误差可估,比,较,牛顿,插值,法,一阶均差,二阶均差,三阶均差,四阶均差,差商计算表,Newdon,插值,的计算,例,2.4,解题步骤,(1),完成差商表；,(2),求出插值多项式；,(3),求出插值；,(4),估计误差；,典型例题分析,依照 的函数表，求,5,次牛顿插值多项式，,并由此计算 的近似值,.,0.40,0.41075,0.55,0.57815,0.65,0.69675,0.80,0.88811,0.90,1.02652,1.05,1.25382,(5),给出直观解释。,解析,(1),完成差商表；,(2),求出插值多项式；,(3),求出插值；,(4),估计误差；,Newdon,插值,的例题,一阶均差,二阶均差,三阶均差,四阶均差,五阶均差,0.40,0.41075,0.55,0.57815,1.11600,0.65,0.69675,1.18600,0.28000,0.80,0.88811,1.27573,0.35893,0.197333,0.90,1.02652,1.38410,0.43348,0.212952,0.0312381,1.05,1.25382,1.51533,0.52493,0.228667,0.0314286,0.000380952,(5),给出直观解释。,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,例,2.4,Newdon,插值,的例题,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,第1步,完成差商表,一阶均差,二阶均差,三阶均差,四阶均差,五阶均差,0.40,0.41075,0.55,0.57815,1.11600,0.65,0.69675,1.18600,0.28000,0.80,0.88811,1.27573,0.35893,0.197333,0.90,1.02652,1.38410,0.43348,0.212952,0.0312381,1.05,1.25382,1.51533,0.52493,0.228667,0.0314286,0.000380952,例,2.4,Newdon,插值,的例题,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,一阶均差,二阶均差,三阶均差,四阶均差,五阶均差,0.40,0.41075,0.55,0.57815,1.11600,0.65,0.69675,1.18600,0.28000,0.80,0.88811,1.27573,0.35893,0.197333,0.90,1.02652,1.38410,0.43348,0.212952,0.0312381,1.05,1.25382,1.51533,0.52493,0.228667,0.0314286,0.000380952,第2步,求出插值多项式,例,2.4,1,基函数,Newdon,插值,的例题,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,第3步,求出插值,N(0.596)=0.6319174965773844,例,2.4,例,2.4,Newdon,插值,的例题,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,第4步,估计误差,N(0.596)=0.6319174965773844,例,2.4,解题步骤,(1),完成差商表；,(2),求出插值多项式；,(3),求出插值；,(4),估计误差；,典型例题的求解实验,依照 的函数表，求,5,次牛顿插值多项式，,并由此计算 的近似值,.,0.40,0.41075,0.55,0.57815,0.65,0.69675,0.80,0.88811,0.90,1.02652,1.05,1.25382,(5),几何直观验证。,第1步,完成差商表,Clearx,k,y,x0,x1,x2,x3,x4,x5=0.4,0.55,0.65,0.80,0.90,1.05;,y0,y1,y2,y3,y4,y5=0.41075,0.57815,0.69675,0.88811,1.02652,1.25382;,Tableyk,k,0,5/N;,fi_,j,_:=(,yj-yi)/(xj-xi,),Tablefi,i+1,i,0,4/N;,fi_,j_,k,_:=(,fj,k-fi,j)/(xk-xi,),Tablefi,i+1,i+2,i,0,3/N;,fi_,j_,k_,l,_:=(,fj,k,l-fi,j,k)/(xl-xi,),Tablefi,i+1,i+2,i+3,i,0,2/N;,fi_,j_,k_,l_,m,_:=(,fj,k,l,m-fi,j,k,l)/(xm-xi,),Tablefi,i+1,i+2,i+3,i+4,i,0,1/N;,fi_,j_,k_,l_,m_,p,_:=(,fj,k,l,m,p-fi,j,k,l,m)/(xm-xi,),Tablefi,i+1,i+2,i+3,i+4,i+5,i,0,0/N;,程序设计,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,例,2.4,典型例题的求解实验,程序设计,A=y0,y1,y2,y3,y4,y5,0,f0,1,f1,2,f2,3,f3,4,f4,5,0,0,f0,1,2,f1,2,3,f2,3,4,f3,4,5,0,0,0,f0,1,2,3,f1,2,3,4,f2,3,4,5,0,0,0,0,f0,1,2,3,4,f1,2,3,4,5,0,0,0,0,0,f0,1,2,3,4,5;,TransposeA,/N;,MatrixForm,%,0.41075 0 0 0 0 0,0.57815,1.116,0 0 0 0,0.69675 1.186,0.28,0 0 0,0.88811 1.27573 0.358933,0.197333,0 0,1.02652 1.3841 0.433467 0.212952,0.0312381,0,1.25382 1.51533 0.524933 0.228667 0.0314286,0.000380952,第1步,完成差商表,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,例,2.4,典型例题的求解实验,程序设计,0.41075 0 0 0 0 0,0.57815,1.116,0 0 0 0,0.69675 1.186,0.28,0 0 0,0.88811 1.27573 0.358933,0.197333,0 0,1.02652 1.3841 0.433467 0.212952,0.0312381,0,1.25382 1.51533 0.524933 0.228667 0.0314286,0.000380952,第2步,求出插值多项式,a0=y0;,a1=f0,1;,a2=f0,1,2;,a3=f0,1,2,3;,a4=f0,1,2,3,4;,a5=f0,1,2,3,4,5;,NUx,=a0+Sumak*,Product(x-xm,),m,0,k-1,k,1,5/N,Expand%,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,例,2.4,典型例题的求解实验,第3步,求出插值,N%145/.x-0.596,20,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,例,2.4,典型例题的求解实验,估计误差,第4步,程序设计,ClearA,g1,g2,xx=0.40,0.55,0.65,0.80,0.90,1.05;,yy,=0.41075,0.57815,0.69675,0.88811,1.02652,1.25382;,A=Tablexxk,yyk,k,1,6;,g1=,ListPlotA,Prolog,-AbsolutePointSize15;,InterpolationA,InterpolationOrder,-5,g2=Plot%x,x,0.40,1.05,Showg1,g2,N%0.596,20,插值原理计算的数值,牛顿插值数值,依照 的函数表，求,5,次牛顿插值多项式，并由此计算 的近似值,.,例,2.4,典型例题的求解实验,ClearA,g1,g2,xx=0.40,0.55,0.65,0.80,0.90,1.05;,yy,=0.41075,0.57815,0.69675,0.88811,1.02652,1.25382;,A=Tablexxk,yyk,k,1,6;,g1=,ListPlotA,Prolog,-AbsolutePointSize15;,InterpolationA,InterpolationOrder,-5,g2=Plot%x,x,0.40,1.05,Showg1,g2,程序设计,几何直观验证