第二章數(shù)值微分和數(shù)值積分

1、數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS第二章 數(shù)值微分和數(shù)值積分?jǐn)?shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS數(shù)值微分hhxfhxfhhxfxfhxfhxfxfhhh2)()(lim)()(lim)()(lim)( 0001. 函數(shù)f(x)以離散點(diǎn)列給出時(shí)，而要求我們給出導(dǎo)數(shù)值，2. 函數(shù)f(x)過于復(fù)雜這兩種情況都要求我們用數(shù)值的方法求函數(shù)的導(dǎo)數(shù)值微積分中，關(guān)于導(dǎo)數(shù)的定義如下：自

2、然，而又簡(jiǎn)單的方法就是，取極限的近似值，即差商數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICShxfhxfxf)()()( 000向前差商x0 x0+h數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS由Taylor展開hxxfhxhfxfhxf002000),( ! 2)( )()(因此，有誤差)()( ! 2)()()( )(000hOfhhxfhxfxfxR數(shù) 學(xué) 系University

3、of Science and Technology of ChinaDEPARTMENT OF MATHEMATICShhxfxfxf)()()( 000向后差商x0-hx0數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS由Taylor展開hxxfhxhfxfhxf002000),( ! 2)( )()(因此，有誤差)()( ! 2)()()( )(000hOfhhhxfxfxfxR數(shù) 學(xué) 系University of Science and Technology of ChinaDEPART

4、MENT OF MATHEMATICShhxfhxfxf2)()()( 000中心差商x0-hx0 x0+h數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS由Taylor展開23000010102300002020()()()()( ),2!3!()()()()(),2!3!hhf xhf xhfxfxfxxhhhf xhf xhfxfxfxhx因此，有誤差)()( 6)( )( 12 2)()()( )(22212000hOfhffhhhxfhxfxfxR數(shù) 學(xué) 系University of

5、 Science and Technology of ChinaDEPARTMENT OF MATHEMATICSf(x)=exp(x)hf(1.15)R(x)hf(1.15)R(x)0.103.1630-0.00480.053.1590-0.00080.093.1622-0.00400.043.1588-0.00060.083.1613-0.00310.033.1583-0.00010.073.1607-0.00250.023.1575-0.00070.063.1600-0.00180.013.1550-0.0032例：數(shù) 學(xué) 系University of Science and Techn

6、ology of ChinaDEPARTMENT OF MATHEMATICS由誤差表達(dá)式，h越小，誤差越小，但同時(shí)舍入誤差增大，所以，有個(gè)最佳步長(zhǎng)我們可以用事后誤差估計(jì)的方法來確定設(shè)D(h),D(h/2) 分別為步長(zhǎng)為h,h/2 的差商公式。則)2()(hDhD時(shí)的步長(zhǎng)h/2 就是合適的步長(zhǎng)( )( )( )( )( /2)( /2)fxD hO hfxD hO h( )( )( )2( )( /2)( /2)fxD hO hfxD hO h( )( )2( )2 ( /2)fxD hfxD h( )( /2)( )( /2)fxD hD hD h數(shù) 學(xué) 系University of Sci

7、ence and Technology of ChinaDEPARTMENT OF MATHEMATICS 插值是建立逼近函數(shù)的手段，用以研究原函數(shù)的性質(zhì)。因此，可以用插值函數(shù)的導(dǎo)數(shù)近似為原函數(shù)的導(dǎo)數(shù))()()()(xLxfknk誤差插值型數(shù)值微分用Taylor展開分析數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS給定點(diǎn)列20)(,(iiixfx且hxxxx0112，求)( ),( ),( 012xfxfxf解：)(2)()()()(2)()(2210122002212xfhxxxxxfhx

8、xxxxfhxxxxxL)(2)()()()(2)()(2210122002212xfhxxxxxfhxxxxxfhxxxxxL例：數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS20200121() ()3 ()4 ()()( )23hfxLxf xf xf xfh2121021() ()()()( )26hfxLxf xf xfh22220121() ()()4 ()3 ()( )23hfxLxf xf xf xfhTaylor展開分析，可以知道，它們都是)(2hO稱為三點(diǎn)公式三點(diǎn)公式02

9、001221() ()()2 ()()fxLxf xf xf xh11201221() ()()2 ()()fxLxf xf xf xh22201221() ()()2 ()()fxLxf xf xf xh誤差？數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS數(shù)值積分)()()(aFbFdxxfba關(guān)于積分，有Newton-Leibniz 公式但是，在很多情況下，還是要數(shù)值積分：1、函數(shù)有離散數(shù)據(jù)組成2、F(x)求不出3、F(x)非常復(fù)雜定義數(shù)值積分如下：是離散點(diǎn)上的函數(shù)值的線性組合)()(0

10、iniinxfafI稱為積分系數(shù)積分系數(shù)，與f(x)無關(guān)，與積分區(qū)間和積分點(diǎn)有關(guān)數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS)()()(0afabfI例：)2()()()(2)()(00bafabfIafabfI數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS)()(0iniinxfafI為數(shù)值積分，badxxffI)()(為積分，則稱數(shù)值積分有k階代數(shù)精度階代數(shù)精度是指：)()(;, 0

11、),()(11kkniinxIxIkixIxI問題：如果判斷好壞？代數(shù)精度代數(shù)精度 對(duì)任意次數(shù)不高于k次的多項(xiàng)式f(x)，數(shù)值積分沒有誤差數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS用插值函數(shù)的積分，作為數(shù)值積分)()()()()()(00inibaiibaniibannxfdxxldxxfxldxxLfI ia代數(shù)精度代數(shù)精度由Lagrange插值的誤差表達(dá)式，)()!1()()()1(xnfxRnnn，有dxxnfdxxRfIfIbannbann)()!1()()()()()1(可以看

12、出，至少n 階代數(shù)精度nkxxfxfkn,)(, 0)()1(插值型數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS121111122101010nabababaaaxxxxxxnnnnnnnnVandermonde行列式使用盡可能高的代數(shù)精度mixIxIiin, 0),()( niia0已知求系數(shù)所以，如果mn，則系數(shù)唯一前面得到的系數(shù)是最好的嗎？數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATI

13、CS若數(shù)值積分至少n階代數(shù)精度，則系數(shù)唯一( ),0,biiaal x dx in誤差誤差dxxnxfdxxRfIfIbannbann)()!1()()()()()1(數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSNewton-Cotes 積分若節(jié)點(diǎn)可以自由選取，則，一個(gè)自然的辦法就是取等距節(jié)點(diǎn)。對(duì)區(qū)間做等距分割。該數(shù)值積分稱為Newton-Cotes積分?jǐn)?shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHE

14、MATICSniihaxnabhi,0,dtntititttnininhdthinintititttdxxlaninninthaxbaii00)() 1)(1() 1()!( !) 1( ) 1()!( !)() 1)(1() 1()()()(niiCaba設(shè)節(jié)點(diǎn)步長(zhǎng))( niC(b-a)與步長(zhǎng)h無關(guān)，可以預(yù)先求出數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSN1時(shí)2121) 1(10)1(110)1(0dttCdttC)(21)()(21)()(1bfabafabfI梯形公式數(shù) 學(xué) 系Un

15、iversity of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSN2 時(shí)61) 1(4164)2(2161)2)(1(4120)2(220)2(120)2(0dtttCdtttCdtttC)(61)()2(64)()(61)()(2bfababfabafabfISimpson公式數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS1、梯形公式1( )( )()()2!bxafE fx a x bdx此處用了積分中值定理誤差誤差

16、3( )()()()( )2!12bafb ax a x bdxf 數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS2、Simpson 公式)()()()()()()(332fSPSPIfIfSfIfE 注意到，Simpson 公式有3 階代數(shù)精度，因此為了對(duì)誤差有更精確地估計(jì)，我們用3 次多項(xiàng)式估計(jì)誤差)2()2(),()(),()(333bafbaPbfbPafaP為0)(2880)()(2)(! 4)( )(2)(! 4)( )4(52)4(2)4(fabdxbxbaxaxfdxbxba

17、xaxxfbaba)2( )2( ,3bafbaP數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS2)4(32)(! 4)()(! 3)( )(2)( )( )()(cxxfcxcfcxcfxxcfcfxf數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS一般的有kndxxxKfnKfEbannnnn2, 0)(),()!2()()2(12, 0)(),()!1()()1(kndxxKfnKfE

18、bannnnn因此，N-C積分，對(duì)偶數(shù)有n+1 階代數(shù)精度，而奇數(shù)為n 階代數(shù)精度數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS復(fù)化積分?jǐn)?shù)值積分公式與多項(xiàng)式插值有很大的關(guān)系。因此Runge現(xiàn)象的存在，使得我們不能用太多的積分點(diǎn)計(jì)算。采用與插值時(shí)候類似，我們采用分段、低階的方法數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS)( 12)()(2)(311iiixxfhxfxfhxfii1031

19、11031)( 12)(21)()(21 )( 12)()(2)(niiniiniiiinfhbfxfafhfhxfxfhfT誤差誤差做等距節(jié)點(diǎn)，niihaxnabhi, 0,復(fù)化梯形公式數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS由均值定理知)( )( .,.,102nfftsbabaCfnii)( 12)()( )(12)( 12)(2323fnabfabhfnhfEn可以看出，復(fù)化梯形公式是收斂的。如果節(jié)點(diǎn)不等距，還可以做復(fù)化積分嗎？怎么處理？數(shù) 學(xué) 系University of S

20、cience and Technology of ChinaDEPARTMENT OF MATHEMATICS)(2880)2()()(4)(62)()4(522122222iiiixxfhxfxfxfhxfii10)4(5112101210)4(522122)(2880)2()()(2)(4)(3 )(2880)2()()()(62)(miimiimiiniiiiinfhbfxfxfafhfhxfxfxfhfS誤差誤差做等距節(jié)點(diǎn)，mnniihaxnabhi2;, 0,復(fù)化Simpson公式數(shù) 學(xué) 系University of Science and Technology of ChinaDE

21、PARTMENT OF MATHEMATICS由均值定理知)(180)()(2880)()(2880)2()()4(45)4(45)4(5fnabfmabfmhfEn可以看出，復(fù)化Simpson公式是收斂的。數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS定義定義 若一個(gè)積分公式的誤差滿足若一個(gè)積分公式的誤差滿足 且且C 0，則，則稱該公式是稱該公式是 p 階收斂階收斂的。的。 ChfRphlim0)(,)(,)(642hOChOShOTnnn例：例：計(jì)算計(jì)算dxx 10142 解：解： )1

22、()(2)0(161718fxffTkk8kxk 其中其中= 3.138988494 )1()(2)(4)0(241oddeven4fxfxffSkk8kxk 其中其中= 3.141592502運(yùn)算量基運(yùn)算量基本相同本相同數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSLab03 復(fù)化積分1.分別編寫用復(fù)化Simpson積分公式和復(fù)化梯形積 分公式計(jì)算積分的通用程序2.用如上程序計(jì)算積分51( )sin( )I fx dx取節(jié)點(diǎn)xi , i=0,N，N 為 2k,k=0,1,12 ，并計(jì)算誤差

23、，同時(shí)給出誤差階3.簡(jiǎn)單分析你得到的數(shù)據(jù)數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS誤差階：記步長(zhǎng)為h時(shí)的誤差為e,步長(zhǎng)為h/k時(shí)的誤差為ek則，相應(yīng)的誤差階為：)ln()ln(keeokk數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSSample Output ( represents a space)復(fù)化梯形積分，誤差和誤差階為k=0,0.244934066848e00k=1,0.5

24、34607244904 , 1.90.復(fù)化Simpson積分，誤差和誤差階為k=1,0.244934066848e00k=2,0.534607244904e-01 , 4.01.數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS 函數(shù)變化有急有緩，為了照顧變化劇烈部分的誤差，我們需要加密格點(diǎn)。對(duì)于變化緩慢的部分，加密格點(diǎn)會(huì)造成計(jì)算的浪費(fèi)。以此我們介紹一種算法，可以自動(dòng)在變化劇烈的地方加密格點(diǎn)計(jì)算，而變化緩慢的地方，則取稀疏的格點(diǎn)。積分的自適應(yīng)計(jì)算積分的自適應(yīng)計(jì)算數(shù) 學(xué) 系University o

25、f Science and Technology of ChinaDEPARTMENT OF MATHEMATICS先看看事后誤差估計(jì)事后誤差估計(jì)以復(fù)化梯形公式為例)( 12)()()(2fhabfTfIn)( 212)()()(22fhabfTfInn等分區(qū)間2n等分區(qū)間近似有：)( )( ff)()(31)()(22fTfTfTfInnn)()(151)()(22fSfSfSfInnn類似，復(fù)化Simpson公式數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS自適應(yīng)計(jì)算記為復(fù)化一次，2次的

26、Simpson公式,21baSbaS0控制dxxffIba)()(求數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS,21baSbaS15,21baSbaS,)(2baSfI是215,2,22152,2,2121bbaSbbaSbaaSbaaS數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS)()(31)()(22fTfTfTfInnn由前面的事后誤差估計(jì)式，)()()(31)()(222fS

27、fTfTfTfInnnn則，這啟發(fā)我們，可以用低階的公式組合后成為一個(gè)高階的公式。)()()(151)()(222fCfSfSfSfInnnn類似，Romberg積分積分?jǐn)?shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS記為以步長(zhǎng)為h的某數(shù)值積分公式，有)(hI)()()(mmhochhIfImmhohchIfI22)2()(12)(22)(mhIhIhIfI 1222)(mhIhIhIfI數(shù) 學(xué) 系University of Science and Technology of ChinaDEP

28、ARTMENT OF MATHEMATICS有如下的Euler-Maclaurin定理)(12)()2()2()2(222)()()()1(mmmmmmhOhIhIhIhI若)()()(2)(mmhOhIfI為2m階公式，則Romberg 積分就是不斷地用如上定理組合低階公 式為高階公式，進(jìn)而計(jì)算積分 Romberg 算法：算法： ? ? ? T1 =)0(0T T8 =)3(0T T4 =)2(0T T2 =)1(0T S1 =)0(1T R1 =)0(3T S2 =)1(1T C1 =)0(2T C2 =)1(2T S4 =)2(1T數(shù) 學(xué) 系University of Science a

29、nd Technology of ChinaDEPARTMENT OF MATHEMATICS重積分的計(jì)算 badcdxdyyxf),( 在微積分中，二重積分的計(jì)算是用化為累次積分的方法進(jìn)行的。計(jì)算二重?cái)?shù)值積分也同樣采用累次積分的計(jì)算過程。簡(jiǎn)化起見，我們僅討論矩形區(qū)域上的二重積分。對(duì)非矩形區(qū)域的積分，大多可以變化為矩形區(qū)域上的累次積分。a,b,c,d 為常數(shù)，f 在D 上連續(xù)。將它變?yōu)榛鄞畏e分 dcbabadcbadcdydxyxfdxdyyxfdxdyyxf),(),(),(首先來看看復(fù)化梯形公式的二重推廣數(shù) 學(xué) 系University of Science and Technology

30、of ChinaDEPARTMENT OF MATHEMATICS做等距節(jié)點(diǎn)，x軸，y軸分別有：ncdkmabh,dcdyyxf),(先計(jì)算),(21),(),(21),(110nnjjdcyxfyxfyxfkdyyxf，將x作為常數(shù)，有再將y作為常數(shù)，在x方向，計(jì)算上式的每一項(xiàng)的積分),(21),(),(212),(210110000yxfyxfyxfhdxyxfmmiiba),(21),(),(212),(21110nmmininbanyxfyxfyxfhdxyxf二重積分的復(fù)化梯形公式二重積分的復(fù)化梯形公式數(shù) 學(xué) 系University of Science and Technology

31、 of ChinaDEPARTMENT OF MATHEMATICS 1111110111101111),(),(21),(21 ),(21),(),(21 ),(),(njmijinjjmjnjjmmijijnjbajbanjjyxfhyxfyxfhyxfyxfyxfhdxyxfdxyxf數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS 1111,1111111101111000000),( ),( ),(),(),(),(21 ),(),(),(),(41),(njmijijinjmiji

32、njjmnjjminimiimmnbadcyxfchkyxfyxfyxfyxfyxfyxfyxfyxfyxfhkdxdyyxf系數(shù)，在積分區(qū)域的四個(gè)角點(diǎn)為1/4，4個(gè)邊界為1/2，內(nèi)部節(jié)點(diǎn)為1),(),(12)()(222222fykfxhabcdfE誤差數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS類似前面有： minjjijibadcyxfhkdxdyyxf00,),(),(記TnTmvvvVuuuU31,34,32,34,32,34,31,31,34,32,34,32,34,31,10

33、10jijivu ,二重積分的復(fù)化二重積分的復(fù)化Simpson公式公式做等距節(jié)點(diǎn)，x軸，y軸分別有：ncdkmabh,m，n為偶數(shù)數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS),(),(180)()(444444fykfxhabcdfE誤差數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSGauss型積分公式 Newton-Cotes 積分公式，可以知道n為偶數(shù)時(shí)，n+1個(gè)點(diǎn)數(shù)值積分公式有n

34、+1階精度。是否有更高的代數(shù)精度呢？n個(gè)點(diǎn)的數(shù)值積分公式，最高可以到多少代數(shù)精度？本節(jié)會(huì)解決這個(gè)問題。數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS例：在兩點(diǎn)數(shù)值積分公式中，如果積分點(diǎn)也作為未知量，則有4個(gè)未知量，可以列出4個(gè)方程： （以f(x)在-1,1為例）032021111133113001122112001111001110dxxxaxadxxxaxaxdxxaxadxaa可解出：31,31, 1, 11010 xxaa數(shù)值積分公式)31()31(11fffdx具有3階代數(shù)精度，比梯

35、形公式1階代數(shù)精度高數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSbaiiniiinbadxxlaxfafIdxxffI)(, )()()()(0證明：取)()()()()(222120 xxxxxxxxpnn易知：0)(0)(xpIxpIn也就是說，數(shù)值積分公式，對(duì)一個(gè)2n+2階的多項(xiàng)式是有誤差的，所以，n+1個(gè)點(diǎn)的數(shù)值積分公式不超過2n+1階n 個(gè)積分點(diǎn)的數(shù)值積分公式，最高2n1階定理如何構(gòu)造如何構(gòu)造最高階精度的公式？數(shù) 學(xué) 系University of Science and Tech

36、nology of ChinaDEPARTMENT OF MATHEMATICS一般性，考慮積分：0)(,)()()(xWdxxfxWfIba稱為權(quán)函數(shù)定義兩個(gè)可積函數(shù)的內(nèi)積為：badxxgxfxWgf)()()(),(兩個(gè)函數(shù)正交，就是指這兩個(gè)函數(shù)的內(nèi)積為0數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS利用Schmidt 正交化過程，nxx, 1變?yōu)檎换?010( )( )( ),( )( )( )( )( ),( )nninniiiigxfxfx g xgxfxg xg x g x)(

37、,),(),(10 xpxpxpn就可以將多項(xiàng)式基函數(shù)數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS以n階正交多項(xiàng)式的n個(gè)零點(diǎn)為積分點(diǎn)的數(shù)值積分公式有2n1階的代數(shù)精度Gauss點(diǎn)Gauss積分，記為Gn(f)證明：bannndxxWxxxxxffIfIfE)()(,)()()(21,21xxxxfn若 f 為 2n1 次多項(xiàng)式，則為 n1 次多項(xiàng)式又，)(),(xpxnn僅差一個(gè)常數(shù)（零點(diǎn)相同）0)(fE1)(nnPfp具有一個(gè)很好的性質(zhì)：nfIfGn),()(數(shù) 學(xué) 系Universit

38、y of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS(2)求出pn(x)的n n個(gè)零點(diǎn)x1 , x2 , xn 即為Gauss點(diǎn). (1)求出區(qū)間a,b上權(quán)函數(shù)為W(x)的正交多項(xiàng)式pn(x) .(3)計(jì)算積分系數(shù) Gauss型求積公式的構(gòu)造方法數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS)()(),()(,()(00001xpxpxpxpxxxp)()(),()(,()()(),()(,()(111120000222x

39、pxpxpxpxxpxpxpxpxxxp5321141151121142xxdxxdxxdxxdxxx解解 按 Schemite 正交化過程作出正交多項(xiàng)式: 的2點(diǎn)Gauss公式.求積分dxxfx)(112例：0( )1p x 數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS故兩點(diǎn)Gauss公式為 積分系數(shù)為31)(11212211121dxxxxxxdxxlxA112212211211( )3xxAx lx dxxdxxx)()()(535331112ffdxxfxP2(x)的兩個(gè)零點(diǎn)為 ,

40、532531xx數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICS 區(qū)間-1,1上權(quán)函數(shù)W(x)=1的Gauss型求積公式,稱為Gauss-Legendre求積公式求積公式,其Gauss點(diǎn)為L(zhǎng)egendre多項(xiàng)式的零點(diǎn). (1) Gauss-Legendre求積公式求積公式公式的Gauss點(diǎn)和求積系數(shù)可在數(shù)學(xué)用表中查到 .幾種幾種Gauss型求積公式型求積公式由因此,a,b上權(quán)函數(shù)W(x)=1的Gauss型求積公式為batabbaxdttabbafabdxxf)2)()()22(2)(11ban

41、iiixabbafAabdxxf1)22(2)(數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSnxkAknxkAk10260.93246951420.66120938650.2386191861036076157300.467913934620.5773502692130.774596669200.55555555560.888888888970.94910791230.74153118560.405845151400.12948496620.27970539150

42、.38183005050.417959183740.86113631160.33998104360.34785484510.652145154980.96028985650.79666647740.5255324099010122853630.22238103450.31370664590.362683783450.90617984590.538469310100.23692688510.47862867050.5688888889數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHE

43、MATICS 區(qū)間0,)上權(quán)函數(shù)W(x)=e-x的Gauss型求積公式,稱為Gauss-Laguerre求積公式求積公式,其Gauss點(diǎn)為L(zhǎng)aguerre多項(xiàng)式的零點(diǎn). (2) Gauss-Laguerre求積公式求積公式公式的Gauss點(diǎn)和求積系數(shù)可在數(shù)學(xué)用表中查到 .由所以,對(duì)0, + )上權(quán)函數(shù)W(x)=1的積分,也可以構(gòu)造類似的Gauss-Laguerre求積公式:00)()(dxxfeedxxfxx01)()(niixixfeAdxxfi數(shù) 學(xué) 系University of Science and Technology of ChinaDEPARTMENT OF MATHEMATICSnxkAknxkAk20.58588643763.41421356230.85355339050.146446609450.26356031971.41340305913.5964