计算方法之数值微分与积分

1、hxfhxfxfh)()(lim)(0000 000)()(lim)(0 xxxfxfxfxx hxfhxfxfiii)()()( 2)(! 2)()()()(iiiixxfxxxfxfxf hxxfhxfhxfhxfiiiiiii )(! 2)()()(2)()(2)()()()(hOfhhxfhxfxfxRiiiii hhxfxfxfiii)()()( hhxfhxfxfiii2)()()( )(! 3)(! 2)()()()(! 3)(! 2)()()(232132fhxfhxfhxfhxffhxfhxfhxfhxfiiiiiiii hxhxhOfhffhhhxfhxfxfxRiiiii

2、i )()(6)()(122)()()()(222120 .1020. 0/ )5 . 050. 1()20. 0(0 .1510. 0/ )00. 250. 0()0( ff0 .2510. 0/ )50. 100. 1()40. 0( fnixLxfini, 2 , 1)()( )()()()(11111 iiiiiiiixfxxxxxfxxxxxLxfiiiiiiiiiixxhhxfxfxxxfxxxfxf 11111,)()()()()( 12121111,)(2)()()()(2)()()(iiiiiiiixxfhhxfxfxffhhxfxfxf )(,)(2)()()()(2)()

3、()(1212211222212 iiiiiiiiiiiiixxxxhxfhxxxxxfhxxxxxfhxxxxxLxf )(3)(3)(4)(21)()(6)()(21)()(3)()(4)(321)(3221222211221 fhxfxfxfhxffhxfxfhxffhxfxfxfhxfiiiiiiiiiii312231218! 3)(4! 2)()(2)()(! 3)(! 2)()()()(hfhxfxfhxfxfhfhxfxfhxfxfiiiiiiii )(3)()(4)(321)(1221 fhxfxfxfhxfiiii )()()(aFbFdxxfba niiibanxxfdxx

4、ffI1)(lim)()( baniiinxxffIdxxffI1)()()()( niiinxfAfI0)()( niiinxfxlxL0)()()( nibaiibabaniiinxfdxxldxxfxldxxLfI00)()()()()()( baiidxxlA)( baniiinnxfAdxxLfI0)()()( baiidxxlA)( 20200031)0 . 20 . 0)(5 . 00 . 0()0 . 2)(5 . 0()(dxxxdxxlA 202011916)0 . 25 . 0)(0 . 05 . 0()0 . 2)(0 . 0()(dxxxdxxlA 20202295)

5、5 . 00 . 2)(0 . 00 . 2()5 . 0)(0 . 0()(dxxxdxxlA)0 . 2(5)5 . 0(16)0 . 0(391)(2ffffI niiinxfAfI0)()(mkxIxIxEknkkn, 1 , 0,0)()()( )()()!1()()()()()()(0)1(badxxxnfdxxLxfdxxLdxxffRbaniinbanbanban nkkkxaxf0)( 3011)(dxxfI)()()()()(332211003xfAxfAxfAxfAfI 30300083)30)(20)(10()3)(2)(1()(dxxxxdxxlA 30301189)

6、31)(21)(01()3)(2)(0()(dxxxxdxxlA 30302289)32)(12)(02()3)(1)(0()(dxxxxdxxlA 30303383)23)(13)(03()2)(1)(0()(dxxxxdxxlA)3(83)2(89)1(89)0(83)(3fffffI )()()()()(3322110030 xfAxfAxfAxfAdxxf 8389898327894321321030332130232130321303210AAAAdxxAAAdxxAAAxdxAAAdxAAAA babaiiibabaxfAdxxfxldxxfxldxxfxlxfxldxxLfI10

7、110011001)()()()()()()()()()()()1(010100)()(21)(CababdxxxxxdxxlAbaba )1(101011)()(21)(CababdxxxxxdxxlAbaba 21,21)1(1)1(0 CC babfafabdxxffI)()(2)()()()(2)(bfafabfT )()(12)()(! 2)()()()()()(3111bafabdxbxaxfdxxLxfdxxLdxxffRbabababa 202211002211002)()()()()()()()()()()()()()()()(iiibabababababaxfAdxxfxl

8、dxxfxldxxfxldxxfxlxfxlxfxldxxLdxxffI)2(020102100)()(61)()()(CababdxxxxxxxxxdxxlAbaba )2(121012011)()(64)()()(CababdxxxxxxxxxdxxlAbaba )2(212021022)()(61)()()(CababdxxxxxxxxxdxxlAbaba 61,64,61)2(2)2(1)2(0 CCC)()2(4)(6)(bfbafafabfS )2()2(bafbaP )2()2(bafbaP )()2)(! 4)()(2)4(bxbaxaxfxR )()2)()()(2bxbax

9、axxkxR 0)2(;0)2(;0)(;0)( baR ba R b R aR)()2)()()()()(2btbatatxktPtft x ba b a0)(;0)2(;0)(;0)( a2ba xb0)(1 0)(2 0)(3 0)2( ba 0)()4( ! 4)()()4( fxk )()()()()()()(2fSPIPIfIfSfIfR )()()2(4)(6)()(fSbPbaPaPabPSPI )(2880)()()2)(! 4)()()()()()()4(52)4(2 fabdxbxbaxaxfdxxPxfPIfIfRbaba ),2(, )(2880)()(90)()4(

10、5)4(52baabhfabfhfR 1011)(dxxfI.693147. 02ln| )1ln()1(1111)(101010 xxdxdxxfIxxf 11)(75. 0)5 . 00 . 1(201)()(2)( bfafabfT694444. 0)213241(61)()2(4)(6)( bfbafafabfS3)1(2)(xxf 5)4()1(24)(xxf 1666667. 0122)(12)()(31 fabfR0083333. 0288024)(2880)()()4(52 fabfRnC0nC1nC2nC3nC4nC5nC6)(7)(32)(12)(32)(790)(4321

11、0 xfxfxfxfxfabfC badxxxnffRbaniinn 0)1()()!1()()( baniniiixxxfxfhdxxfdxxfii10101)()(2)()(1 11)()(2)(2)(niinbfxfafhfT)()(12)()(31bafabfR )()(12)(113 ikinkknxxfhfR)(max)(1)(min,1,xffnxfbaxnkkbax nkkfnf1)(1)()()(12)(12)()(12)(2233bafhabfnabfnhfRn )()(4)(62)()(4)(6)(2212222122222222 iiiiiiiixxxfxfxfhxfx

12、fxfxxdxxfii bamiiiixfxfxfhdxxf1022122)()(4)(62)( )()(2)(4)(3)(1011212 mimiiinbfxfxfafhfS),2(, )(90222)4(5 ikikbxmabhfh)(, )(180)(180)()(90)(90)()()4(4)4(45)4(51)4(5bafhabfnabfmhfhfSfImkkn 708333. 0)213221(41 )()(2)(2)(112 iibfxfafhfT693254. 021322)7454(41 121)()(2)()(4)(3 )()(2)(4)(3 )()(2)(4)(3)(23

13、1112101211210124 bfxfxfxfafhbfxfxfafhbfxfxfafhfSiiiimiimiixexf )(4223102112)(12)()()( nefnabfTfIn exfxf )()()4(xexfxf )()()4(44102128801)()( emfSfIn)(21)()(21)(11bfxfafhfTniinn )()(2)()(2)(12/12102/122 iinniiinnxfxfhxfxfhfT)()(21)()(21)()(2)(2)(2)(102/1112102/11122 niiniinniiniinnxfbfxfafhbfxfxfafhf

14、T)()(21)(2fHfTfTnnn 102/1)()(niinnxfhfH)(12)()(2fhabfTfInn )()2(12)()(22fhabfTfInn ,ba )()(ff )()( 4)()(2fTfIfTfInn )()(31)()(22fTfTfTfInnn fTfIn )()(2fTfTnn3)()(2 )()(151)()(22fSfSfSfInnn 10sindxxx)()(31)()(22fTfTfTfInnn )(31)(34)()(31)()(2222fTfTfTfTfTfTnnnnnn )()(2)(1bfafabfT )()2(2)(4)(2bfbafafa

15、bfT )()2(4)(6)(1bfbafafabfS )(31)(34)(121fTfTfS )(31)(34)(242fTfTfS )(31)(34)(2fTfTfSnnn )()(151)()(22fSfSfSfInnn )(151)(1516)()(151)()(2222fSfSfSfSfSfSnnnnnn )(151)(1516)(2fSfSfCnnn )(631)(6364)(2fCfCfRnnn )(0kT)(1kT)(2kT)(3kT)0(0T)1(0T)1(1T)2(0T)2(1T)2(2T)3(0T)3(1T)3(2T)3(3T)4(0T)4(1T)4(2T)4(3T )3 , 2 , 1(,144, 3 , 2 , 1,212)21(21,)()(2)1(1)(1)(1211110)(000)0(01mmkTTTkhabhhiafhTTabhbfafhTmkmkmmkmkkkikkkkk 121)5 . 0(01(210kihiafhkTkT141114 mmmkTmkTmkT 10sindxxx)(0kT)(1kT)(2kT)(3kT niiinxfAfI0)()( bammmbabadxxxAxAdxxxAxAdxxxAxA101010101111000100 11)()(dxxfdttgbaxabbat22 03202313021