数值微分与数值积分-总结复习+习题课(陈)

1、hxfhxfxfiihi)()(lim)(0hhxfxfiih)()(lim0hhxfhxfiih20)()(limjijixxxxxfxfij)()(limhxfhxfxfiii)()()(hhxfxfii)()(hhxfhxfii2)()()()(2)()()()(hOfhhxfhxfxfxRiiiii hxhxhOfhhhxfhxfxfxRiiiiii )()()()()()(2262)()()()()()(hOfhhhxfxfxfxRiiiii 2hxxfhxfhxfhxfiiiiiii )(!2)()()(20 .1020. 0/ )5 . 050. 1()20. 0(0 .1510

2、. 0/ )00. 250. 0()0( ff0 .2510. 0/ )50. 100. 1()40. 0( fnixLxfini, 2 , 1)()( 12121111,)(2)()()()(2)()()(iiiiiiiixxfhhxfxfxffhhxfxfxf )()()()(11111 iiiiiiiixfxxxxxfxxxxxLxfiiiiiiiiiixxhhxfxfxxxfxxxfxf 11111,)()()()()()(,)(2)()()()(2)()()(1212211222212 iiiiiiiiiiiiixxxxhxfhxxxxxfhxxxxxfhxxxxxLxf )(3)(

3、3)(4)(21)()(6)()(21)()(3)()(4)(321)(3221222211221 fhxfxfxfhxffhxfxfhxffhxfxfxfhxfiiiiiiiiiiixi1.01.11.2f (xi)0.250000.62267570.2066122)1 (1)(xxfhxfxfxfxf2)()(4)(3)( 2100670454. 71 . 02206612. 06226757. 0425. 03)0 . 1 ( fhxfxfxf2)()()( 20121694.01 .02206612.025.0) 1 .1 ( fhxfxfxfxf2)(3)(4)()( 2102354

4、334. 41 . 02206612. 036226757. 040 . 1)2 . 1 ( f baiidxxlA)()(xf)(xLndxxfba)(dxxLban)(niiixfA0)(43,21,41210 xxx102dxx43,21,41210 xxx32)()()()(10434121414321102010210dxxxdxxxxxxxxxA31)()()()(10432141214341102101201dxxxdxxxxxxxxxA)()()()(433221314132ffffI32)()()()(10214341432141101202102dxxxdxxxxxxxxx

5、A1)( xf1323132xxf)(2121102x21433221314132)()()()(433221314132ffffI3)(xxf2)(xxf414332213141323333131103102xdxx314332213141322224141104103xdxx4)(xxf5151105104xdxx1923743322131413244431433221314132222102dxx细分插值型求积公式)(xf)(xLndxxfba)(dxxLban)(bababfafabdxxLdxxff)()()()()()(2121T1科特斯系数21211110)()(,CC个插值节点

6、21,n梯形求积公式bababadxxLxfdxxLdxxffR)()()()()(111badxxR)(1 badxbxaxf)(!)(2)()()(bafab 123)(xf)(xLndxxfba)(dxxLban)(个插值节点32,n)(三点公式辛普森求积公式)()()()()(bfbafafabfS612646161,64,61)2(2)2(1)2(0 CCCbababadxxLxfdxxLdxxffR)()()()()(222badxxR)(2badxbxbaxaxf)()(!)()(233)()(222bafbaL)()(22添加插值条件bafbaPbadxbxbaxaxffR)(

7、)(!)()()(24224辛普森公式截断误差)()()(452880fab 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 fabfR分段数值积分复化梯形求积公式 baniniiixxxf

8、xfhdxxfdxxfii10101)()(2)()(1 11)()(2)(2)(niinbfxfafhfT)()(12)()(31bafabfR )()()()(bafhabfnhfRn 231212)()()(11312 ikinkknxxfhfR等分2分段数值积分mn复化辛普森求积公式 )()(2)(4)(3)(1011212 mimiiinbfxfxfafhfS)()()()(knkiinfxxfR4152222880)()()()(4522880fabfR),(,)()(222145290ikinkkbxmabhfh)(, )()()()()()()()()()()(bafmabfh

9、abfmhfhfSfImkkn445444514528801809090 xexf)(xexfxf )()()4(1)()(1) 4( exfxf10dxex 1011)(dxxfI8 .212n213n521444510288012880emMmabfSfIn)()()(7 . 3m4m52122233101211212 enMnabfnhfRn)()()(复化梯形公式区间对分复化积分自适应算法)()(21)(2fHfTfTnnn 102/1)()(niinnxfhfH 10sindxxx外推法-复化梯形公式龙贝格积分)(31)(34)(2fTfTfSnnn )(151)(1516)(2fS

10、fSfCnnn )(631)(6364)(2fCfCfRnnn 10dxxxxf)(5 . 0 01 )1 () 0 (21211ffT6380. 01312341TTS6036. 05 . 0211212fTT4341,6433. 0)()(4341412214ffTT6565. 02314342TTS6578. 01151215161SSC 6582. 0)()(87858381814218ffffTT87858381,6632. 04318344TTS6536. 02151415162SSC161516131611169167165163161,6636. 016T6654. 08S6656. 04C6658. 02R6536