數值積分基本概念插值型求積公式求積公式的代數精度復合梯-資料下載頁
2025-08-23 10:37本頁面
【導讀】數值積分基本概念。求積公式的代數精度。格林公式中曲線積分處理。對[a，b]做分劃:a≤x0<x1<x2<……(n+1)點插值型求積公式代數精度至少為n階.使代數精度盡可能高.容易驗證,對f=x3求積公式式不能準確成立.
【正文】 階向前差商 )()(!3)(2)()()( 4)3(32hOafhafhafhafhaf ?????????)()()()( hOh hafafaf ?????一階向后差商 29 )()()()()( 222 hOh hafafhafaf ????????二階中心差商 ?????????? )(!3)(2)()()( )3(32afhafhafhafhaf)()()()( 22 hOh hafhafaf ??????一階中心差商 ?????????? )(!3)(2)()()( )3(32afhafhafhafhaf)()(3)(2)()( 5)3(3hOafhafhhafhaf ???????f(x) x f(x) x f(x) x f(x) x true derivative forward finite divided difference approx. backward finite divided difference approx. centered finite divided difference approx. ? ? ? ? ? ?h xfxfxf iii 1 ???? ? ? ? ? ?h xfxfxf iii ?? ? 1 ? ? ? ? ? ?h xfxfxf iii 2 11 ?? ??30 31 )()(62 )()()( 4)3(211 hOxfhhxfxfxfkkkk ????? ??)()()(2)()( 22 11)3( hOh xfxfxfxf kkkk ??????? ??設 xk= a + k h ,( k =0,1,,n) 值 令 mk = f ’ (xk) ( k=0,1,,n) )]()([ 1111 34 ???? ???? kkkkk xfxfhmmmk=1,2,,n1 隱式方法 32 Lagrange插值函數方法 ???20)()()(kkk xfxlxf ?????20)()()(kkjkj xfxlxfxk = x0 + kh (k=0， 1， 2) ???????????????????))(()())(()())(()(10222021212021121xxxxhxlxxxxhxlxxxxhxl2210 22hxxxxl )()( ????2102 22hxxxxl )()( ????22012hxxxxl )()( ?????33 201hxl ??? )( 212hxl ???? )( 221hxl ??? )(??????????????????????)]()()([)()]()([)()]()()([)(210220121003421214321xfxfxfhxfxfxfhxfxfxfxfhxf2210 2hxfxfxfxf )()()()( ?????221022hxxxxl )()( ????2102 22hxxxxl )()( ????2201 2hxxxxl )()( ??????????20)()()(kkjkj xfxlxf一階導數近似 二階 導數近似 34 外推算法 )]()([)( hxfhxfhhG ???? 2 1?????? 4221 hhxfhG ??)()(1424 11?????mmmmmhGhGhG)()()()()()( )( 12 ???? mm hOhGxf?????? 16/4/)()2/( 4221 hhxfhG ????3 )()2/(4 hGhG ???? 4/3)( 42 hxf ?)()()()( 22 hOh hafhafaf ?????
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