【正文】 jj ????. ,0,)( 1 jjjjj xxhnjMxs ?????? ??三彎矩法：令）（則 ].,[ ,)( 111 ??? ??????? jjjjjjjj xxxMhxxMhxxxsP42 ,2)(2)()(11221 cMhxxMhxxxsjjjjjj ????????? ,6)(6)()(211331 cxcMhxxMhxxxsjjjjjj ????????,61)( ,61)( 1211121212 ???? ???????? jjjjjjjjjj ycxcMhxsycxcMhxs.)(61),(61 11112111 ?????? ???????? jjjjjjjjjjjjjjjj MxMxhhxyxycMMhhyyc)( ,)6()6( 6)(6)()(211121331jjjjjjjjjjjjjjjjhxxhMyhxxhMyMhxxMhxxxs???????????????jjjjjjjjjjjj hMMhyyMhxxMhxxxs62)(2)()( 111221 ?????????? ????)0()0(: , 0 ????? jjn xsxsMM 要用導數(shù)連續(xù)條件為了求 ?,63)0( 11jjjjjjjjhyyMhMhxs ??????? ??,36)0( 111jjjjjjjjhyyMhMhxs ?????? ???,1,1 ,6361111111 ???????????????? njhyyhyyMhMhhMhjjjjjjjjjjjjj ?.36)0( 11111????? ??????jjjjjjjjhyyMhMhxs].,[6, ,1,1 ,2 1111111?????????????????jjjjjjjjjjjjjjjjjjxxxfdhhhhhhnjdMMM????其中?00001010000110000:)(6263)0(dfhyyhMMfhyyMhMhxs???????????????nnnnnnnn dhyyfhMM :)(62 1111 ??????????在第一邊界條件下：nnnnnnnnn fhyyMhMhxs ?????????????1111136)0()( ,2122121101101111?????????????????????????????????????????????????????nnnnnnddddMMMM?????????.,00 nn fMfM ??????在第二邊界條件下：)( ,222211220111221122221????????????????????????????????????????????????????????????????nnnnnnnnnfdddfdMMMM?????????????].,[6,2 )()(110100111100xxxfdhhhhhhdMMMMMxSxSnnnnnnnnnnnnnnn???????????????????其中，并且，就有意在第三邊界條件下：注)( ,2222121121112211?????????????????????????????????????????????????????nnnnnnnnddddMMMM?????????????系數(shù)矩陣為嚴格對角占優(yōu)陣，方程組有唯一解。 ? Newton ? Ln(x)， 只是形式不同；節(jié)點等距或漸增節(jié)點時方便處理。 由該區(qū)間上的 S[j](x) 算出 f(x) 的近似值。 7 樣條插值 問題背景 ? .,)(, 2)。 111111????????????????kmkmkmkmkmkmffffffm 階差分一般地可定義.: , , :21212121211????????????kkkkkkkkkfffffffff?????二階中心差分一階中心差分?.: , 1??? kkkk fEfEfIfI 移位算子：引進不變算子. ,)(IEΔfIEIfEfΔf kkkk?????? 可得到則. , 2/12/11 ?? ????? EEEI ?同理，差分的基本性質(zhì) : ( 4 . 4 ) ,)1()1()( ( 1 )00 fjnfEjnfIEfnjjknjnjkjnjknkn??????????????????????????如：差分可用函數(shù)值表示，( 4 . 5 ) ,)1( )1()(001 fjn fEjnfEIfnjnjkjnnjknjjnknkn?????????????????????????????）（如：函數(shù)值可用差分表示， .)( ( 2 )00????? ??????????????????????????njkjknjjknknkn fjn fjnfIfEf,2],[],[],[ ,],[ ( 3 )22212121111hfxxxxfxxfxxxfhfxxffxxfkkkkkkkkkkkkkkkkk???????????????????差商與差分關系，如：）（一般地， .!1],[ kmmmkk fhmxxf ????）（ .!1],[ kmmmkk fhmxxf ??? ??）（以及 ).,(),( )( nkknnkn xxfhf ???? ??差分表 : ?2f0 ?2f1 ? ┆ ?2f2 ┆ ┆ ?f0 ?f1 ?f2 ?f3 ┆ f0 f1 f2 f3 f4 ┆ 0 1 2 3 4 ┆ ?2 ?3 ? ? fk k )(?)( 1f?)( 2f?)( 3f?)( 4f?)( 2? )( 3?)( 22 f?)( 32 f?)( 42 f?)( 3303 ff ??)( 4313 ff ??)( 4404 ff ??)( 44 ??( 4 . 1 1 ) ),( ),()!1( )()1()( 0)1(1 nnnn ,xxfhn ntttxR ?? ??? ?? ???時，當函數(shù)值上的個等距節(jié)點在已知)10( ,)(),1,0(1)(00?????????tthxxfxfniihxxnxfiii ?二、等距節(jié)點插值公式 ,)()1()()( 101???????? ? kkj jkhktttxxx ??],[)())(( ],[))((],[)(][)(10110210101000nnnxxxfxxxxxxxxxfxxxxxxfxxxfxN????????????????,!1],[ kmmmkk fhmxxf ????( 4 . 1 0 ) . ,!)1()1(!2)1()( 002022牛頓前插公式fnntttfttftfthxN nn ???????????????P34 ],[)())((