大二下課件-數(shù)值計算方法chap5插值法

第五 ：已給sin0.3367高度0壓強(qiáng)kgf1200例5-3：已知直升飛機(jī)旋轉(zhuǎn)機(jī)翼外形輪廓線上某些點的kk1082-93-4-5-6-0700平均增加節(jié)點至121個，在xoy平面上繪制出較光滑的機(jī)翼外形曲線例5-4：已知函數(shù)ux(x1yy1，定義域0,1]2，依據(jù)不同的離散剖分?jǐn)?shù)據(jù)（88，161632326464試?yán)L出各個函數(shù)圖 近似函數(shù)圖xy(x?1)01

y

0x

0101

10誤差函數(shù)x01

1問題的在實際問題中常遇到這樣的

yfx)某個區(qū)間[a,b]上是存在的。但是,通過觀量或試驗只能得到在[a,b]區(qū)間上有限個點x0,, 上的函數(shù)值

f(xi),(i0,1,,或者yfx)的函數(shù)表達(dá)式是已知的,但卻很x y 函數(shù)yf()表達(dá)式未知在一些點處測得函數(shù)值:yx y 求一個簡單易算的函數(shù)g(x 使g(xi) i0,1,在x0xn]上,gxfx)gx)稱為fx)的插值函數(shù)g(x)g(x)

§5.1Lagrange多項式一 二 三x y fx)在[ax y yif(xi innx

axa n(xi)

(i0,1,2,n)

由插值條件nxi

(i0,1,2,n)nai(i0,1n應(yīng)滿足方程a axax2axna axax2axn a ax2axn 系數(shù)矩陣的行列式為范德蒙特行列式，n1個互異節(jié)點上的n次插值多項式存在唯一插值多項式的誤差5.1x0x1,xn是[ab]上n1個互異節(jié)點,nx)是fx)過這組節(jié)點的n次插值多項式若fxC(n1)[ab],則對任意的x[ab有RR(x)f(x)n(n((nn1((a,nn1xxxiif(n1)(f(x)T(x) (xx

O(x, (n 證明：RnxfxnRn(xi)0(i

nRn(x)K(x)(xxinin任意固xxi(i構(gòu)造(tRn(tKx(txii(t)有n2個不同的根x,x0,，反復(fù)利 定理，存(x(a,b)使得 (x)(即R(n1)()K(x)(n1)!即 f(n1)()(n1)()K(x)(n1)! f(n1)( f(n1)(

K(x) (nR(x) x(xx (n1)

i當(dāng)xxi(i0,1,,n)時顯然成立 證畢 當(dāng)n1 l(x)x x 1

l1(x)1

xx0x1x0x xy

x x

x x 1L(x)1

y1

(xx)x1x0 x x x x x x x x 000100 li(xj) in

i,j Ln(x)yili(iLn(x)為滿足插值條件(*)的n次插值多項式li有n個 x0x1?inlixCixx)xx)xxn)Cixxjn j

jjli(xi)1

jji(xixjjl(x)

(xxjnnjnnnj n

(xxjijj

(

xj

jj

(

xj稱lix)(i0,1n)為n次Lagrange插值基函數(shù)nLn(x)yili(xni稱Lnx)為n次Lagrange插值多項式n

(xxjnjn注：1xi)xjj

xj

li(x)

j'n1(xiLagrangel(x)

(xx0)(xx2

l(x)(xx0)(x (xx)(xx

(xx)(xx l1(x

l2(

l0(xl(x)

(xx1)(xx20 (0

)(

x2LagrangeL(x)

(xx1)(xx2

f(x)

(xx0)(xx2

f(x)

(xx0)(xx1

f(x

0x1)(x0x2

(xx)(x

(

x)(xx n=1,2,31次Lagrange插值（插值節(jié)點x0,x）線性LL(x)Lf(x)L(x)f(x)x10011xf(x)x0f(x101x102次Lagrange插值（插值節(jié)點x0x1x）拋物線LL(x)2(xx)(xx(xx1)(xx2 f(x)(xx0)(xx20 (xx)(xxf(x)(xx0)(xx1)11012(xx)(xxf(x220213次Lagrange插值（插值節(jié) x0,x1,x2,xLL(x)3(xx1)(xx2)(xx3(x0x1)(x0x2)(x0x3(xx0)(xx1)(xx3f(x)0(xx0)(xx2)(xx3(x1x0)(x1x2)(x1x3(xx0)(xx1)(xx2f(x1(xx)(xx)(xxf(x)2f(x3202123(xx)(xx)(xx303132xyln 取x0 L(x)2.3979x122.4849x 11 12ln11.5L(11.5)2.397911.5122.484911.511Rn(x) (n1)!Rn(x) (n1)!f(n1)(R1(x)

(x11)(x

1

(ln(ln

x0 x1 x2L(x)(x12)(x13)2.397(x11)(x13) (1211)(12(x11)(x12)ln11.5L2(11.5)R2(11.5)

(lnx)

2 取x0R1

(11.511)(11.5

2

0.5 ...18 取點x011,x112,x2f(3)(x)x3R2

(11.511)(11.512)(11.5f(3)(6 0.5f(3)(6

10413

8 例已知函數(shù)fx56x324x25在21227的函數(shù)值，解：設(shè)其三次插值多項式為L3(xR3(x)f(x)L3(x)

f(4)()

4(x) L3(x)fn例2 設(shè)x0,x1,,xn為n1個互異節(jié)點，li(x)(i0,1,,n)為這組節(jié)點上的Lagrange插值基函數(shù)，試證明nli(x)i 證明：取f(x) Ln(x)f(xi)li(x)li 且f(n1)(nRn(x)f(x)Ln(x) (n1)!n1(x)n Ln(x)li(x)f(x)i§5.2Newton P(x)ff1f0(xx (ff(x)y x1

Pn(x)a0a1(xx0)a2(xx0)(xx1an(xx0)(xxn1其中ajj0,1,n)為待定系數(shù),可由Pxjfj確定Pn(x)a0a1(xx0)a2(xx0)(xan(xx0)(x n(0)a0 P(x)aa(xx) P(x)aa(xx)a(xx)(xx)

a0 f a1

x1xa f2 f0 f1 fa x2x x1x2 x2x12 依次可得到a3,a4,,an 定義：差商設(shè)由函數(shù)fx一組互異的節(jié)點x0x1, f(xi)f(xj)fx,x xixfx)關(guān)于點xixj的一階差商(均差fxi,xjfxj,xkfx,x,xxi

k為fx)關(guān)于點xi,xj,xk的二階差商(均差

fx,x fx,x,xfx,x,x x0

為fx差商的性質(zhì)

x0x1xk的k線性性質(zhì)fxaxbxfx0,x1,,xkax0,x1,,xkbx0,x1,,xkk差商可以表示成函數(shù)值的線性組合,即kfx,x,,x

f(xi k

(xik

k 其中k1xixixjji

,n次多項式函數(shù)關(guān)于點x,xi的一階差商為n-1次多項式。事實上設(shè)Pnx)是n次多項式,PxPnxPnxi)仍是n次多項式,且Pxi)0故P(x)(xxi)Pn1(

Px,x

(x)i xi

計算各階差商可用差商表f(xi一階差二階差三階差商xfxfxf(x2fx1,x2xf(x3f由差商的f(x)f(x0)(xx0)fx,x0f[x,x]f[x,x](xx)fx,x,x f[x,x,x]f[x,x,x](xx)fx,x,x,x f[x,x,, ]f[x,x,x](xx)fx,x,,x

f(x)(xx)(xx)(xx)fx,x,x,x Rn(Nn( f(xi)Nn(xi)R(xi)Rn(Nn(Nn(x)f(x0)(xx0)fx0,x1(xx0)(xx1)fx0,x1,x2(xx)(xx)(x

)fx,x,x

ff(x)(xx)fx,x(xx)(xx)fx,x,x00 01 (xx)(xx)(x01)fx,x,x nNn(x)稱為f(x)的n f(x)的n 插值多項式Nn(x)N(x)f(x)(xx)fx,x(xx)(xx)fx,x,x (xx)(xx)(x

) x,x Newton

R(x)(xx)(xx)(xn01f(n1)(xn)(nR(x)(xx)(xx)(xn01f(n1)(xn)(n Lagrange插值余項Ln(x)Nn(xn1(x)fx,x0,x1,xn

(n1(n

n1(xff(n)(fx0,x1,,xn其 minxi maxxi0i f(xixf(0)1xf(1 (xx0xf(x2)fx,x fx,x,x 1(xxjxf(x3fx,x fx,x,x fx,x,x,x 2(xxjjxyln試 法求ln11.5近似yiln1x--N1(x)2.39790.0870(xln11.52.39790.0870(11.511)N2(x)2.39790.0870(x11)0.0035(x11)(x例:已知f(x)的函數(shù)表,求四次 插值多項式,計算f(0.596)的近似值,并估計誤差.做差商表一階差二階三階差四階差N4(x)0.410751.116(x0.4)0.28(x0.4)(x0.19733(x0.4)(x0.55)(x0.03134(x0.4)(x0.55)(x0.65)(xf(0.596)N4(x)|R4(0.596)

f[0.4,0.55,0.65,0.8,0.9,0.596](0.596xii一階二階三一階二階三階差四階差-

-差

xix0i

(i0,..., fi

fi

fkfi k1kifik

fif

fikfk1fk1iiik

1

其中 f(i

h2i ii ikfkifik12i2f(xixf(0)xf(1xf(x2)2f(2f xf(x32f(2f 例 (afbg)afbk2k

fk

2fk

f

f f

2fk

f n

nnn (1)j

fk(1)

fk nk

j

j其中

jn

C

n(n1)...(njj j j f[x,x]

kfk!hk

f[xn,

,,

nk]

kfnkf(k)() 由導(dǎo)數(shù)和差商的關(guān)系fx0,xk

(k即得k等距節(jié)點插值Nn(x)f(x0)f[x0,x1](xx0)...f[x0,...,xn](xx0)...(xf(n1)(Rn(x)

(n n1(xxx0thN(x)N(xth)f

...t(t1)(tn1)n

Rn(x)

t(t1)...(tn)(n1)!

f(n1)() (x,xNn(x)f(x0)f[x0,x1](xx0)...f[x0,...,xn](xx0)...(xf(n1)(Rn(x)

(n n1(x后差公式，將節(jié)點順序倒置設(shè)xxntNn(x)Nn(xnf

t(t1)2ft(t1)(tn1)n

Rn(x)

t(t1)...(tn)(n1)!

f

(x0,xnf(xixf(0)1xf(1txf(x2)2f(2f (tj)2!j0xf(x32f(2f (tj)3!j01t1(t (t小日型插值余項(導(dǎo)數(shù)型)f(n1)Rn(x)(n1)! 型插值余項(差商型)

Ln(x)f(xi)li(xiLn(x)f(xi)li(xin 日插值 插值的比 型插值余項公式對fx)是由離散點給出或f多項式插值理論的深入日插值多項式和插值多項式是否有收斂性R(x)R(x)f(x)Pn(x)f(n1)((nin(xxi插值就是數(shù)值近的一種，而數(shù)值近，為得在代數(shù)插值中,給定函數(shù)fx)在n1個不同的節(jié)點,得n次插值多項式的余插值多f(x)=Interpolationat[0二次插值多f(x)=Interpolationat[01三次插值多f(x)=Interpolationat[014Runge函f(x)

125x2Runge函f(x)

125x20附近插值效果是好的，即余項而振動。 而且高階插值還是不穩(wěn)定的。(版數(shù)值分析有證明) 問題：設(shè)在a,b上給定n1個不同的節(jié)點ax0x1xn1xnyif(xi)(i求函數(shù)(x)使得(xi) (i0,1,,稱x)為fx)過節(jié)點x0x1,xn的分段線性插值函數(shù)表達(dá)(x)xxixyxiyixx,iiixiii(xi) f(x)

1

x (ii f i012345f對x(x)x11.00000x00.5000010.5x01 10類似地10.8(x)0.40.22354

xxx(2,x(3,x(4,f(4.5)(4.5)0.140260.020364.50.04864與真值f(4.50.04706誤差估定理5.2設(shè)ax0x1xnb,x)為fx)過這組節(jié)點的分段線性插值多項式,若fxC2ab則對任意x[a,b],RR(x)f(x)(x)hmaxf(28axhmaxxi1xi(xxi)(x證明(xxi)(xR(x)( x

xix

f

f(

f(22

xix

8a 另一個思 特（Hermite）問題

,

,

為n1個互異節(jié)點

y y y y y yyif(xi (i求至2n1次多項式Hx使其滿足插值條件H(xi)yi, H(xi) (iHx)為fx)的Hermite插值多項式

h(x) j

hi(xj)

j

hi(xj)

j

H

)

j

Hi(x)

H'(x)

j

1 j

H(x)inH(x)in h(x)yii(x)g(a)

g'(a)0g(x)(xa)2h(x)設(shè)hx)abxx)l2

hi(x)

由h(x) h(x) 由 a b2li(xi (i0,1,,h(x)12l(x)(xx)l(2iiiii(i0,1,,

H(x)C(xx)l2(

Hi(x) Hi(

)

C

H(x)(xx)l2(i (i0,1,hh(xi12l(x)(xx)l(xi2i i(i0,1,,HH(x)(xx)l2(iii(i0,1,,n)0 特別地,n1 l(x)x l(x)0 xx0 0x0h(x)(1-0

x

)(xx1

0H(0

(xx)

x x0

x0

0x001l(x)1

x

l

) x1 x1 h(x)(1

x

)(xx0

H(

(xx

x x x x 0h(x)(1-0

xx0)(

xx1

H(x)(xx)(

xx1x0

x0

x01h(x)(11

x

)(xx0

H(x)(xx

xx00000x1

x1

x111∴兩11HH(x)(12xx0)(xx1)2y(12xx1)(xx0)2xx0xx110010110(xx)(xx1)2y(xx)(xx0)20x0101x110求法二（降階法Hx)Nnx) H(x)Nn(x)Pn(x)n1(x)n其中n1(x)(xxi N(x)為f(x)過x0,x1,,xniNn(xi)f(xi) (i0,1,,Pnx)n再由導(dǎo)數(shù)條H(xi)確定Pnx)的系數(shù)事實上，Pn(x)只需滿

H(x)Nn(x)Pn(x)n1(H(xi)Nn(xi)Pn(xi)n1(xi)i0,1,,

(ixiyixiyi003119h(i12l(x)(iix)l(2ii(i0,1,,HH(x)(xx)l2(i (i0,1,

l0(x)1 l1(x)1h(x)(12(x1))x23x22x31H0(x)x(1x) H1(x)(x1)xH(x)h1(x)3H0(x)9H1(x)10

12x23例:Hermitexi yi

N(x)010(x0) 1設(shè)Hxx(abxx0x1)H(x)1bx(x1)(abx)(x1H(0)1a H(1)1aba2,b H(x)x(10x2)x(x1)10x312x2誤差定理5.3設(shè)x0x1,xn為[ab]上的互異節(jié)點Hx)為fx)過這組節(jié)點的Hermite插值多項式若fxC(2n2a則對任意的x[abf(2n2)(R(x)f(x)H(x) (x (a,b)(2n2)! n1f(4)(R(x)

(xx)2(xx R

)0,R(

)

RxKx)

(

x

令(t

f(t)H(t)

顯然(t)C(2n2)

i且(x)0,(x) (ii(t)f(t)H(t)K( (t (t (xi)

f(xi)H(xi(i因此(t)在[a,b]上有2n2個零點.依次類推 定理，知(2n2)(x)在(a,b)內(nèi)至少存在一個零點(t)f(t)H(t)K(x)

(t(2n2)(t)f(2n2)(t)K(x)(2n(a 使(2n2)()f(2n2)()K(x)(2n2)! K(x)

f(2n2)((2n R(x)

f(2n2)((2n

2

(定理5.4 nfxC1[a,b則fx)在互異節(jié)點xx, n處的2n1次Hermite插值多項式唯一Hx)是Hx)的2n1次HermiteH(2n2)( H(x)H(x)

(2n2)!n1(x)因此Hermite插值多項式是唯一若fx)是次數(shù)不超過2n的多項式,Hx)fx)的過任意n1個節(jié)點的Hermite插值多項式Hx)fx)nhi(x)n分段三次Hermite x0x1 yif(xi H(xi)yi H(xi) (i0,1,,在每個小區(qū)間 xi1](i0,1,,n)上，H(x)H(x)(1

x

)(xxi1)2y(12xxi1 x

)2 x

x

ii (xx)(xxi1)2y(x

)(x

i i )2ixii

i

i

xi1

ix[xi,xi1 i0,1,n1誤差估計R(x) f(x)H(x)

4hm f(4)(xhhmax(xi1xi0in

384ax事實上，當(dāng)xxi,xi1](xx)2(x R(x)

f(x)H(

i

xix

f(4)(同時|(xx)(x )|(xi1xi i Hermite插值的一般fx)在n1個互異節(jié)點處x0x1,xn的函數(shù)值yifxi)(i0,1,n)yif(xi)(k H(xi)iH( )i

(i0,1,,(k0,1,,m) 設(shè)H(x)Nn(x)Pm(x)n1(Pmx)iH(x)yik

(k0,1,,m)5.5:(誤差估計若fx)C(nm2)nm1次多項式Hx)滿足插值條件iH(xi)yiH(x )yi 則x[a,

(i0,1,,(k0,1,,f(nm2) R(x)f(x)H(x)

(nm2)!n1(x)(xxik(a,m證明：R(xi)0,R(xi) 令R(x)K(x)n1(x)(xxi k mixx(i0,1,n)(tf(tH(tK(x)n1(t(txii顯然(t)C

(x0,(xi

(i(t)在[ab]上至少有n1個異于x和xi(i0,1,n)的零(t)(t ) (t)(t (t)f(t)H(t)K(x) k l0k

f(xi)H(xi

即(xi0kk因此(t)在[a,b]上有nm2個零點.反復(fù) 定 (a, 使f

(nm R(x)

f(nm2)()(nm2)!

n1(x)(xxikk例求滿足下列表中插值條件xi012f(xi13f'(xi15的四次插值多項式P4x并導(dǎo)出其余項的表達(dá)式設(shè)fx)在插值區(qū)間上具有直到五階連續(xù)導(dǎo)數(shù))思路用降階法先求過x0,x1,x2的插 多項式N2(設(shè)P4xN2x(axbxx0xx1xx2其中系數(shù)a、b,由導(dǎo)數(shù)條件確解:降階 先作插商fxi 一階差商二階差商0111x2343x(xN2(x)f(x0)f[x0,x1](xx0)f[x0,x1,x2](xx0)(x12x3x(x1)3x25x4Px3x25x1(axb)xx1)(x44P(x)4ax33(b3a)x22(2a3b3)x(2b4

012f(xi13f'(xi15由

即P(2) 即

ab02ab1

a得b得 P4(x)x44x32x23x余項表達(dá)式R(x)f(x)P4(x)

f(5)()

x(x1)2(x2)

(0,2)例：求符合表值的插值多項式并給出插值余項012212---6個條件,可確定惟一一個5次多項式P5(x滿足函數(shù)值條件的2次插值多項式

N2(x)x22x用降階法

設(shè)Px)Nx(ax2bxcxx1x 5P(x)x22x2(ax2bxc)x(x1)(x5P5(x)(2x2)(2axb)x(x1)(x(ax2bxc)(3x265P(x)20ax3(12b36a)x2(12a4b6c)54b6c由P(1

c1得 abc

a0122012212yy

3c1

ci5P(x)x22x2(4x23x)x(x1)(x54x515x417x35x22x余項為：f(x)P(x)f(6)()x(x1)(x

(0,§5.5樣條從六十年始，首先由于航空、造船等工程設(shè)計樣條函數(shù)的已知[ab]的一個分劃ax0x1xnf(xi)yi(i函數(shù)S(x)滿在每個小區(qū)間xi,xi1](i0,1,n1)上Sx)是m次多項式S(x)C(m1)[a,則稱S(x)為關(guān)于分劃的m次樣條函數(shù),其圖形稱為樣條曲線三次樣條已知ax0x1xnf(xi)yi(i求函數(shù)S(x),使S(xi) (i在每個小區(qū)間xi,xi1](i0,1,n1)上S(x)是三次多項式記為Sjx即Sx)Sjx),xxj,xSjx)三次多S(x)C(2)[a,則稱S(x)為關(guān)于f(x)的三次樣條函設(shè)SxjMjj0,1,n),hjxj1xjj0,1,n1)Sj(x)

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