荊 科,朱功勤

(1.阜陽師范學(xué)院 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,安徽 阜陽 236037;2.合肥工業(yè)大學(xué) 數(shù)學(xué)學(xué)院,合肥 230009)

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一種高階導(dǎo)數(shù)有理插值算法

荊 科1,2,朱功勤2

(1.阜陽師范學(xué)院 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,安徽 阜陽 236037;2.合肥工業(yè)大學(xué) 數(shù)學(xué)學(xué)院,合肥 230009)

針對(duì)目前高階導(dǎo)數(shù)切觸有理插值方法計(jì)算復(fù)雜度較高的問題,利用多項(xiàng)式插值基函數(shù)和多項(xiàng)式插值誤差的性質(zhì),給出一種不僅滿足各點(diǎn)插值階數(shù)不相同且插值階數(shù)最高為2的切觸有理插值算法,并將其推廣到向量值切觸有理插值中.解決了切觸有理插值函數(shù)的存在性及算法復(fù)雜性問題,并通過數(shù)值實(shí)例證明了算法的有效性.

切觸有理插值；高階導(dǎo)數(shù)；Hermite插值；基函數(shù)

0 引 言

切觸有理插值是類似于Hermite多項(xiàng)式插值的一種插值方法,是有理插值的自然延伸.所謂切觸有理插值問題,即給定n+1個(gè)互異的點(diǎn){xi}：

(1)

(2)

目前,關(guān)于切觸有理插值的研究已有許多結(jié)果:文獻(xiàn)[1]將式(2)的非線性問題轉(zhuǎn)化為線性問題;文獻(xiàn)[2]證明了切觸有理插值函數(shù)的存在性及唯一性;文獻(xiàn)[3]利用連分式的方法給出了一種切觸有理插值算法;文獻(xiàn)[4]給出一種具有重節(jié)點(diǎn)的Pade逼近與切觸有理插值有關(guān)的算法;文獻(xiàn)[5-6]給出了一種基于Newton-Pade逼近方法的切觸有理插值算法;文獻(xiàn)[7]利用Hermite-Newton插值公式給出了一個(gè)判斷切觸有理插值函數(shù)存在的充要條件;文獻(xiàn)[8]給出了切觸有理插值的Lagrange遞推算法;文獻(xiàn)[9]利用Hermite插值基函數(shù)和多項(xiàng)式插值誤差公式給出了一種切觸有理插值算法,降低了有理插值函數(shù)的次數(shù);文獻(xiàn)[10-11]研究了多元切觸有理插值及多元矩陣值切觸有理插值.上述算法的結(jié)果雖然較好,但均為針對(duì)一階導(dǎo)數(shù)插值情形的,算法的可行性大多數(shù)有限制條件,且計(jì)算量較大.

所謂向量值切觸有理插值問題,就是尋求向量值有理函數(shù)：

使之滿足下列條件：

(3)

其中Q(x)和nj(x)(j=0,1,…,t)是實(shí)系數(shù)多項(xiàng)式.

文獻(xiàn)[12-13]利用分段組合方法和Newton插值思想,構(gòu)造了一種向量值切觸有理插值算法；文獻(xiàn)[14]利用連分式及Samelson逆給出了一種向量值切觸有理插值算法;文獻(xiàn)[15]給出了一種Thiele-Werner型向量值切觸有理插值插值算法.這些算法的可行性都有條件限制,計(jì)算量較大,且大多數(shù)針對(duì)一階導(dǎo)數(shù)的情形.

本文利用多項(xiàng)式插值基函數(shù)和多項(xiàng)式插值誤差的性質(zhì),構(gòu)造一種不僅滿足各點(diǎn)插值階數(shù)不相同且插值階數(shù)最高可為2的切觸有理插值算法,并將其推廣到向量值切觸有理插值的情形,解決了該類切觸有理插值函數(shù)的存在性及算法的復(fù)雜性問題.相比于其他算法,具有計(jì)算量較低的優(yōu)點(diǎn).

1 Hermite插值公式

設(shè)在插值節(jié)點(diǎn)(1)上,

yi=f(xi),mi=f″(xi),ni=f″(xi),

求插值多項(xiàng)式H(x)滿足條件:

(4)

這里給出了3n+3個(gè)插值條件,根據(jù)文獻(xiàn)[16]中定理1可確定唯一次數(shù)不超過3n+2的多項(xiàng)式,其形式為H(x)=a0+a1x+…+a3n+2x3n+2.如果根據(jù)條件(4)確定3n+3個(gè)系數(shù)a0,a1,…,a3n+2,顯然計(jì)算復(fù)雜度較高,因此本文采用求Hermite插值基函數(shù)的方法.設(shè)插值基函數(shù)為αi(x),βi(x),γi(x)(i=0,1,…,n),共有3n+3個(gè),每個(gè)基函數(shù)都屬于P3n+2(P3n+2表示所有次數(shù)不高于3n+2的(實(shí)系數(shù))多項(xiàng)式集合),且滿足下列條件：

(5)

其中i,k=0,1,…,n.于是滿足插值條件(4)的插值多項(xiàng)式H(x)可寫成用插值基函數(shù)表示的形式,即

(6)

由式(5)可見,式(6)滿足插值條件(4).下面求滿足條件(5)的基函數(shù)αi(x),βi(x),γi(x)(i=0,1,…,n).根據(jù)文獻(xiàn)[17]的Hermite插值公式可得

(7)

(8)

(9)

2 切觸有理插值公式

為了建立切觸有理插值公式,定義如下有理函數(shù):

其中:

qi=q(xi);pi=q′(xi);ti=q″(xi).

命題1qi(x)具有如下性質(zhì)：

證明：3)由

當(dāng)i=k時(shí),

當(dāng)i≠k時(shí),

當(dāng)i=k時(shí),

當(dāng)i≠k時(shí),

先討論數(shù)量切觸有理插值問題.引入插值算子

(10)

(11)

由qi(x)的性質(zhì)2)可知：

(12)

(13)

(14)

(15)

(16)

顯然,由式(11)得到的切觸有理插值函數(shù)滿足插值條件(17),但該有理函數(shù)的次數(shù)太高.

(17)

其中si最大值為3.

定理1設(shè)H(x)是在節(jié)點(diǎn)(1)上滿足條件(4)的插值多項(xiàng)式,若f(x)∈C3n+2[a,b],f(3n+3)(x)在(a,b)內(nèi)存在,則對(duì)任意給定的x∈[a,b],存在ξ∈(a,b),使得

(18)

證明:仿照文獻(xiàn)[18]中Hermite插值余項(xiàng)的證明方法即可.

推論1如果f(x)∈P3n+2,則f(x)在節(jié)點(diǎn)(1)上滿足條件(4)的插值多項(xiàng)式H(x)∈P3n+2,恒等于f(x).

證明：因?yàn)閒(3n+3)(ξ)=0對(duì)一切ξ∈(a,b)都成立,故由式(18)得f(x)-H(x)=0.證畢.

(19)

由式(19)可見,本文方法可以把有理插值函數(shù)分母多項(xiàng)式的次數(shù)降低到需要的任意次數(shù).

下面討論向量值切觸有理插值算法.引入插值算子:

(20)

顯然

(21)

利用插值算子Ni(x)和qi(x)做線性組合：

(22)

由qi(x)的性質(zhì)及Ni(x)的定義,用類似于數(shù)量值的證明方法即可證明

(23)

其中si最大值為3.

推論2如果V(x)是次數(shù)不超過3n+2次的向量值多項(xiàng)式,則V(x)在節(jié)點(diǎn)(1)上滿足插值條件

(24)

的插值多項(xiàng)式N(x)=V(x).

(25)

由式(25)可見,本文算法可以把有理插值函數(shù)的分母多項(xiàng)式次數(shù)降低到需要的任意次數(shù).

3 數(shù)值實(shí)例

下面通過數(shù)值實(shí)例進(jìn)一步驗(yàn)證本文有理插值算法的可行性是無條件的,且具有計(jì)算量較低、無極值和有理函數(shù)次數(shù)較低等優(yōu)點(diǎn).

解：由式(10)知插值算子為

p0(x)=-0.5x2+0.5,p1(x)=2x+1,p2(x)=2,

根據(jù)定義分母多項(xiàng)式q(x)=2x2+2,得

q0=4,q1=2,q2=4,s0=-4,s1=0,s2=4,t0=t1=t2=4;

由式(7)～(9)得：

再由式(11)得

(26)

易驗(yàn)證式(26)不僅滿足插值條件,且分母多項(xiàng)式在實(shí)數(shù)范圍內(nèi)無極點(diǎn),次數(shù)僅為2.

綜上可見,本文利用多項(xiàng)式插值基函數(shù)和多項(xiàng)式插值誤差公式的性質(zhì),給出了一種針對(duì)最高階導(dǎo)數(shù)為2的切觸有理插值算法.雖然解決了該類切觸有理插值函數(shù)的存在性問題,降低了切觸有理插值函數(shù)的分母多項(xiàng)式次數(shù),但對(duì)于更高階導(dǎo)數(shù)插值條件的切觸有理插值情況并未解決.

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(責(zé)任編輯：趙立芹)

ARationalInterpolationAlgorithmofHigherOrderDerivative

JING Ke1,2,ZHU Gongqin2

(1.SchoolofMathematicsandStatistics,FuyangTeachersCollege,Fuyang236037,AnhuiProvince,China;2.SchoolofMathematics,HefeiUniversityofTechnology,Hefei230009,China)

In view of the higher computational complexity of the osculatory rational interpolation method of higher derivative mostly based on the idea of generalized vandermonde matrix,by means of basis function of polynomial interpolation and error nature of polynomial interpolation,we proposed an osculatory rational interpolation algorithm that not only satisfies different interpolation order but also makes the toppest of interpolation order equal 2,and it also meets the vector-valued osculatory rational interpolation.It solves the problem of the existence of osculatory rational interpolation function and complexity of algorithm.In the end,we illustrated the effectiveness of the algorithm with a numerical example.

osculatory rational interpolation;higher order derivative;Hermite interpolation;basis function

10.13413/j.cnki.jdxblxb.2015.03.08

2014-07-17.

荊 科(1983—),男,漢族,博士研究生,講師,從事應(yīng)用數(shù)值逼近的研究,E-mail:jingxuefei296@sina.com.

國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):71371062)、安徽省自然科學(xué)基金(批準(zhǔn)號(hào):1408085MD70)和安徽省高校自然科學(xué)研究項(xiàng)目(批準(zhǔn)號(hào):2014KJ011).

O241.3

：A

：1671-5489(2015)03-0389-06