二元矩陣有理插值函數(shù)的構(gòu)造

杜偉偉

(安徽教育出版社,安徽合肥 230601)

二元矩陣有理插值函數(shù)的構(gòu)造

杜偉偉

(安徽教育出版社,安徽合肥 230601)

一般構(gòu)造矩陣值有理函數(shù)的方法是利用連分式給出的,其算法的可行性不易預(yù)知,且計(jì)算量大.本文對(duì)于二元矩陣值有理插值的計(jì)算,通過(guò)引入多個(gè)參數(shù),定義一對(duì)二元多項(xiàng)式：代數(shù)多項(xiàng)式和矩陣多項(xiàng)式,利用兩多項(xiàng)式相等的充分必要條件通過(guò)求解線性方程組確定參數(shù),并由此給出了矩陣值有理插值公式.該公式簡(jiǎn)單,具有廣闊的應(yīng)用前景.

二元矩陣值;有理插值;參數(shù);方程組

1 引 言

設(shè)x0＜x1＜…＜xn;y0＜y1＜…＜yn,給定矩陣插值節(jié)點(diǎn)陣(xi,yj)及相應(yīng)矩陣Aij=A(xi,yj)∈Rd1×d2(i=0,1,…,n;j=0,1,…,m).

所謂二元矩陣值有理插值,就是尋求矩陣值有理函數(shù)

使之滿足條件

其中N(x,y)是矩陣多項(xiàng)式,D(x,y)是實(shí)系數(shù)多項(xiàng)式,Aij=A(xi,yj).

文獻(xiàn)[1]給出了二元Thiele型矩陣值有理插值的有關(guān)結(jié)果,也可以利用矩陣的行向量列展開概念,將二元向量值有理插值結(jié)果應(yīng)用到矩陣值有理插值情形,給出二元矩陣值有理插值的算法.上述結(jié)果雖然很好,但不便于實(shí)際應(yīng)用.

顯然wi(x)是n次多項(xiàng)式,wj(y)是m次多項(xiàng)式.記wij(x,y)=wi(x)wj(y).

定義2 對(duì)給定的(xi,yj)及相應(yīng)的矩陣

2 二元矩陣值有理插值公式的構(gòu)造

定理1 對(duì)于給定的插值節(jié)點(diǎn)(xi,yj)及相應(yīng)的矩陣

且D(x,y)是n+m次二元代數(shù)多項(xiàng)式.

通過(guò)觀察,容易看出這種構(gòu)造方式類似于二元Lagrange插值.下面通過(guò)例子給出算法的具體步驟.

表1

顯然,由(8)式構(gòu)造的二元矩陣值有理插值函數(shù)次數(shù)較高.為了降低次數(shù),引入?yún)?shù)αij(i=0,1,…,n; j=0,1,…,m),重新定義并仍記D(x,y),N(x,y)和βij(x,y).

下面給出降低分母多項(xiàng)式D(x,y)次數(shù)的方法.由(9)式知D(x,y)最高次項(xiàng)為axnym(a為常數(shù)),如果要降低1次,由多項(xiàng)式相等的充分必要條件,可令xnym項(xiàng)系數(shù)為零,便得方程(αij為未知量)

由此便得定理2.

定理2 對(duì)于給定的節(jié)點(diǎn)(xi,yj)及相應(yīng)的矩陣值

且D(x,y)的類型和次數(shù)可根據(jù)需要確定.

注意,在構(gòu)造二元矩陣值有理插值函數(shù)R(x,y)時(shí),如果求解過(guò)程中存在i0,j0使得αi0j0=0,則R(x,y)在該點(diǎn)(xi0,yj0)處可能不會(huì)插值了;如果αij均為非零實(shí)數(shù),則該R(x,y)必插值所有節(jié)點(diǎn).

解 在例1中,如果要降低分母次數(shù),希望D(x,y)=x+y-4.由分析知

需要指出的是,本文只給出了降低有理函數(shù)分母多項(xiàng)式次數(shù)的方法.分子是矩陣多項(xiàng)式,如何降低其次數(shù)是值得深入研究的.

例3 就例1已知條件,判斷分母形如D(x,y)=x+y+1的二元矩陣值有理插值函數(shù)是否存在.

解 與例2類似,考慮方程組但是發(fā)現(xiàn)α00=0時(shí),R(x,y)不能插值其所對(duì)應(yīng)的節(jié)點(diǎn)(x0,y0),而在其他點(diǎn)滿足插值條件.

由定義1易知,由(5),(6)和(8)式定義的矩陣值有理插值函數(shù)是[m+n,m+n]型的,由文獻(xiàn)[1]中

3 結(jié)束語(yǔ)

本文所構(gòu)造的二元矩陣值有理插值函數(shù)類似二元Lagrange插值函數(shù),其構(gòu)造只考慮節(jié)點(diǎn),不必關(guān)心所給的矩陣值.文中構(gòu)造二元矩陣有理插值方法,具有直觀、簡(jiǎn)單的特點(diǎn),并且還可根據(jù)需要降低分母多項(xiàng)式的次數(shù).以往的二元矩陣有理插值計(jì)算大多數(shù)是從連分式入手,從而構(gòu)造出二元Thiele型矩陣有理插值,并且次數(shù)較高.本文在計(jì)算中都是使用簡(jiǎn)單的代數(shù)知識(shí),便于實(shí)際應(yīng)用.

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Method of Constructing Bivariate Matrix-valued Rational Interpolation Functions

DU Wei-wei
(Anhui Education Press,Hefei,Anhui 230601,China)

The well-known algorithms of constructing matrix-valued rational interpolations use continued fractions. Their applicability is not easily forecast and they need a large amount of calculation.In this paper,for calculation of bivariate matrix-valued rational interpolations,multi-parameters are introduced and a group of polynomials with two elements,that is an algebraic polynomial and matrix-valued polynomials,are defined.By using the necessary and sufficient conditions for polynomials identity,linear equations are solved to determine the parameters and the formula of the matrix-valued rational interpolation is given.The formula is simple,so that it has abright application future.

bivariate matrix-valued rational interpolation;parameter;system of equations

O241.3

A

1672-1454(2011)03-0110-05

2008-05-15