Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 642818, 10 pages
http://dx.doi.org/10.1155/2013/642818
Research Article
A New Algorithm to Approximate Bivariate Matrix
Function via Newton-Thiele Type Formula
Rongrong Cui1,2 and Chuanqing Gu1
1
2
Department of Mathematics, Shanghai University, Shanghai 200444, China
School of Mathematical Science, Yancheng Teachers University, Yancheng 224000, China
Correspondence should be addressed to Chuanqing Gu; cqgu@staff.shu.edu.cn
Received 22 October 2012; Revised 6 December 2012; Accepted 10 December 2012
Academic Editor: Juan Manuel Peña
Copyright © 2013 R. Cui and C. Gu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new method for computing the approximation of bivariate matrix function is introduced. It uses the construction of bivariate
Newton-Thiele type matrix rational interpolants on a rectangular grid. The rational interpolant is of the form motivated by Tan and
Fang (2000), which is combined by Newton interpolant and branched continued fractions, with scalar denominator. The matrix
quotients are based on the generalized inverse for a matrix which is introduced by C. Gu the author of this paper, and it is effective
in continued fraction interpolation. The algorithm and some other important conclusions such as divisibility and characterization
are given. In the end, two examples are also given to show the effectiveness of the algorithm. The numerical results of the second
example show that the algorithm of this paper is better than the method of Thieletype matrix-valued rational interpolant in Gu
(1997).
1. Introduction
Matrix-valued rational interpolation and approximation theory have further practical application in many fields, such
as in automatic control theory, computer science, and elementary particle physics [1]. Kuchminska et al. proved an
analog of the van Vlerk theorem and constructed an interpolation formular of the Newton-Thiele type in [2]. Tan and
Fang in [3] put emphasis on the study of Newton-Thiele
bivariate rational interpolants. Graves-Morris provided a
practical Thiele-fraction method for rational interpolation
of vectors, based on the Samelson inverse in [4]. Gu et al.
generalized the definition of Samelson inverse to the case of
matrices and applied it to deal with the problems of rational
interpolation of matrices [5–9]. Bose and Basu gave the
existence, nonuniqueness, and recursive computation of the
two-dimensional matrix Padé approximants in [10].
In this paper, we introduce a bivariate matrix-valued
rational interpolant, with scalar denominator, in NewtonThiele form motivated by [3]. The construction of the
interpolant is combined by the classic methods: Newton
interpolant and branched continued fractions. As we know,
branched continued fraction has been studied by Cuyt and
Verdonk and Siemaszko [11, 12] and many other authors. The
interpolant of this paper is based on using the generalized
inverse of matrices. A new algorithm to approximate bivariate
matrix function is given and some examples are also put to
show the effectiveness of the algorithm which is much better
than the method mentioned in [9].
First, we will give the definition of the so-called generalized inverse of a matrix. Let 𝐶𝑢×𝑣 consists of all 𝑢 × 𝑣
matrices with their elements in the complex plan 𝐶, and let
𝐴 = (𝑎𝑖𝑗 ), 𝐵 = (𝑏𝑖𝑗 ), and 𝐴, 𝐵 ∈ 𝐶𝑢×𝑣 .
Definition 1 (see [8]). The scalar product of matrices 𝐴 and 𝐵
is defined by
𝑢 𝑣
𝐴 ⋅ 𝐵 = ∑∑ 𝑎𝑖𝑗 𝑏𝑖𝑗 = tr (𝐴𝐵∗ ) ,
𝑖=1 𝑗=1
(1)
where 𝐵∗ denotes the transpose of 𝐵 and the Euclidean norm
of 𝐴 is given by
𝑢 𝑣
1/2
󵄨 󵄨2
‖𝐴‖ = (∑∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 )
𝑖=1 𝑗=1
.
(2)
2
Journal of Applied Mathematics
𝜑𝑙,1 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 , 𝑦𝑟 , 𝑦𝑠 )
It follows from Definition 1 and (2) that
= (𝑦𝑠 − 𝑦𝑟 ) × (𝜑𝑙,0 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 , 𝑦𝑠 )
𝑢 𝑣
∗
󵄨 󵄨2
𝐴 ⋅ 𝐴 = ∑ ∑󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 = ‖𝐴‖2 = tr (𝐴𝐴 ) ,
(3)
𝑖=1 𝑗=1
∗
where 𝐴 denotes the complex conjugate matrix of 𝐴 and 𝐴
denotes the complex conjugate transpose matrix of 𝐴. On the
basis of (2) and (3), the generalized inverse of the matrix 𝐴 is
defined as
𝐴−1
𝑟
𝐴
1
= =
,
𝐴 ‖𝐴‖2
−1
−𝜑𝑙,0 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 , 𝑦𝑟 )) ,
𝜑𝑙,𝑘 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 , 𝑦𝑎 , . . . , 𝑦𝑏 , 𝑦𝑟 , 𝑦𝑠 )
= (𝑦𝑠 − 𝑦𝑟 )
× (𝜑𝑙,𝑘−1 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 , 𝑦𝑎 , . . . , 𝑦𝑏 , 𝑦𝑠 )
−1
𝑢×𝑣
𝐴 ≠ 0, 𝐴 ∈ 𝐶
,
−𝜑𝑙,𝑘−1 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 , 𝑦𝑎 , . . . , 𝑦𝑏 , 𝑦𝑟 )) ,
(4)
2
in particular, for 𝐴 ∈ 𝑅𝑢×𝑣 , 𝐴−1
𝑟 = 1/𝐴 = 𝐴/‖𝐴‖ , 𝐴 ≠ 0.
By means of generalized inverse 𝐴−1
𝑟 for matrices, we want
to define bivariate Newton-Thiele rational interpolants and
give the algorithm and some properties on a rectangular grid.
2. Newton-Thiele Interpolation Formula
Let Λ 𝑚,𝑛 = {(𝑥𝑙 , 𝑦𝑘 ) : 𝑙 = 0, 1, . . . , 𝑚, 𝑘 = 0, 1, . . . , 𝑛, (𝑥𝑙 , 𝑦𝑘 ) ∈
R2 }.
A matrix-valued set
Π𝑚,𝑛 = {𝑌𝑙,𝑘 : 𝑌𝑙,𝑘 = 𝑌 (𝑥𝑙 , 𝑦𝑘 ) ∈ 𝐶𝑢×𝑣 , (𝑥𝑙 , 𝑦𝑘 ) ∈ Λ 𝑚,𝑛 } . (5)
(8)
where the first subscript 𝑙 of 𝜑 means the number of nodes
𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 minus 1, and the second subscript 𝑘 of 𝜑
means the number of nodes 𝑦𝑎 , . . . , 𝑦𝑏 , 𝑦𝑟 , 𝑦𝑠 minus 1.
For simplicity, we let the nodes 𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 be
replaced by 𝑥0 , . . . , 𝑥𝑙 , and the nodes 𝑦𝑎 , . . . , 𝑦𝑏 , 𝑦𝑟 , 𝑦𝑠 be
replaced by 𝑦0 , . . . , 𝑦𝑘 , then we can get the following definition.
Definition 2. By means of generalized inverse (4), we define
bivariate Newton-Thiele type matrix blending differences as
follows:
𝜑0,0 (𝑥𝑙 , 𝑦𝑘 ) = 𝑌𝑙,𝑘 ,
𝜑1,0 (𝑥0 , 𝑥1 , 𝑦𝑘 ) =
We need to find a bivariate matrix rational function
𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦𝑘 )
𝑁 (𝑥, 𝑦)
𝑅𝑚,𝑛 (𝑥, 𝑦) =
,
𝐷 (𝑥, 𝑦)
(6)
𝜑1,0 (𝑥𝑖 , 𝑥𝑗 , 𝑦𝑘 ) =
(𝑥𝑙 , 𝑦𝑘 ) ∈ Λ 𝑚,𝑛 .
𝜑𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦1 )
𝜑0,0 (𝑥𝑗 , 𝑦𝑘 ) − 𝜑0,0 (𝑥𝑖 , 𝑦𝑘 )
(7)
(𝑦1 − 𝑦0 )
,
𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦1 ) − 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 )
𝜑0,𝑘 (𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 )
= (𝑦𝑘 − 𝑦𝑘−1 )
× ( 𝜑0,𝑘−1 (𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘 )
∀ (𝑥𝑙 , 𝑦𝑘 ) ∈ Λ 𝑚,𝑛 ,
𝑥𝑗 − 𝑥𝑖
−𝜑𝑙−1,0 (𝑥0 , . . . , 𝑥𝑙−2 , 𝑥𝑙−1 , 𝑦𝑘 )) × (𝑥𝑙 − 𝑥𝑙−1 ) ,
=
First, some notations need to be given as follows:
𝜑0,0 (𝑥𝑙 , 𝑦𝑘 ) = 𝑌𝑙,𝑘 ,
= (𝜑𝑙−1,0 (𝑥0 , . . . , 𝑥𝑙−2 , 𝑥𝑙 , 𝑦𝑘 )
−1
whose numerator 𝑁(𝑥, 𝑦) is a complex or real polynomial
matrix and denominator 𝐷(𝑥, 𝑦) is a real polynomial, and
𝑁 (𝑥𝑙 , 𝑦𝑘 )
= 𝑌𝑙,𝑘 ,
𝑅𝑚,𝑛 (𝑥𝑙 , 𝑦𝑘 ) =
𝐷 (𝑥𝑙 , 𝑦𝑘 )
𝜑0,0 (𝑥1 , 𝑦𝑘 ) − 𝜑0,0 (𝑥0 , 𝑦𝑘 )
,
𝑥1 − 𝑥0
−1
−𝜑0,𝑘−1 (𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘−1 )) ,
,
𝜑𝑙,𝑘 (𝑥0 , 𝑥1 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 )
= (𝑦𝑘 − 𝑦𝑘−1 )
𝜑𝑙,0 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑥𝑗 , 𝑦𝑘 )
× (𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘 )
= (𝜑𝑙−1,0 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑗 , 𝑦𝑘 )
−1
−1
− 𝜑𝑙−1,0 (𝑥𝑝 , . . . , 𝑥𝑞 , 𝑥𝑖 , 𝑦𝑘 )) × (𝑥𝑗 − 𝑥𝑖 ) ,
−𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘−1 )) .
(9)
Journal of Applied Mathematics
3
We assume that for all 𝑙, 𝑥𝑙 ≠ 𝑥𝑙−1 and
𝜓𝑙,𝑘 (𝑥0 , 𝑥1 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 )
= (𝑥𝑙 − 𝑥𝑙−1 )
𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘 )
≠ 𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘−1 ) ,
∀𝑙, 𝑘.
(10)
× (𝜓𝑙−1,𝑘 (𝑥0 , . . . , 𝑥𝑙−2 , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 )
−1
From (9) matrix-valued Newton-Thiele type continued
fractions for two-variable function can be constructed as
follows:
𝑅𝑚,𝑛 (𝑥, 𝑦)
−𝜓𝑙−1,𝑘 (𝑥0 , . . . , 𝑥𝑙−1 , 𝑦0 , . . . , 𝑦𝑘 )) .
(13)
From (13) we can get the antithetical formula for twovariable function
̂ 𝑚,𝑛 (𝑥, 𝑦) = 𝑉0 (𝑥) + 𝑉1 (𝑥) (𝑦 − 𝑦0 )
𝑅
= 𝑈0 (𝑦) + 𝑈1 (𝑦) (𝑥 − 𝑥0 ) + ⋅ ⋅ ⋅ + 𝑈𝑚 (𝑦) (𝑥 − 𝑥0 ) (11)
+ ⋅ ⋅ ⋅ + 𝑉𝑛 (𝑥) (𝑦 − 𝑦0 ) (𝑦 − 𝑦1 )
× (𝑥 − 𝑥1 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑚−1 ) ,
(14)
⋅ ⋅ ⋅ (𝑦 − 𝑦𝑛−1 ) ,
where for 𝑙 = 0, 1, . . . , 𝑚
𝑦 − 𝑦0
𝑈𝑙 (𝑦) = 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 ) +
𝜑𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦1 )
𝑦 − 𝑦𝑛−1 |
+ ⋅⋅⋅ +
.
| 𝜑𝑙,𝑛 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑛 )
where for 𝑘 = 0, 1, . . . , 𝑛
𝑉𝑘 (𝑥) = 𝜓0,𝑘 (𝑥0 , 𝑦0 , . . . , 𝑦𝑘 )
(12)
Remark 3. Because of the construction, 𝐷(𝑥, 𝑦) in 𝑅𝑚,𝑛 (𝑥, 𝑦)
mentioned as (6) is actually 𝐷(𝑦) and each 𝑈𝑙 (𝑦) depends on
𝑛.
We can also construct the antithetical form of bivariate
matrix continued fraction in light of Definition 4.
+
+ ⋅⋅⋅ +
(𝜓0,0 (𝑥𝑙 , 𝑦1 ) − 𝜓0,0 (𝑥𝑙 , 𝑦0 ))
𝜓0,1 (𝑥𝑙 , 𝑦0 , 𝑦1 ) =
(𝑦1 − 𝑦0 )
(15)
𝑥 − 𝑥𝑚−1 |
.
| 𝜓𝑚,𝑘 (𝑥0 , . . . , 𝑥𝑚 , 𝑦0 , . . . , 𝑦𝑘 )
Now we will give two theorems about 𝑅𝑚,𝑛 (𝑥, 𝑦) as in (11)
̂ 𝑚,𝑛 (𝑥, 𝑦) as in (14) and (15). First of all, some
and (12) and 𝑅
notations need to be defined.
In (12), for 𝑙 = 0, 1, . . . , 𝑚, let
𝑈𝑙(𝑠) (𝑦) = 𝜑𝑙,𝑠 (𝑥0 , 𝑥1 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑠 )
Definition 4. By means of generalized inverse (4), we define
bivariate antithetical Newton-Thiele type matrix blending
differences
𝜓0,0 (𝑥𝑙 , 𝑦𝑘 ) = 𝑌𝑙,𝑘 ,
𝑥 − 𝑥0
𝜓1,𝑘 (𝑥0 , 𝑥1 , 𝑦0 , . . . , 𝑦𝑘 )
+
𝑦 − 𝑦𝑠
𝑈𝑙(𝑠+1)
(𝑦)
,
𝑠 = 0, 1, . . . , 𝑛 − 1,
(16)
where
𝑈𝑙(𝑛) (𝑦) = 𝜑𝑙,𝑛 (𝑥0 , 𝑥1 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑛 ) ,
𝑈𝑙(0) (𝑦) = 𝑈𝑙 (𝑦) .
𝜓0,𝑘 (𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 )
(17)
Similarly, in (15), for 𝑘 = 0, 1, . . . , 𝑛, let
= (𝜓0,𝑘−1 (𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘 )
𝑉𝑘(𝑠) (𝑥) = 𝜓𝑠,𝑘 (𝑥0 , 𝑥1 , . . . , 𝑥𝑠 , 𝑦0 , . . . , 𝑦𝑘 )
−𝜓0,𝑘−1 (𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘−1 ))
−1
× (𝑦𝑘 − 𝑦𝑘−1 ) ,
+
𝜓1,𝑘 (𝑥0 , 𝑥1 , 𝑦0 , . . . , 𝑦𝑘 )
(𝑥1 − 𝑥0 )
=
,
𝜓0,𝑘 (𝑥1 , 𝑦0 , . . . , 𝑦𝑘 ) − 𝜓0,𝑘 (𝑥0 , 𝑦0 , . . . , 𝑦𝑘 )
𝑥 − 𝑥𝑠
𝑉𝑘(𝑠+1) (𝑥)
,
𝑠 = 0, 1, . . . , 𝑚 − 1,
where
𝑉𝑘(𝑚) (𝑥) = 𝜓𝑚,𝑘 (𝑥0 , 𝑥1 , . . . , 𝑥𝑚 , 𝑦0 , . . . , 𝑦𝑘 ) ,
𝑉𝑘(0) (𝑥) = 𝑉𝑘 (𝑥) .
𝜓𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦𝑘 )
= (𝑥𝑙 − 𝑥𝑙−1 )
(18)
(19)
Theorem 5. Let
× (𝜓𝑙−1,0 (𝑥0 , . . . , 𝑥𝑙−2 , 𝑥𝑙 , 𝑦𝑘 )
−1
−𝜓𝑙−1,0 (𝑥0 , . . . , 𝑥𝑙−1 , 𝑦𝑘 )) ,
(i) 𝜑𝑙,𝑘 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 ), 𝑙 = 0, 1, . . . , 𝑚; 𝑘 = 0,
1, . . . , 𝑛 exist and be nonzero (except for 𝜑𝑙,0 (𝑥0 ,
. . . , 𝑥𝑙 , 𝑦0 )),
4
Journal of Applied Mathematics
(ii) 𝑈𝑙(𝑠) (𝑦) = 𝜑𝑙,𝑠 (𝑥0 , 𝑥1 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑠 ) + ((𝑦 −
𝑦𝑠 )/𝑈𝑙(𝑠+1) (𝑦)) satisfy 𝑈𝑙(𝑠+1) (𝑦𝑠 ) ≠ 0, 𝑠 = 0, 1, . . . , 𝑛 − 1,
then 𝑅𝑚,𝑛 (𝑥, 𝑦) as in (11) and (12) exists such that
𝑅𝑚,𝑛 (𝑥𝑙 , 𝑦𝑘 ) = 𝑌𝑙𝑘 , (𝑥𝑙 , 𝑦𝑘 ) ∈ Λ m,𝑛 .
if the conditions hold, thus for 𝑙 = 0, 1, . . . , 𝑚; 𝑘 = 0, 1, . . . , 𝑛
𝑈𝑙 (𝑦𝑘 ) = 𝑈𝑙(0) (𝑦𝑘 )
= 𝜑𝑙,0 +
Proof. For simpleness, let
+
𝜑𝑙,𝑘 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 ) = 𝜑𝑙,𝑘
(20)
𝑦𝑘 − 𝑦0
𝑦 − 𝑦𝑘−1 |
+ ⋅⋅⋅ + 𝑘
𝜑𝑙,1
| 𝜑𝑙,𝑛
(21)
𝑦𝑘 − 𝑦𝑘 |
| 𝑈𝑙(𝑘+1) (𝑦𝑘 )
(𝑦𝑘 −𝑦𝑘 )/𝑈𝑙(𝑘+1) (𝑦𝑘 ) = 0, since 𝑈𝑙(𝑘+1) (𝑦𝑘 ) ≠ 0, then we get
𝑈𝑙 (𝑦𝑘 )
= 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 ) +
+
𝑦𝑘 − 𝑦𝑘−1 |
| 𝜑𝑙,𝑘 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 )
= 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 ) +
+
+
𝑦𝑘 − 𝑦0
𝑦𝑘 − 𝑦𝑘−2 |
+ ⋅⋅⋅ +
𝜑𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦1 )
| 𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−1 )
𝑦𝑘 − 𝑦0
+ ⋅⋅⋅
𝜑𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦1 )
𝑦𝑘 − 𝑦𝑘−2 |
| 𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−1 )
(22)
𝑦𝑘 − 𝑦𝑘−1 |
−1
| (𝑦𝑘 − 𝑦𝑘−1 ) × (𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘 ) − 𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−1 ))
= 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 ) +
𝑦𝑘 − 𝑦0
𝑦𝑘 − 𝑦𝑘−2 |
+ ⋅⋅⋅ +
𝜑𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦1 )
| 𝜑𝑙,𝑘−1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘−2 , 𝑦𝑘 )
= ⋅ ⋅ ⋅ = 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 ) +
𝑦𝑘 − 𝑦0
𝜑𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦𝑘 )
= 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦𝑘 ) .
We can use (9), (11), and (22) to find that
Now we give the Newton-Thiele Matrix Interpolation
Algorithm (NTMIA).
𝑅𝑚,𝑛 (𝑥𝑙 , 𝑦𝑘 )
= 𝑈0 (𝑦𝑘 ) + 𝑈1 (𝑦𝑘 ) (𝑥𝑙 − 𝑥0 )
+ ⋅ ⋅ ⋅ + 𝑈𝑙 (𝑦𝑘 ) (𝑥𝑙 − 𝑥0 ) (𝑥𝑙 − 𝑥1 ) ⋅ ⋅ ⋅ (𝑥𝑙 − 𝑥𝑙−1 )
(23)
= 𝑌 (𝑥𝑙 , 𝑦𝑘 ) .
We can similarly prove the following result.
Theorem 6. Let
(i) 𝜓𝑙,𝑘 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 ), 𝑙 = 0, 1, . . . , 𝑚; 𝑘 = 0,
1, . . . , 𝑛 exist and be nonzero (except for 𝜓0,𝑘 (𝑥0 ,
𝑦0 , . . . , 𝑦𝑘 )),
(ii) 𝑉𝑘(𝑠) (𝑥) = 𝜓𝑠,𝑘 (𝑥0 , 𝑥1 , . . . , 𝑥𝑠 , 𝑦0 , . . . , 𝑦𝑘 ) + ((𝑥 −
𝑥𝑠 )/𝑉𝑘(𝑠+1) (𝑥)) satisfy 𝑉𝑘(𝑠+1) (𝑥𝑠 ) ≠ 0, 𝑠 = 0, 1, . . . , 𝑚−1,
̂ 𝑚,𝑛 (𝑥, 𝑦) as in (14) and (15) exists such that
then 𝑅
̂
𝑅𝑚,𝑛 (𝑥𝑙 , 𝑦𝑘 ) = 𝑌𝑙𝑘 , (𝑥𝑙 , 𝑦𝑘 ) ∈ Λ 𝑚,𝑛 .
Algorithm 7 (NTMIA). Input: {(𝑥𝑖 , 𝑦𝑗 ), 𝑌𝑖,𝑗 } (𝑖 = 0, 1, . . . ,
𝑚; 𝑗 = 0, 1, . . . , 𝑛)
̂ 𝑚 (𝑥, 𝑦).
Output: 𝑅
Step 1. For all (𝑥𝑖 , 𝑦𝑗 ) ∈ Λ 𝑚,𝑛 , let 𝜑(𝑥𝑖 , 𝑦𝑗 ) = 𝑌𝑖,𝑗 .
Step 2. For 𝑗 = 0, 1, . . . , 𝑛; 𝑝 = 1, 2, . . . , 𝑚; 𝑖 = 𝑝, 𝑝 +
1, . . . , 𝑚, compute
𝜑𝑝,0 (𝑥0 , . . . , 𝑥𝑝−1 , 𝑥𝑖 , 𝑦𝑗 )
=
𝜑𝑝−1,0 (𝑥0 , . . . , 𝑥𝑝−2 , 𝑥𝑖 , 𝑦𝑗 )−𝜑𝑝−1,0 (𝑥0 , . . . , 𝑥𝑝−1 , 𝑦𝑗 )
𝑥𝑖 −𝑥𝑝−1
.
(24)
Journal of Applied Mathematics
5
Step 3. For 𝑖 = 0, 1, . . . , 𝑚; 𝑞 = 1, 2, . . . , 𝑛; 𝑗 = 𝑞, 𝑞 + 1, . . . , 𝑛,
compute
𝜑𝑖,𝑞 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑞−1 , 𝑦𝑗 )
= (𝑦𝑗 − 𝑦𝑞−1 ) × (𝜑𝑖,𝑞−1 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑞−2 , 𝑦𝑗 )
−1
−𝜑𝑖,𝑞−1 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑞−1 ))
= ((𝑦𝑗 − 𝑦𝑞−1 ) (𝜑𝑖,𝑞−1 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑞−2 , 𝑦𝑗 )
−𝜑𝑖,𝑞−1 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑞−1 )) )
󵄩
× (󵄩󵄩󵄩󵄩𝜑 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑞−2 , 𝑦𝑗 )
󵄩2
−𝜑 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑞−1 )󵄩󵄩󵄩󵄩 ) .
−1
Definition 9. 𝑅(𝑥, 𝑦) = 𝑁(𝑥, 𝑦)/𝐷(𝑥, 𝑦) is defined to be of
type [𝑙/𝑡] if deg{𝑁𝑖𝑗 } ≤ 𝑙 for 1 ≤ 𝑖 ≤ 𝑢, 1 ≤ 𝑗 ≤ 𝑣, deg{𝑁𝑖𝑗 } = 𝑙
for some (𝑖, 𝑗) and deg{𝐷} = 𝑡, where 𝑁(𝑥, 𝑦) = (𝑁𝑖𝑗 (𝑥, 𝑦)) ∈
𝐶𝑢×𝑣 .
Lemma 10 (see [5]). Let 𝑈0 (𝑦) be defined as in Lemma 8, and
𝑈0 (𝑦) = 𝐸0 (𝑦)/𝐹0 (𝑦); if 𝑛 is even, 𝑈0 (𝑦) is of type [𝑛/𝑛]; if 𝑛 is
odd, 𝑈0 (𝑦) is of type [𝑛/𝑛 − 1].
Theorem 11. Let 𝜑𝑙,𝑘 ∈ 𝐶𝑢×𝑣 , (𝑥𝑙 , 𝑦𝑘 ) ∈ Λ 𝑚,𝑛 , and 𝑥, 𝑦 ∈
𝑅. Define 𝑈𝑙 (𝑦) 𝑜𝑓 𝑅𝑚𝑛 (𝑥, 𝑦) as in (7) by a tail-to-head
rationalization using generalized inverse and suppose every
intermediate denominator be nonzero in the operation, then
a square polynomial matrix 𝑁(𝑥, 𝑦) and a real polynomial
𝐷(𝑥, 𝑦) exist such that
(i) 𝑅𝑚,𝑛 (𝑥, 𝑦) = 𝑁(𝑥, 𝑦)/𝐷(𝑥, 𝑦),
(25)
Step 4. For 𝑖 = 0, 1, . . . , 𝑚; 𝑗 = 0, 1, . . . , 𝑛, compute 𝑎𝑖,𝑗 =
𝜑𝑖,𝑗 (𝑥0 , . . . , 𝑥𝑖 , 𝑦0 , . . . , 𝑦𝑗 ).
Step 5. For 𝑖 = 0, 1, . . . , 𝑚; 𝑗 = 1, . . . , 𝑛 judge if 𝑎𝑖,𝑗 ≠ 0; if yes
go to Step 6, otherwise exist and show “the algorithm is not
valid.”
(ii) 𝐷(𝑥, 𝑦) ≥ 0,
(iii) 𝐷(𝑥, 𝑦) | ‖𝑁(𝑥, 𝑦)‖2 .
Proof. Consider the following algorithm for the construction
of 𝑁(𝑥, 𝑦), 𝐷(𝑥, 𝑦), and 𝑅𝑚𝑛 (𝑥, 𝑦).
Initialization: let 𝐷0 (𝑥, 𝑦) = 1, and
Step 6. For 𝑖 = 0, 1, . . . , 𝑚; 𝑗 = 𝑛 − 2, 𝑛 − 3, . . . , 0
𝑄𝑖,𝑛 = 1,
𝑃𝑖,𝑛 = 𝑎𝑖,𝑛 ,
󵄩 󵄩2
𝑄𝑖,𝑛−1 = 󵄩󵄩󵄩𝑎𝑖,𝑛 󵄩󵄩󵄩 ,
𝑃𝑖,𝑛−1 = 𝑎𝑖,𝑛−1 𝑄𝑖,𝑛−1 + (𝑦 − 𝑦𝑛−1 ) 𝑃𝑖,𝑛 ,
∗
󵄩2
󵄩
𝑄𝑖,𝑗 = 󵄩󵄩󵄩󵄩𝑎𝑖,𝑗+1 󵄩󵄩󵄩󵄩 𝑄𝑖,𝑗+1 + 2 (𝑦 − 𝑦𝑗+1 ) tr (𝑎𝑖,𝑗+1 𝑃𝑖,𝑗+2 ) (26)
2
+ (𝑦 − 𝑦𝑗+1 ) 𝑄𝑖,𝑗+2 ,
𝑃𝑖,𝑗 = 𝑎𝑖,𝑗 𝑄𝑖,𝑗 + (𝑦 − 𝑦𝑗 ) 𝑃𝑖,𝑗+1 .
Step 7. For 𝑖 = 0, 1, . . . , 𝑚; 𝑗 = 𝑛, 𝑛 − 1, . . . , 1, let 𝐵𝑖,𝑗 (𝑦) =
𝑃𝑖,𝑗 (𝑦)/𝑄𝑖,𝑗 (𝑦); if 𝐵𝑖,𝑗 (𝑦𝑗−1 ) ≠ 0 then go to Step 8, otherwise
exist and show “the algorithm is not valid.”
Step 8. For 𝑖 = 0, 1, . . . , 𝑚, let 𝐵̂𝑖 (𝑦) = 𝐵𝑖,0 (𝑦) = 𝑃𝑖,0 (𝑦)/
𝑄𝑖,0 (𝑦).
̂ 0 (𝑥, 𝑦) = 𝐵̂0 (𝑦), then compute
Step 9. For 𝑘 = 1, . . . , 𝑚 let 𝑅
̂ 𝑘 (𝑥, 𝑦) = 𝑅
̂ 𝑘−1 (𝑥, 𝑦) + ⋅ ⋅ ⋅ + (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑘−1 )𝐵̂𝑘 (𝑦).
𝑅
3. Some Properties
Lemma 8 (see [5]). Define 𝑈0 (𝑦) = 𝜑0,0 (𝑥0 , 𝑦0 ) + (𝑦 −
𝑦0 )/𝜑0,1 (𝑥0 , 𝑦0 , 𝑦1 ) + ⋅ ⋅ ⋅ + (𝑦 − 𝑦𝑛−1 )/𝜑0,𝑛 (𝑥0 , 𝑦0 , . . . , 𝑦𝑛 ); if we
use generalized inverse by a tail-to-head rationalization, then
a polynomial matrix 𝐸0 (𝑦) and a real polynomial 𝐹0 (𝑦) exist
such that
(i) 𝑈0 (𝑦) = 𝐸0 (𝑦)/𝐹0 (𝑦), 𝐹0 (𝑦) ≥ 0,
(ii) 𝐹0 (𝑦) | ‖𝐸0 (𝑦)‖2 ,“|” means the sign of divisibility.
𝑁0 (𝑥, 𝑦) = 𝑈0 (𝑦) = 𝜑0,0 +
𝑦 − 𝑦0
𝑦 − 𝑦𝑛−1 |
+ ⋅⋅⋅ +
. (27)
𝜑0,1
| 𝜑0,𝑛
By Lemma 8, a polynomial matrix 𝐸0 (𝑦) and a real polynomial 𝐹0 (𝑦) exist such that
(a1) 𝑁0 (𝑥, 𝑦) = 𝐸0 (𝑦)/𝐹0 (𝑦),
(a2) 𝐹0 (𝑦) ≥ 0,
(a3) 𝐹0 (𝑦) | ‖𝐸0 (𝑦)‖2 .
Recursion: for 𝑗 = 1, 2, . . . , 𝑚 let
𝑆(𝑗) (𝑥, 𝑦) = 𝑆(𝑗−1) (𝑥, 𝑦) + 𝑈𝑗 (𝑦) (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗−1 )
(28)
with 𝑈𝑗 (𝑦) = 𝐸𝑗 (𝑦)/𝐹𝑗 (𝑦), 𝐹𝑗 (𝑦) | ‖𝐸𝑗 (𝑦)‖2 having the
representation
(b1) 𝑆(𝑗) (𝑥, 𝑦) = 𝑁𝑗 (𝑥, 𝑦)/𝐷𝑗 (𝑥, 𝑦),
(b2) 𝐷𝑗 (𝑥, 𝑦) ≥ 0,
(b3) 𝐷𝑗 (𝑥, 𝑦) | ‖𝑁𝑗 (𝑥, 𝑦)‖2 ,
where 𝐷𝑗 (𝑥, 𝑦) is a real polynomial.
6
Journal of Applied Mathematics
(i) if n is even, 𝑅 is of type [𝑚𝑛 + 𝑚 + 𝑛/𝑚𝑛 + 𝑛];
(ii) if n is odd, 𝑅 is of type [𝑚𝑛 + 𝑛/𝑚𝑛 + 𝑛 − 𝑚 − 1].
Then we can get
𝑆(𝑗+1) (𝑥, 𝑦) = 𝑆(𝑗) (𝑥, 𝑦) + 𝑈𝑗+1 (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗 )
=
𝑁𝑗 (𝑥, 𝑦)
𝐷𝑗 (𝑥, 𝑦)
+
𝐸𝑗+1 (𝑦)
𝐹𝑗+1 (𝑦)
(𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗 )
= (𝑁𝑗 (𝑥, 𝑦) 𝐹𝑗+1 (𝑦) + 𝐷𝑗 (𝑥, 𝑦) 𝐸𝑗+1 (𝑦)
Proof. The proof is recursive. Let 𝑛 be even.
For 𝑚 = 0, 𝑅0,𝑛 (𝑥, 𝑦) = 𝑈0 (𝑦) = 𝐸0 /𝐹0 , by Lemma 10, we
find that it is of type [𝑛/𝑛].
For 𝑚 = 1, because deg{𝐸0 } = deg{𝐹0 } = 𝑛, deg{𝐸1 } =
deg{𝐹1 } = 𝑛
𝑅1,𝑛 (𝑥, 𝑦) =
× (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗 ))
× (𝐷𝑗 (𝑥, 𝑦) 𝐹𝑗+1 (𝑦))
=
𝑁𝑗+1 (𝑥, 𝑦)
𝐷𝑗+1 (𝑥, 𝑦)
−1
,
(29)
where
𝑁𝑗+1 (𝑥, 𝑦) = 𝑁𝑗 (𝑥, 𝑦) 𝐹𝑗+1 (𝑦)
+ 𝐷𝑗 (𝑥, 𝑦) 𝐸𝑗+1 (𝑦) (𝑥 − 𝑥0 )
⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗 ) ,
(30)
𝐷𝑗+1 (𝑥, 𝑦) = 𝐷𝑗 (𝑥, 𝑦) 𝐹𝑗+1 (𝑦) .
=
𝐸0 𝐸1
+
(𝑥 − 𝑥0 )
𝐹0 𝐹1
=
𝐸0 𝐹1 + 𝐸1 𝐹0 (𝑥 − 𝑥0 )
𝐹0 𝐹1
󵄩󵄩
󵄩2 󵄩
󵄩2 2
󵄩󵄩𝑁𝑗+1 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑁𝑗 (𝑥, 𝑦)󵄩󵄩󵄩 𝐹(𝑗+1)
(𝑦) + 𝐷𝑗2 (𝑥, 𝑦)
󵄩
󵄩
󵄩
󵄩
2
󵄩
󵄩2
2
× 󵄩󵄩󵄩󵄩𝐸(𝑗+1) (𝑦)󵄩󵄩󵄩󵄩 (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗 )
is of type [2𝑛 + 1/2𝑛].
For 𝑚 = 2, from the formula above we know that
deg{𝑁1 } = 2𝑛 + 1, deg{𝐷1 } = 2𝑛, deg{𝐸2 } = 𝑛, and deg{𝐹2 } =
𝑛. one has
𝑁
𝑁
𝐸
𝑅2,𝑛 (𝑥, 𝑦) = 2 = 1 + 2 (𝑥 − 𝑥0 ) (𝑥 − 𝑥1 )
𝐷2 𝐷1 𝐹2
(35)
𝑁1 𝐹2 + 𝐸2 𝐷1 (𝑥 − 𝑥0 ) (𝑥 − 𝑥1 )
=
.
𝐷1 𝐹2
(31)
=
𝑁𝑘 𝐹𝑘+1 + 𝐸𝑘+1 𝐷𝑘 (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑘 )
(36)
𝐷𝑘 𝐹𝑘
=
𝑁𝑘+1
,
𝐷𝑘+1
× (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗 )
+ 𝑁𝑗 (𝑥, 𝑦) ⋅ 𝐸𝑗+1 (𝑦) 𝐹𝑗+1 (𝑦) 𝐷𝑗 (𝑥, 𝑦)
where
× (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑗 )
deg {𝑁𝑘+1 } = (𝑘 + 1) (𝑛 + 1) + 𝑛
since
󵄩
󵄩2
𝐷𝑗 (𝑥, 𝑦) | 󵄩󵄩󵄩󵄩𝑁𝑗 (𝑥, 𝑦)󵄩󵄩󵄩󵄩 ,
󵄩
󵄩2
𝐹𝑗+1 (𝑦) | 󵄩󵄩󵄩󵄩𝐸𝑗+1 (𝑦)󵄩󵄩󵄩󵄩 ,
󵄩
󵄩2
𝐷𝑗+1 (𝑥, 𝑦) | 󵄩󵄩󵄩󵄩𝑁𝑗+1 (𝑥, 𝑦)󵄩󵄩󵄩󵄩 .
(34)
It is easy to find that 𝑅2,𝑛 (𝑥, 𝑦) is of type [3𝑛 + 2/3𝑛].
For 𝑚 = 𝑘, let 𝑅𝑘,𝑛 (𝑥, 𝑦) = 𝑁𝑘 /𝐷𝑘 be of type [(𝑘 + 1)𝑛 +
𝑘/(𝑘 + 1)𝑛].
For 𝑚 = 𝑘 + 1, we consider that
𝐸
𝑁
𝑅𝑘+1,𝑛 (𝑥, 𝑦) = 𝑘 + 𝑘+1 (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑘 )
𝐷𝑘 𝐹𝑘+1
Obviously,
+ 𝑁𝑗 (𝑥, 𝑦) ⋅ 𝐸𝑗+1 (𝑦) 𝐹𝑗+1 (𝑦) 𝐷𝑗 (𝑥, 𝑦)
𝑁1
= 𝑈0 (𝑦) + 𝑈1 (𝑦) (𝑥 − 𝑥0 )
𝐷1
= (𝑘 + 2) 𝑛 + (𝑘 + 1) ,
(32)
Termination: 𝑆(𝑚) (𝑥, 𝑦) = 𝑅𝑚,𝑛 (𝑥, 𝑦).
Theorem 12 (Characterization). Let 𝑅𝑚𝑛 (𝑥, 𝑦) = 𝑁(𝑥, 𝑦)/
𝐷(𝑥, 𝑦) be expressed as
𝑅 = 𝑅𝑚,𝑛 (𝑥, 𝑦) = 𝑈0 (𝑦) + 𝑈1 (𝑦) (𝑥 − 𝑥0 )
+ ⋅ ⋅ ⋅ + 𝑈𝑚 (𝑦) (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑚−1 ) ,
(33)
where 𝑈𝑙 (𝑦) = 𝐸𝑖 /𝐹𝑖 , as in (12) we assume 𝐹𝑖 , 𝐹𝑗 have no
common factor, 𝑖, 𝑗 = 0, 1, . . . , 𝑚, 𝑖 ≠ 𝑗, then
(37)
deg {𝐷𝑘+1 } = (𝑘 + 1) 𝑛 + 𝑛 = (𝑘 + 2) 𝑛.
Therefore, when 𝑛 is even, 𝑅 is of type [𝑚𝑛 + 𝑚 + 𝑛/𝑚𝑛 + 𝑛].
Similarly, if 𝑛 is odd, for 𝑚 = 0, 𝑅0,𝑛 (𝑥, 𝑦) = 𝑈0 (𝑦) =
𝐸0 /𝐹0 , by Lemma 10, we find that it is of type [𝑛/𝑛 − 1].
For 𝑚 = 1, because deg{𝐸0 } = deg{𝐸1 } = 𝑛, deg{𝐹0 } =
deg{𝐹1 } = 𝑛 − 1
𝑁
𝑅1,𝑛 (𝑥, 𝑦) = 1 = 𝑈0 (𝑦) + 𝑈1 (𝑦) (𝑥 − 𝑥0 )
𝐷1
=
𝐸0 𝐸1
+
(𝑥 − 𝑥0 )
𝐹0 𝐹1
=
𝐸0 𝐹1 + 𝐸1 𝐹0 (𝑥 − 𝑥0 )
𝐹0 𝐹1
is of type [2𝑛/2𝑛 − 2].
(38)
Journal of Applied Mathematics
7
For 𝑚 = 2, from the formula above we know that deg{𝑁1 }
= 2𝑛, deg{𝐷1 } = 2𝑛 − 2, deg{𝐸2 } = 𝑛, and deg{𝐹2 } = 𝑛 − 1.
one has
𝑅2,𝑛 (𝑥, 𝑦) =
𝑁2 𝑁1 𝐸2
=
+
(𝑥 − 𝑥0 ) (𝑥 − 𝑥1 )
𝐷2 𝐷1 𝐹2
𝑁 𝐹 + 𝐸2 𝐷1 (𝑥 − 𝑥0 ) (𝑥 − 𝑥1 )
= 1 2
.
𝐷1 𝐹2
𝑁𝑘
𝐷𝑘
∗
(𝑥, 𝑦) = 𝑈0∗ (𝑦) + 𝑈1∗ (𝑦) (𝑥 − 𝑥0 )
𝑅𝑚,𝑛
+ ⋅ ⋅ ⋅ + 𝑈𝑚∗ (𝑦) (𝑥−𝑥0 ) (𝑥−𝑥1 ) ⋅ ⋅ ⋅ (𝑥−𝑥𝑚−1 ) ,
(45)
(39)
where for 𝑙 = 0, 1, . . . , 𝑚
𝑈𝑙∗ (𝑦) = 𝜑𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 ) +
We get that 𝑅2,𝑛 (𝑥, 𝑦) is of type [3𝑛/3𝑛 − 3].
For 𝑚 = 𝑘, let
𝑅𝑘,𝑛 (𝑥, 𝑦) =
Lemma 14. Let
+ ⋅⋅⋅ +
(40)
+
be of type [(𝑘 + 1)𝑛/(𝑘 + 1)(𝑛 − 1)].
For 𝑚 = 𝑘 + 1, we consider that
𝑅𝑘+1,𝑛 (𝑥, 𝑦) =
𝑁𝑘 𝐸𝑘+1
+
(𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑘 )
𝐷𝑘 𝐹𝑘+1
𝑁𝑘+1
,
𝐷𝑘+1
deg {𝑁𝑘+1 } = (𝑘 + 1) (𝑛 − 1) + 𝑘 + 1 = (𝑘 + 2) 𝑛,
deg {𝐷𝑘+1 } = (𝑘 + 1) (𝑛 − 1) + (𝑛 − 1) = (𝑘 + 2) (𝑛 − 1) .
(42)
Therefore, when 𝑛 is odd, 𝑅 is of type [(𝑚+1)𝑛/(𝑚+1)(𝑛−
Definition 13. A matrix-valued Newton-Thiele type rational fraction 𝑅𝑚,𝑛 (𝑥, 𝑦) = 𝑁(𝑥, 𝑦)/𝐷(𝑥, 𝑦) is defined to
be a bivariate generalized inverse and rational interpolant
(BGIRINT ) on the rectangular grid Λ 𝑚,𝑛 if
(i) 𝑅𝑚𝑛 (𝑥𝑙 , 𝑦𝑘 ) = 𝑌𝑙𝑘 , (𝑥𝑙 , 𝑦𝑘 ) ∈ Λ 𝑚,𝑛 ,
(ii) 𝐷(𝑥𝑙 , 𝑦𝑘 ) ≠ 0, (𝑥𝑙 , 𝑦𝑘 ) ∈ Λ 𝑚,𝑛 ,
deg {𝐷 (𝑥, 𝑦)} = 𝑚𝑛+𝑛,
(43)
(b) if n is odd,
deg {𝑁 (𝑥, 𝑦)} = 𝑚𝑛+𝑛,
𝑦 − 𝑦𝑛 |
,
| 𝜑𝑙,𝑛+1 (𝑥0 , . . . , 𝑥𝑙 , y0 , . . . , 𝑦𝑛 , 𝑦)
∗
(𝑥𝑖 , 𝑦) = 𝑓 (𝑥𝑖 , 𝑦) ,
𝑅𝑚,𝑛
∗
𝑅𝑚,𝑛
(𝑥, 𝑦𝑗 ) = 𝑅𝑚,𝑛 (𝑥, 𝑦𝑗 ) ,
𝑖 = 0, 1, . . . , 𝑚,
𝑗 = 0, 1, . . . , 𝑛.
(47)
Now the error estimate Theorem 16 will be put which is
also motivated by [3]. We let
𝑅𝑚,𝑛 (𝑥, 𝑦) =
𝑁 (𝑥, 𝑦)
,
𝐷 (𝑥, 𝑦)
∗
(𝑥, 𝑦) =
𝑅𝑚,𝑛
𝑁∗ (𝑥, 𝑦)
. (48)
𝐷∗ (𝑥, 𝑦)
Theorem 16. Let a matrix 𝑓(𝑥, 𝑦) be max(𝑚 + 1, 𝑛 + 1) times
continuously differentiable on a set 𝐷 = {𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑}
containing the points Λ 𝑚,𝑛 = {(𝑥𝑙 , 𝑦𝑘 ) : 𝑙 = 0, 1, . . . , 𝑚, 𝑘 =
0, 1, . . . , 𝑛, (𝑥𝑙 , 𝑦𝑘 ) ∈ R2 }, then, for any point (𝑥, 𝑦) ∈ 𝐷, there
exists a point (𝜉, 𝜂) ∈ 𝐷 such that
𝑓 (𝑥, 𝑦) − 𝑅𝑚,𝑛 (𝑥, 𝑦)
=
(iii) (a) if 𝑛 is even,
deg {𝑁 (𝑥, 𝑦)} = 𝑚𝑛+ 𝑚+𝑛,
(46)
Remark 15. We delete the proof of the above lemma since it
can be easily generalized from [3].
where
1)].
𝑦 − 𝑦𝑛−1 |
| 𝜑𝑙,𝑛 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑛 )
∗
then 𝑅𝑚,𝑛
(𝑥, 𝑦) satisfies
𝑁 𝐹 + 𝐸𝑘+1 𝐷𝑘 (𝑥 − 𝑥0 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑘 )
(41)
= 𝑘 𝑘+1
𝐷𝑘 𝐹𝑘
=
𝑦 − 𝑦0
𝜑𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦1 )
1
𝐷 (𝑥, 𝑦) 𝐷∗ (𝑥, 𝑦)
×{
𝜛𝑛+1 (𝑦) 𝜕𝑛+1
𝜔𝑚+1 (𝑥) 𝜕𝑚+1
𝐸
(𝜉,
𝑦)
+
𝐸 (𝑥, 𝜂)} ,
(𝑚 + 1)! 𝜕𝑥𝑚+1 1
(𝑛 + 1)! 𝜕𝑦𝑛+1 2
(49)
where
deg {𝐷 (𝑥, 𝑦)} = 𝑚𝑛+𝑛−𝑚−1,
(44)
(iv) 𝐷(𝑥, 𝑦) | ‖𝑁(𝑥, 𝑦)‖2 ,
(v) 𝐷(𝑥, 𝑦) is real, and 𝐷(𝑥, 𝑦) ≥ 0.
Now let us turn to the error estimate of the BGIRINT .
Firstly, we give Lemma 14.
𝜔𝑚+1 (𝑥) = (𝑥 − 𝑥0 ) (𝑥 − 𝑥1 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑚 ) ,
𝜛𝑛+1 (𝑦) = (𝑦 − 𝑦0 ) (𝑦 − 𝑦1 ) ⋅ ⋅ ⋅ (𝑦 − 𝑦𝑛 ) ,
∗
𝐸1 (𝑥, 𝑦) = 𝐷 (𝑥, 𝑦) 𝐷∗ (𝑥, 𝑦) [𝑓 (𝑥, 𝑦) − 𝑅𝑚,𝑛
(𝑥, 𝑦)] ,
∗
(𝑥, 𝑦) − 𝑅𝑚,𝑛 (𝑥, 𝑦)] .
𝐸2 (𝑥, 𝑦) = 𝐷 (𝑥, 𝑦) 𝐷∗ (𝑥, 𝑦) [𝑅𝑚,𝑛
(50)
8
Journal of Applied Mathematics
Proof. Let
Based on generalized inverse (4), we obtain
∗
𝐸1 (𝑥, 𝑦) = 𝐷 (𝑥, 𝑦) 𝐷∗ (𝑥, 𝑦) [𝑓 (𝑥, 𝑦) − 𝑅𝑚,𝑛
(𝑥, 𝑦)] ,
𝑅2,2 (𝑥, 𝑦)
𝑎 𝑎
= ( 11 12 ) × (70𝑦6 − 368𝑦5 + 892𝑦4
𝑎21 𝑎22
∗
(𝑥, 𝑦) − 𝑅𝑚,𝑛 (𝑥, 𝑦)] ,
𝐸2 (𝑥, 𝑦) = 𝐷 (𝑥, 𝑦) 𝐷∗ (𝑥, 𝑦) [𝑅𝑚,𝑛
𝐸 (𝑥, 𝑦) = 𝐸1 (𝑥, 𝑦) + 𝐸2 (𝑥, 𝑦) .
−1
−1240𝑦3 +1040𝑦2 −512𝑦+128)
(51)
From Lemma 14, we know
𝐸1 (𝑥𝑖 , 𝑦) = 0,
=
𝑖 = 0, 1, . . . , 𝑚,
(52)
which results in
󵄨󵄨
𝜔 (𝑥) 𝜕𝑚+1
󵄨󵄨
󵄨󵄨 ,
𝐸
(𝑥,
𝑦)
𝐸1 (𝑥, 𝑦) = 𝑚+1
1
𝑚+1
󵄨󵄨
(𝑚 + 1)! 𝜕𝑥
󵄨𝑥=𝜉
𝑗 = 0, 1, . . . , 𝑛,
(53)
(54)
(55)
𝐸 (𝑥, 𝑦) = 𝐸1 (𝑥, 𝑦) + 𝐸2 (𝑥, 𝑦)
󵄨󵄨
𝜔𝑚+1 (𝑥) 𝜕𝑚+1
󵄨󵄨
󵄨󵄨
𝐸
(𝑥,
𝑦)
󵄨󵄨
(𝑚 + 1)! 𝜕𝑥𝑚+1 1
󵄨𝑥=𝜉
(56)
𝑎21 = 108𝑦5 − 200𝑦4 + 200𝑦3 − 112𝑦2 + 32𝑦
− 28𝑦6 + 60𝑦6 𝑥 − 344𝑦5 𝑥 + 736𝑦4 𝑥 − 784𝑦3 𝑥
+ 408𝑥𝑦2 − 32𝑥𝑦 − 64𝑥 − 24𝑦5 𝑥2 + 16𝑦4 𝑥2
+ 32𝑦 − 14𝑦6 − 50𝑦6 𝑥2 + 270𝑦5 𝑥2 − 640𝑦4 𝑥2
+ 820𝑦3 𝑥2 − 560𝑥2 𝑦2 + 160𝑥2 𝑦 + 50𝑦6 𝑥 − 270𝑦5 𝑥
+ 640𝑦4 𝑥 − 820𝑦3 𝑥 + 560𝑥𝑦2 − 160𝑥𝑦.
The proof is thus completed.
(60)
4. Numerical Examples
Example 17. Let (𝑥𝑖 , 𝑦𝑗 ) and 𝑌𝑖,𝑗 , 𝑖, 𝑗 = 0, 1, 2 be given in
Table 1:
(57)
Using Algorithm 7, we can get
From Definition 13, we know that 𝑅2,2 (𝑥, 𝑦) is a BGIRINT
since 𝑅2,2 (𝑥𝑖 , 𝑦𝑗 ) = 𝑌𝑖,𝑗 and deg 𝑁(𝑥, 𝑦) = 8, deg 𝐷(𝑥, 𝑦) =
6, and 𝐷(𝑥, 𝑦) | ‖𝑁(𝑥, 𝑦)‖2 .
In paper [9], a bivariate Thieletype matrix-valued rational
interpolant 𝑅𝑇 (𝑥, 𝑦) = 𝐺0,0 (𝑦) + (𝑥 − 𝑥0 )/𝐺1,0 (𝑦) + ⋅ ⋅ ⋅ + (𝑥 −
𝑥𝑚−1 )/𝐺𝑛,0 (𝑦) (page 73), where for 𝑙 = 0, 1, . . . , 𝑛,
𝐺𝑙,0 (𝑦) = 𝐵𝑙,0 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 )
𝑦−1|
𝑦
1 0
,
𝑈0 (𝑦) = (
)+ 00 +
0
0
0 0
( 0 1 ) | ( −2/5
−1/5 )
0 1
𝑦−1|
𝑦
.
𝑈2 (𝑦) = ( 1 ) + 0 −3/5 +
0 −15/14
0
(
)
|
(
)
1/5
0
−5/14
−5/7
2
− 1776𝑥𝑦2 + 1024𝑥𝑦 − 256𝑥 − 38𝑦5 𝑥2
𝑎22 = 68𝑦5 − 140𝑦4 + 160𝑦3 − 96𝑦2
󵄨󵄨
𝜛 (𝑦) 𝜕
󵄨󵄨
󵄨󵄨 .
+ 𝑛+1
𝐸
(𝑥,
𝑦)
󵄨󵄨
(𝑛 + 1)! 𝜕𝑦𝑛+1 2
󵄨𝑦=𝜂
𝑦−1|
𝑦
0 −1
𝑈1 (𝑦) = (
) + 0 1/2 + 0 1 ,
0 0
( −1/2 0 ) | ( 1 0 )
𝑎12 = 178𝑦5 𝑥 − 858𝑦4 𝑥 + 1688𝑦3 𝑥
− 96𝑥2 𝑦 + 10𝑦6 𝑥2 + 40𝑥2 𝑦2 + 64𝑥2 ,
𝑛+1
𝑅2,2 (𝑥, 𝑦) = 𝑈0 (𝑦) + 𝑈1 (𝑦) 𝑥 + 𝑈2 (𝑦) 𝑥 (𝑥 − 1) .
+ 1040𝑦2 − 512𝑦 + 128,
− 5𝑦6 𝑥2 + 5𝑦6 𝑥 + 800𝑥2 𝑦2 + 128𝑥2 ,
where 𝜛𝑛+1 (𝑦) = (𝑦−𝑦0 )(𝑦−𝑦1 ) ⋅ ⋅ ⋅ (𝑦−𝑦𝑛 ), and 𝜂 is a number
contained in the interval (𝑐, 𝑑) which may depend on 𝑥.
Therefore,
=
𝑎11 = 70𝑦6 − 368𝑦5 + 892𝑦4 − 1240𝑦3
+ 262𝑦4 𝑥2 − 640𝑦3 𝑥2 − 512𝑥2 𝑦
we can get that
󵄨󵄨
𝜛 (𝑦) 𝜕𝑛+1
󵄨󵄨
󵄨󵄨 ,
𝐸2 (𝑥, 𝑦) = 𝑛+1
𝐸
(𝑥,
𝑦)
2
󵄨󵄨
(𝑛 + 1)! 𝜕𝑦𝑛+1
󵄨𝑦=𝜂
(59)
where
where 𝜔𝑚+1 (𝑥) = (𝑥 − 𝑥0 )(𝑥 − 𝑥1 ) ⋅ ⋅ ⋅ (𝑥 − 𝑥𝑚 ), and 𝜉 is a
number contained in the interval (𝑎, 𝑏) which may depend
on 𝑦. Similarly, from
𝐸2 (𝑥, 𝑦𝑗 ) = 0,
𝑁 (𝑥, 𝑦)
,
𝐷 (𝑥, 𝑦)
+
(58)
𝑦 − 𝑦0
𝐵𝑙,1 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , 𝑦1 )
+ ⋅⋅⋅ +
(61)
𝑦 − 𝑦𝑚−1 |
.
| 𝐵𝑙,𝑚 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑚 )
Here 𝐵𝑙,𝑘 (𝑥0 , . . . , 𝑥𝑙 , 𝑦0 , . . . , 𝑦𝑘 ), 𝑙 = 0, 1, . . . , 𝑛, 𝑘 = 0, 1, . . . , 𝑚
is defined different from that of this paper (see [9]). We
Journal of Applied Mathematics
9
Table 1
𝑦𝑗 /𝑥𝑖
𝑥0 = 0
1 0
(
)
0 0
1 0
(
)
0 1
1 0
(
)
−1 0
𝑦0 = 0
𝑦1 = 1
𝑦2 = 2
𝑥1 = 1
1 −1
(
)
0 0
1 0
(
)
−1 1
1 1
(
)
−1 0
𝑥2 = 2
1 0
(
)
1 0
1 −1
(
)
0 1
1 1
(
)
0 −1
Table 2
𝑌(1.3, 1.3)
−0.856888 0.5155013 3.669296
(
)
1.30000
3.66929
2.600
𝑌(1.5, 1.5)
−0.989999 0.1411200 4.481689
(
)
1.50000 4.481689
3.00
𝑅NT (1.3, 1.3)
−0.856888 0.5155010 3.669297
(
)
1.30000
3.66929
2.600
𝑅𝑇 (1.3, 1.3)
−0.98999 0.14111 5.47394
(
)
1.29999 5.47394 2.999
𝑅NT (1.5, 1.5)
−0.989992 0.1411203 4.481688
(
)
1.50000 4.481688
3.00
𝑅𝑇 (1.5, 1.5)
−0.99829372 −0.05837 5.47394
(
)
1.49999
5.47394 3.1999
now give another numerical example to compare the two
algorithms, which shows that the method of this paper is
better than the one of [9].
Example 18. Let
cos (𝑥 + 𝑦) sin (𝑥 + 𝑦) 𝑒𝑦
)
𝑌 (𝑥, 𝑦) = (
𝑥
𝑒𝑦
𝑥+𝑦
‖𝑅NT − 𝑌‖𝐹
1.09𝑒 − 006
‖𝑅𝑇 − 𝑌‖𝐹
2.613697
‖𝑅NT − 𝑌‖𝐹
9.15𝑒 − 007
‖𝑅𝑇 − 𝑌‖𝐹
1.43144
Acknowledgments
The work is supported by Shanghai Natural Science Foundation (10ZR1410900) and by Key Disciplines of Shanghai
Municipality (S30104).
References
(62)
and we suppose {𝑥0 , 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 } = {𝑦0 , 𝑦1 , 𝑦2 , 𝑦3 , 𝑦4 , 𝑦5 }
= {1.0, 1.2, 1.4, 1.6, 1.8, 2.0}. The numerical results 𝑅NT (𝑥, 𝑦)
in Step 9 of Algorithm 7 and 𝑅𝑇 (𝑥, 𝑦) in [9] are given in
Table 2.
[1] G. A. Baker and P. R. Graves-Morris, Padé Approximation, Part
II, Addison-Wesley Publishing Company, Reading, Mass, USA,
1981.
[2] I. Kh. Kuchminska, O. M. Sus, and S. M. Vozna, “Approximation
properties of two-dimensional continued fractions,” Ukrainian
Mathematical Journal, vol. 55, pp. 36–54, 2003.
[3] J. Tan and Y. Fang, “Newton-Thiele’s rational interpolants,”
Numerical Algorithms, vol. 24, no. 1-2, pp. 141–157, 2000.
Remark 19. In Table 2, from the F-norm of 𝑅NT −𝑌 and 𝑅𝑇 −𝑌,
we can see that the error using Newton-Thiele type formula
is much less than the one using Thieletype formula.
[4] P. R. Graves-Morris, “Vector valued rational interpolants I,”
Numerische Mathematik, vol. 42, no. 3, pp. 331–348, 1983.
5. Conclusion
[6] G. Chuan-qing, “Matrix Padé type approximat and directional
matrix Padé approximat in the inner product space,” Journal of
Computational and Applied Mathematics, vol. 164-165, pp. 365–
385, 2004.
In this paper, the bivariate generalized inverse Newton-Thiele
type matrix interpolation to approximate a matrix function
𝑓(𝑥, 𝑦) is given, and we also give a recursive algorithm
accompanied with some other important conclusions such as
divisibility, characterization and some numerical examples.
From the second example, it is easy to find that the approximant using Newton-Thiele type formula is much better than
the one using Thieletype formula.
[5] G. Chuan-qing and C. Zhibing, “Matrix valued rational interpolants and its error formula,” Mathematica Numerica Sinica,
vol. 17, pp. 73–77, 1995.
[7] C. Gu, “Thiele-type and Lagrange-type generalized inverse
rational interpolation for rectangular complex matrices,” Linear
Algebra and Its Applications, vol. 295, no. 1–3, pp. 7–30, 1999.
[8] C. Gu, “A practical two-dimensional Thiele-type matrix Padé
approximation,” IEEE Transactions on Automatic Control, vol.
48, no. 12, pp. 2259–2263, 2003.
10
[9] C. Gu, “Bivariate Thiele-type matrix-valued rational interpolants,” Journal of Computational and Applied Mathematics,
vol. 80, no. 1, pp. 71–82, 1997.
[10] N. K. Bose and S. Basu, “Two-dimensional matrix Padé approximants: existence, nonuniqueness and recursive computation,”
IEEE Transactions on Automatic Control, vol. 25, no. 3, pp. 509–
514, 1980.
[11] A. Cuyt and B. Verdonk, “Multivariate reciprocal differences
for branched Thiele continued fraction expansions,” Journal of
Computational and Applied Mathematics, vol. 21, no. 2, pp. 145–
160, 1988.
[12] W. Siemaszko, “Thiele-tyoe branched continued fractions for
two-variable functions,” Journal of Computational and Applied
Mathematics, vol. 9, no. 2, pp. 137–153, 1983.
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