A New Algorithm to Approximate Bivariate Matrix Function via Newton-Thiele Type Formula
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).
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 . Kuchminska et al. proved an analog of the van Vlerk theorem and constructed an interpolation formular of the Newton-Thiele type in . Tan and Fang in  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 . 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 .
In this paper, we introduce a bivariate matrix-valued rational interpolant, with scalar denominator, in Newton-Thiele form motivated by . 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 .
First, we will give the definition of the so-called generalized inverse of a matrix. Let consists of all matrices with their elements in the complex plan , and let .
Definition 1 (see ). The scalar product of matrices and is defined by where denotes the transpose of and the Euclidean norm of is given by It follows from Definition 1 and (2) that where denotes the complex conjugate matrix of and denotes the complex conjugate transpose matrix of . On the basis of (2) and (3), the generalized inverse of the matrix is defined as in particular, for .
By means of generalized inverse 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
A matrix-valued set
We need to find a bivariate matrix rational function whose numerator is a complex or real polynomial matrix and denominator is a real polynomial, and
First, some notations need to be given as follows: where the first subscript of means the number of nodes minus 1, and the second subscript of means the number of nodes minus 1.
For simplicity, we let the nodes be replaced by , and the nodes be replaced by , 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:
We assume that for all and
From (9) matrix-valued Newton-Thiele type continued fractions for two-variable function can be constructed as follows: where for
Remark 3. Because of the construction, in mentioned as (6) is actually and each depends on .
We can also construct the antithetical form of bivariate matrix continued fraction in light of Definition 4.
Definition 4. By means of generalized inverse (4), we define bivariate antithetical Newton-Thiele type matrix blending differences
From (13) we can get the antithetical formula for two-variable function where for
We can similarly prove the following result.
Now we give the Newton-Thiele Matrix Interpolation Algorithm (NTMIA).
Algorithm 7 (NTMIA). Input:
Step 1. For all , let .
Step 2. For , compute
Step 3. For , compute
Step 4. For , compute .
Step 5. For judge if ; if yes go to Step 6, otherwise exist and show “the algorithm is not valid.”
Step 6. For
Step 7. For , let ; if then go to Step 8, otherwise exist and show “the algorithm is not valid.”
Step 8. For , let .
Step 9. For let , then compute .
3. Some Properties
Lemma 8 (see ). Define ; if we use generalized inverse by a tail-to-head rationalization, then a polynomial matrix and a real polynomial exist such that(i), (ii),“∣” means the sign of divisibility.
Definition 9. is defined to be of type if for for some and , where .
Theorem 11. Let , , and . Define 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 and a real polynomial exist such that(i), (ii), (iii).
Proof. Consider the following algorithm for the construction of , , and .
Initialization: let , and By Lemma 8, a polynomial matrix and a real polynomial exist such that(a1), (a2), (a3).
Recursion: for let with having the representation(b1), (b2), (b3), where is a real polynomial.
Then we can get where Obviously, since Termination: .
Theorem 12 (Characterization). Let be expressed as where , as in (12) we assume have no common factor, , then(i)if n is even, is of type ;(ii)if n is odd, is of type .
Proof. The proof is recursive. Let be even.
For , by Lemma 10, we find that it is of type .
For , because is of type .
For , from the formula above we know that . one has It is easy to find that is of type .
For , let be of type .
For , we consider that where Therefore, when is even, is of type .
Similarly, if is odd, for , by Lemma 10, we find that it is of type .
For , because is of type .
For , from the formula above we know that ,. one has We get that is of type .
For , let be of type .
For , we consider that where
Therefore, when is odd, is of type .
Definition 13. A matrix-valued Newton-Thiele type rational fraction is defined to be a bivariate generalized inverse and rational interpolant on the rectangular grid if(i),
(iii)(a) if is even,
(b) if n is odd,
(v) is real, and .
Now let us turn to the error estimate of the . Firstly, we give Lemma 14.
Lemma 14. Let where for then satisfies
Remark 15. We delete the proof of the above lemma since it can be easily generalized from .
Theorem 16. Let a matrix be times continuously differentiable on a set containing the points , then, for any point , there exists a point such that where
From Lemma 14, we know
which results in
where , and is a number contained in the interval which may depend on . Similarly, from
we can get that
where , and is a number contained in the interval which may depend on .
Therefore, The proof is thus completed.
4. Numerical Examples
Example 17. Let and be given in Table 1:
Using Algorithm 7, we can get
Based on generalized inverse (4), we obtain
From Definition 13, we know that is a since and , and .
In paper , a bivariate Thieletype matrix-valued rational interpolant (page 73), where for , Here is defined different from that of this paper (see ). We now give another numerical example to compare the two algorithms, which shows that the method of this paper is better than the one of .
Remark 19. In Table 2, from the F-norm of and , we can see that the error using Newton-Thiele type formula is much less than the one using Thieletype formula.
In this paper, the bivariate generalized inverse Newton-Thiele type matrix interpolation to approximate a matrix function 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.
The work is supported by Shanghai Natural Science Foundation (10ZR1410900) and by Key Disciplines of Shanghai Municipality (S30104).
G. A. Baker and P. R. Graves-Morris, Padé Approximation, Part II, Addison-Wesley Publishing Company, Reading, Mass, USA, 1981.
J. Tan and Y. Fang, “Newton-Thiele's rational interpolants,” Numerical Algorithms, vol. 24, no. 1-2, pp. 141–157, 2000.View at: Google Scholar
G. Chuan-qing and C. Zhibing, “Matrix valued rational interpolants and its error formula,” Mathematica Numerica Sinica, vol. 17, pp. 73–77, 1995.View at: Google Scholar
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.View at: Google Scholar
C. Gu, “Bivariate Thiele-type matrix-valued rational interpolants,” Journal of Computational and Applied Mathematics, vol. 80, no. 1, pp. 71–82, 1997.View at: Google Scholar
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.View at: Google Scholar
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.View at: Google Scholar
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.View at: Google Scholar