A Fast Algorithm of Moore-Penrose Inverse for the Loewner-Type Matrix
In this paper, we present a fast algorithm of Moore-Penrose inverse for Loewner-type matrix with full column rank by forming a special block matrix and studying its inverse. Its computation complexity is , but it is by using .
Loewner matrix was first studied by Loewner in 1934 in . He studied the relations of various Loewner matrices via the characteristic of the monotone matrix function and the problem of rational interpolation at that time. Since then, more further studies had been carried out by many scientists in [2–5], such as Belevitch, Donoghue Jr., Fiedler, Chen and Zhang. From their mentioned works, we can find various properties of Loewner matrix and its application in the rational interpolation. In 1984, Vavrín presented a fast algorithm for the inverse of Loewner matrix in , then the fast algorithm for the Loewner system was got accordingly. Rost and Vavrín presented a fast algorithm for the system whose coefficient is a Loewner-Vandermonde matrix from 1995 to 1996 in [7, 8]. Lu gave a fast triangular factorization algorithm for the symmetrical Loewner-type matrix in 2003 in . Xu et al. and so forth gave a fast triangular factorization algorithm for the inverse of symmetrical Loewner-type matrix in 2003 in . In 2009, Tong et al. gave a fast algorithm of the Moore-Penrose inverse for symmetrical Loewner-type matrix . In this paper, we generalize the fast algorithm of symmetrical Loewner-type matrix to the Loewner-type matrix. The theory and computation of generalized inverse matrix arise in various applications such as optimization theory, control theory, computation mathematics, and mathematical statistics. Therefore, there are very important theoretical and practical significance when we study the fast algorithm of Moore-Penrose inverse for the Loewner-type matrix.
This paper is organized as follows: in Section 1, we present some preliminaries results. The fast algorithm of Moore-Penrose inverse for Loewner-type matrix is driven in Section 2. We give some numerical examples to illustrate the fast algorithm obtained in Section 3.
A Loewner matrix is a matrix of the form , where () are given numbers and . A Loewner-type matrix is a matrix which satisfies where , , , and . Loewner matrix is a special case of Loewner-type matrix, and it satisfies .
Let be the rank of Loewner-type matrix . Forming an matrix we obtain
It is obvious that we may obtain the Moore-Penrose inverse of Loewner-type matrix by (2.3) if we can get .
Now let us begin to find .
The following result will be useful for getting .
Lemma 2.1 (see ). Let all the leading principal submatrices of matrix whose order is be invertible. Linear system be given, where . Note that , and let , be solution vectors of linear systems , differently, then where .
3. Fast Algorithm of the Moore-Penrose Inverse for Loewner-Type Matrix
By using (2.1), we know that satisfies
Let all the leading principal submatrices ?? of be invertible. If , by virtue of (3.1), we have
Let , be solution vectors of linear systems differently, then using , we obtain , . By virtue of Lemma 2.1 we have where , , and is a solution vector of . Multiplying (3.2) by on the left and on the right differently, we obtain Multiplying (3.5) by and noting that , we have Substituting (3.4) in (3.6) and noting that , , we have and hence, Now, let us look for . Choosing the th of , namely and using (3.8), we have
Note that , where is the th column of , then Letting in (3.5) and multiplying it by on the right, we have
For , Step 2.
For , then
The algorithm requires multiplication and division operations and addition and subtraction operations, and the computation complexity is , but it is by using .
4. Numerical Examples
We get the Moore-Penrose inverse matrix of Loewner-type matrix with Fortran program in computer, and the results of the partial numerical examples are given as follows in Table 1 (the error is measured by the 2-norm of vector, and the time is measured by second).
Example 1. Consider
From above examples and many more examples not given, we find that the stability of the fast algorithm is very good, and it needs shorter time than that of .
This work is being supported by the National Natural Science Foundation Grants of China (no. 60974082).
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