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 .


Introduction
Loewner matrix was first studied by Loewner in 1934 in 1 .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, Vavřín presented a fast algorithm for the inverse of Loewner matrix in 6 , then the fast algorithm for the Loewner system was got accordingly.Rost  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.

Preliminaries
, where α i , β j , ξ i , and η j i 1, 2, . . ., m; j 1, 2, . . ., n are given numbers and α i / β j .A Loewnertype matrix L is a matrix which satisfies , and q j q j 1 , q j 2 , . . ., q j n T .Loewner matrix L 1 is a special case of Loewner-type matrix, and it satisfies Let n be the rank of m × n Loewner-type matrix L. Forming an m n matrix It is obvious that we may obtain the Moore-Penrose inverse L of Loewner-type matrix L by 2.3 if we can get M −1 .Now let us begin to find M −1 .
The following result will be useful for getting M −1 . Lemma where

Fast Algorithm of the Moore-Penrose Inverse for Loewner-Type Matrix
By using 2.1 , we know that M satisfies

3.1
Let all the leading principal submatrices M i i 1, 2, . . ., m n of M be invertible.If i > m, by virtue of 3.1 , we have

T be solution vectors of linear systems
where Multiplying 3.2 by M −1 i on the left and on the right differently, we obtain 3.5

3.6
Mathematical Problems in Engineering Substituting 3.4 in 3.6 and noting that g j ii σ j i u ii , h j ii τ j i u ii , we have and hence,

3.8
Now, let us look for u ii .Choosing the ith of 3.9 and using 3.8 , we have

3.11
Letting i m n in 3.5 and multiplying it by e m n k k > m on the right, we have So, we obtain the fast algorithm of Moore-Penrose inverse L of Loewner-type matrix L by 2.3 , 3.4 , and 3.8 ∼ 3.12 .

3.15
For k m 1, . . .m n − 1, 3.17 The algorithm requires 7l − 3 mn ln 2 O m multiplication and division operations and 5l − 3 mn l/2 n 2 O m addition and subtraction operations, and the computation complexity is O mn O n 2 , but it is O mn 2  O n 3 by using L L T L −1 L T .

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 .q 1 j 1, q 2 j −η j , q 3 j 1, q 4 j −η j .i 1, 2, . . ., m; j 1, 2, . . ., n.

4.1
From above examples 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 L L T L −1 L T .
and Vavřín 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 9 .Xu et al. and so forth gave a fast triangular factorization algorithm for the inverse of symmetrical Loewner-type matrix in 2003 in 10 .In 2009, Tong et al. gave a fast algorithm of the Moore-Penrose inverse for m × n symmetrical Loewner-type matrix 11 .

m 1 ,
. . ., m n 3.3 differently, then using M m −I m , we obtain g j m −p j , h j m 0 m .By virtue of Lemma 2.1 we have