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Let then ( 12) changes to Its inverse matrix is For the values of in ( 2), we specify them by the following case. If we set, then we can verify that the Lagrange interpolation polynomial basis where satisfy ( 5) that is to say, For Wronskian matrix, From ( 6), the inverse matrix of is where are row subscripts and are column subscripts in the above two matrices.ĭenote by the diagonal matrix composed of that is, It is easily proved that Denoted by using inverse matrix formula of matrix multiplication, we have From ( 10) and ( 14), we can get Here we will use a new method entirely different from to deduce formula ( 2).įor a group of linearly independent functions, where has the ( )th derivative, Wronskian matrix is defined as If there is another group of functions which satisfy then the inverse matrix of ( 4) can be obtained as where and. For the inverse of this matrix, Neagoe obtained the calculation formula ( 2) according to relations between the determinant of the matrix ( 1) and the determinant of the matrix which was obtained by adding a row whose elements were and a column whose elements were to ( 1) and based on the calculation formula of inverse matrix by using the matrix determinant: where are column subscripts and are row subscripts and If are different numbers, that is to say, they are not equal to each other, then we define the -order matrix where as a Vandermonde matrix. A New Derivation of Vandermonde Inverse Matrix In Section 4, numerical simulations are presented with some cases in Mathematica and conclusion and prospect are given by comparing and analyzing the numerical results in the last section. In Section 3, a recursive formula of Vandermonde inverse matrix is deduced. Firstly, we summarize the main idea and analytical formulas presented in and give a new derivation process for the formula in Section 2. The rest of this paper is arranged as follows. In this paper, based on results in, a new derivation process of an analytical expression of Vandermonde inverse matrix is presented based on Wronskian matrix and Lagrange interpolation polynomial basis, and recursive formula and implementation cases for the direct formula of Vandermonde inverse matrix are presented based on deducing the unified formula of Wronskian inverse matrix. Eisinberg and Fedele presented a general explicit formula for the elements of inverse matrix and two different algorithms were deduced. Neagoe deduced an analytic formula of complex-type Vandermonde inverse matrix based on symmetric polynomials. Fox example, Tou and Reis obtained the matrix formula from the coefficients of polynomial terms. There are many effective ways to calculate the Vandermonde inverse matrix. Vandermonde matrix and generalized Vandermonde matrix and their inverses are always widely concerned in many research fields, such as numerical analysis, data and signal processing, and control theory.
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Of Vandermonde and generalized Vandermonde inverse matrix. The derivation process and idea both have very important values in theory and practice The process and steps of recursive algorithm are relatively To be more efficient than Mathematica which is good at symbolic computation by comparing Symbol-type Vandermonde inverse matrix, the direct formula and recursive method are verified
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Given based on deriving the unified formula of Wronskian inverse matrix. Recursiveįormula and implementation cases for the direct formula of Vandermonde inverse matrix are On Wronskian matrix and Lagrange interpolation polynomial basis is presented. For an analytical expression of Vandermonde inverse matrix, a new derivation process based