Abstract
For an analytical expression of Vandermonde inverse matrix, a new derivation process based on Wronskian matrix and Lagrange interpolation polynomial basis is presented. Recursive formula and implementation cases for the direct formula of Vandermonde inverse matrix are given based on deriving the unified formula of Wronskian inverse matrix. For the calculation of symbol-type Vandermonde inverse matrix, the direct formula and recursive method are verified to be more efficient than Mathematica which is good at symbolic computation by comparing the computing time in Mathematica. The process and steps of recursive algorithm are relatively simple. The derivation process and idea both have very important values in theory and practice of Vandermonde and generalized Vandermonde inverse matrix.
Highlights
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 [1–8]
Neagoe [3] deduced an analytic formula of complex-type Vandermonde inverse matrix based on symmetric polynomials
In this paper, based on results in [1, 2], 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
Summary
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 [1–8]. Neagoe [3] deduced an analytic formula of complex-type Vandermonde inverse matrix based on symmetric polynomials. Eisinberg and Fedele [4] presented a general explicit formula for the elements of inverse matrix and two different algorithms were deduced. In this paper, based on results in [1, 2], 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.
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