行列[1/(x_i-y_i)]のLU分解

概要

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Let H=LU be the LU decomposition of the general Hilbert matrix H of order n with the entries h_<ij>=1/(x_i-y_i), where x_1, x_2, …x_n, …, y_1, y_2, …, y_n … are distinct numbers, L is a lower triangular matrix with entries l_<ij> such that l_<ij>=0 for i>j, l_<ij>=1 for uniqueness' sake, and U is an upper triangular matrix with u_<ij>, such that u _<ij>=0 for i>j. And let M=L^<-1>, and V=U^<-1>. Simple and n-independent expressions for the entries l_<ij>, m_<ij>, u_<ij> and v_<ij> are obtained in the following way. The relation U=L^<-1>H=MH can be considered as a decomposition of a rationl function u_k(y)=u_<kj> into partial fractions m_<k1>/(x_1-y_i)+m_<k2>/(x_2-y_i)+…+m_<kk>/(x_k-y_i), thus u_k(y) having zeros at y_1, y_2, …, y_<k-1>(for u_k(y_1)=u_<k1>=0, …, u_k(y_<k-1>)=u_<k,k-1>=0) and simple poles at x1, x_2, …, x_k which determine u_k(y) except a constant factor, m_<kj>=lim y→x_k(x_j-y)u_k(y) the coefficients of 1/(x_j-y) in the partial fraction decomposition of u_k(y), and the constant factor is determined by the condition m_<kk>=1. Similarly, l_<ik> and v_<ik> are obtained from the relation L=HV.
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