The relationship between Bézoutian matrix and Newton’s matrix of divided differences
Progress in Analysis and its Applications
Let x 1 ,...,x n be real numbers, P(x)=p n (x-x 1 )⋯(x-x n ), and Q(x) be a polynomial of order less than or equal to n. Denote by Δ(Q) the matrix of generalized divided differences of Q(x) with nodes x 1 ,...,x n and by B(P,Q) the Bézoutian matrix of P and Q. A relationship between the corresponding principal minors of the matrices B(P,Q) and Δ(Q) counted from the right lower corner is established. It implies that if the principal minors of the matrix of divided differences of a function g(x) are positive or have alternating signs then the roots of the Newton’s interpolation polynomial of g are real and separated by the nodes of interpolation.
Hayrapetyan, Ruben G., "The relationship between Bézoutian matrix and Newton’s matrix of divided differences" (2010). Mathematics Publications. 78.