Title: Solution of linear equations with rational Toeplitz matrices
Abstract: We associate a sequence of Toeplitz matrices with the rational formal power series <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. An algorithm for solving linear equations with a Toeplitz matrix from this sequence is given. The algorithm requires <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> operations to solve a set of <italic>n</italic> equations, for <italic>n</italic> sufficiently large.