The Jacobi Matrix for Functions in Noncommutative Algebras

Abstract

We develop a general tool for constructing the exact Jacobi matrix for functions defined in noncommutative algebraic systems without using any partial derivative. The construction is applied to solving nonlinear problems of the form f(x) = 0 with the aid of Newton’s method in algebras defined in ${\mathbb{R}^N}$. We apply this to eight (commutative and noncommutative) algebras in ${\mathbb{R}^4}$. The Jacobi matrix is explicitly constructed for polynomials in x−a and for polynomials in the reciprocals (x−a)¹ such that Jacobi matrices for functions defined by Taylor and Laurent expansions can be constructed in general algebras over ${\mathbb{R}^N}$. The Jacobi matrix for the algebraic Riccati equation with matrix elements from an algebra in ${\mathbb{R}^N}$ is presented, and one particular algebraic Riccati equation is numerically solved in all eight algebras over ${\mathbb{R}^4}$. Another case treated was the exponential function with algebraic variables including a numerical example. For cases where the computation of the exact Jacobi matrix for finding solutions of f(x) = 0 is time consuming, a hybrid method is recommended, namely to start with an approximation of the Jacobi matrix in low precision and only when ${\|f(x)\|}$ is sufficiently small, to switch to the exact Jacobi matrix.