Tridiagonal matrix algorithm

Abstract: Motivated by investigations of the tridiagonal pairs of linear transformations, we introduce the augmented tridiagonal algebra ${\mathcal T}_q$. This is an infinite-dimensional associative ${\mathbb C}$-algebra with 1. We classify the finite-dimensional irreducible representations of ${\mathcal T}_q$. All such representations are explicitly constructed via embeddings of ${\mathcal T}_q$ into the $U_q(sl_2)$-loop algebra. As an application, tridiagonal pairs over ${\mathbb C}$ are classified in the case where $q$ is not a root of unity.

Abstract: This paper presents a new, general method for mathematical simulation of equilibrium stage operations. The procedure solves component material balance equations with a tridiagonal matrix algorithm. Heat balances and summation equations are handled with Broyden's method. The unique feature of this procedure is that, in a mathematical sense, all equations are solved simultaneously. Therefore, the method can be used for all types of equilibrium stage processes. Additionally, the use of Broyden iteration insures solutions which are both stable and more rapid than current techniques. An exact solution for a twenty tray column with twenty components takes approximately 30 sec. on an IBM 360/65 computer. Successful simulations have been made for both absorption and distillation type of operations which have included complex columns with multiple feeds and side product streams. Design applications of the method cover a variety of equilibrium stage processes in the chemical and petroleum industries.