Numerical integration methods based on variation diminishing polynomial approximation for the solution of algebraic differential equations in the context of circuit simulation

Numerical integration techniques based on the Bernstein approximation are presented. A common approach to solving the initial value problems representing the algebraic differential equations is to formulate the solution as a polynomial function. In general, any algorithm capable of calculating the exact value of the unknown state variable for an initial value problem having an exact solution in the form of a kth degree polynomial is called a numerical integration formula of order k. Of course, in almost all cases the exact solution is not a polynomial, and the solution is an approximate one. However, the Weiestrass approximation theorem asserts that any continuous function can be uniformly approximated within any closed interval by a polynomial of sufficiently high degree. Hence, even if the solution is not a polynomial, a numerical integration formula of sufficiently high order can, in principle, be used to evaluate the state variable to any desired accuracy. Not all polynomial interpolants, however, would satisfy Weiestrass approximation theorem. Bernstein polynomials provide a weighted uniform approximation basis, and they are also used to construct one of the many proofs of the Weiestrass approximation theorem. Predictor-based integration formulas are obtained by formulating the solution function as a Bernstein polynomial