An Approximate Solver for Symbolic Equations

This paper describes a program, called NEWTON, that finds approximate symbolic solutions to parameterized equations in one variable. N E W T O N derives an ini t ial approximat ion by solving for the dominant term in the equation, or if this fails, by bisection. It refines this approximation by a symbolic version of Newton's method. It tests whether the first Newton iterate lies closer to the solution than does the in i t ia l solution. If so, it returns this iterate; otherwise, it chooses a new ini t ia l solution and tries again. 1 I n t r o d u c t i o n Research in symbolic equation solving has focused on exact solution methods. The resulting programs, such as MACSYMA [Mathlab Group, 1983] and PRESS [Bundy and Welham, 198l], either return an exact solution or fail. Yet, scientists and engineers routinely must solve problems that have no exact closed-form solution. They need an equation solver that finds adequate approximate solutions to such problems, rather than fail ing. In fact, they often prefer a simple approximate solution even when a complicated exact solution is available. For example, the equation x + x — k cannot be solved for x in closed form, but x = k is an accurate approximation to the solution for k near 0. One might well prefer this approximat ion to the ful l solution of x + x = k, which is very long and complicated. This paper describes an equation solver, called NEWTON, that finds approximate solutions to parameterized equations in one unknown. Numeric analysis provides many algorithms for solving indiv idual equations approximately, but says l i t t le about parameterized equations. One could solve numerically for specific parameter values and interpolate the results. This would provide l i t t le general understanding and require prohibit ive amounts of computation, especially for equations containing mult iple parameters. Instead, NEWTON constructs a single parameterized solut ion, such as x = k above, for all legal parameter values. Users can instantiate a solution w i th mult iple parameter values, rather than rederiving it numerically for each value. They can examine the influence of parameters on a solution analytically instead of by experimentation. For example, the approximate solution x — k increases linearly in k. 2 The A l g o r i t h m N E W T O N takes as input a parameterized function / ( x ) , an interval domain for x, and a set of constraints. Each constraint is a strict or nonstrict inequality between realvalued functions of the parameters, for example k > 1 and k < 2 m . The constraints and f(x) must be extended elementary functions: polynomials and compositions of exponentials, logarithms, trigonometric functions, inverse trigonometric functions, and absolute values. NEWTON assumes that f(x) is d i f ferent iate and has a single, simple root in the domain of x. I discuss mult iple and nonsimple roots in the concluding section. N E W T O N narrows in on the root in stages. First, it brackets the root wi thin an interval on which f(x) is monotonic by the following algorithm: 1. let [a, 6] be the domain of x 2. if f'(x) has a fixed sign on [a, 6] then return [a, 6] 3. let m=(a + b ) /2; if f(a)f(m) < 0 then 6 <m else a <— m