Abstract
Using only algebra, this exposition presents the concept of backward analysis. Numerical examples from finance demonstrate how backward analysis guarantees that the computed value of a polynomial of degree exceeding thirty-one million is accurate to all but the penultimate digit, despite a hundred million intermediate rounding errors. Methods from celestial mechanics then provide initial estimates for the solution of financial polynomial equations. The same analysis applies to the computation of elementary functions from calculus and special functions from differential equations.
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