A Comparison of the Existence Theorems of Kantorovich and Moore
Louis B. Rall · SIAM Journal on Numerical Analysis · 1980
In order to be useful, an approximate solution y of a nonlinear system of equations $f(x) = 0$ in $R^n $ must be close to a solution $x^ * $ of the system. Two theorems which can be used computationally to establish the existence of $x^ * $ and obtain bounds for the error vector $y - x^ * $ are the 1948 result of L. V. Kantorovich and the 1977 interval analytic theorem due to R. E. Moore. The two theorems are compared on the basis of sensitivity (ability to detect a solution $x^ * $ close to y), precision (ability to give sharp error bounds), and computational complexity (cost). A theoretical comparison shows that the Kantorovich theorem has at best only a slight edge in sensitivity and precision, while Moore’s theorem requires far less computation to apply, and thus provides the method of choice. This conclusion is supported by a numerical example, for which available UNIVAC 1108/1110 software is used to check the hypotheses of both theorems automatically, given y and f.