Method and apparatus for solving systems of nonlinear equations using interval arithmetic

Abstract

One embodiment of the present invention provides a computer-based system for solving a system of nonlinear equations specified by a vector function, f, wherein f(x)=0 represents ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0 . . . , ƒn(x)=0, wherein x is a vector (x1, X2, X3, . . . xn). The system operates by receiving a representation of a subbox X=(X1, X2, . . . , Xn), wherein for each dimension, i, the representation of Xi, includes a first floating-point number, ai, representing the left endpoint of Xi, and a second floating-point number, bi, representing the right endpoint of Xi. The system stores the representation in a computer memory. Next, the system applies term consistency to the set of nonlinear equations, ƒ1(x)=0, ƒ2(x)=0, ƒ3(x)=0, . . . , ƒn,(x)=0, over X, and excludes portions of X that violate the set of nonlinear equations. The system also applies box consistency to the set of nonlinear equations over X, and excludes portions of X that violate the set of nonlinear equations. Finally, the system performs an interval Newton step on X to produce a resulting subbox Y, wherein the point of expansion of the interval Newton step is a point x within X, and wherein performing the interval Newton step involves evaluating f(x) using interval arithmetic to produce an interval result fI(x). The system integrates the sub-parts of the process with branch tests designed to increase the overall speed of the process.

Dependencies are inferred from the claim text.