Solving systems of nonlinear equations using interval arithmetic and term consistency

摘要

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, . . . , ƒ_n(x)=0, wherein x is a vector (x₁, x₂, x₃, . . . , x_n). The system operates by receiving a representation of an interval vector X=(X₁, X₂, . . . , X_n), wherein for each dimension, i, the representation of X_i includes a first floating-point number, a_i, representing the left endpoint of X_i, and a second floating-point number, b_i, representing the right endpoint of X_i. For each nonlinear equation ƒ_i(x)=0 in the system of equations f(x)=0, each individual component function ƒ_i(x) can be written in the form ƒ_i(x)=g(x′_j)−h(x) or g(x′_j)=h(x), where g can be analytically inverted so that an explicit expression for x′_j can be obtained: x′_j=g^−1(h(x)). Next, the system substitutes the interval vector element X_j into the modified equation to produce the equation g(X′_j)=h(X), and solves for X′_j=g^−1(h(X)). The system then intersects X′_j with X_j and replaces X_j in the interval vector X to produce a new interval vector X^+, wherein the new interval vector X^+ contains all solutions of the system of equations f(x)=0 within the interval vector X, and wherein the width of the new interval vector X^+ is less than or equal to the width of the interval vector X.