﻿ Two Dimensional Systems
ISSAC 94 - Symbolic-Numeric Nonlinear Equation Solving
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 Two Dimensional Systems

Let be holomorphic on ; be a closed real parallelepiped contained in ; be real valued on ; curves and intersect in a finite number of points; the partial derivatives at each such point; the Jacobian is nonsingular at each such point.

The two dimensional algorithm is like the one dimensional algorithm in that it uses bisection.  However in this case, bisection means cutting rectangles in half, either widthwise or lengthwise depending on which way is most profitable in improving precision.  After each bisection, and are computed.  If or , then can be eliminated.  Otherwise, continued bisection produces such that , in which case the extrema of and must occur at the corners of .  It is then possible to determine if on some edge of and if so to use to perform narrowing.

Bisection and narrowing ultimately produce a parallelepiped such that and both and change sign on .  A nonsingular Jacobian implies that and can intersect in at most one point in .  Each curve intersects the boundary of twice at points and .  Supposing is in the interior of , then all 4 intersection points are distinct and can be isolated from each other.  If the points alternate with the points around the perimeter of , then the two curves actually do intersect and point exists.  Otherwise, the two curves do not intersect and does not exist.

The generalization of Newton's formula to two dimensions is

and bounds and must be determined.