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.
