CL_MATCONT implements a continuation method that is slightly different from
the pseudo-arclength continuation.
Figure 3:
Moore-Penrose continuation
 |
Definition 1
Let
be an
matrix with maximal rank. Then the Moore-Penrose inverse of
is defined by
.
Let
be an
matrix with maximal rank. Consider the following linear system with
:
where
is a point on the curve and
its tangent vector with respect to
, i.e.
.
Since
and
, a solution of this system is
 |
(10) |
Suppose we have a predicted point
using (1). We want to find the point
on the curve which is nearest to
, i.e. we are trying to solve the optimization problem:
 |
(11) |
So, the system we need to solve is:
where
is the tangent vector at point
. In Newton's method this system is solved using a linearization about
. Taylor expansion about
gives:
So when we discard the higher order terms we can see using (8) and (10) that the solution of this system is:
 |
(16) |
However, the null vector of
is not known, therefore we approximate it by
, the tangent vector at
. Geometrically this means
we are solving
in a hyperplane perpendicular to the previous tangent vector. This is illustrated in Figure 3. In other words, the extra function
in (2) becomes:
 |
(17) |
where
for
.
Thus, the Newton iteration we are doing is:
 |
(20) |
 |
(21) |
One can prove that under the same conditions as for the pseudo-arclength continuation,
the Newton iterations (18) and (19) converge to a point on the
curve
and the corresponding tangent vector
, respectively.
In the pseudo-arclength continuation, we had to compute a tangent vector when a new point
was found. In this case however, we already compute the tangent vectors
at each
iterate (19), so we only need to normalize the computed
tangent vectors.