### Moore-Penrose continuation

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 :

|  |  | (8)
|  |  | (9)

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:

|  |  | (12)
|  |  | (13)

where is the tangent vector at point . In Newton's method this system is solved using a linearization about . Taylor expansion about gives:

|  |  | (14)
|  |  | (15)

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:

|  |  | (18)
|  |  | (19)

| (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.