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CL_MatCont implements a continuation method that is slightly different from
the pseudo-arclength continuation.
Definition 1
Let A be an N×(N+1) matrix with maximal rank. Then the Moore-Penrose inverse of A is defined by A+ = AT(AAT)-1.
Let A be an N×(N+1) matrix with maximal rank. Consider the following linear system with x,v Î RN+1,b Î RN:
where x is a point on the curve and v its tangent vector with respect to A, i.e. Av=0.
Since AA+b = b and vTA+b = áAv,(AAT)-1bñ = 0, a solution of this system is
Suppose we have a predicted point X0 using (1). We want to find the point x on the curve which is nearest to X0, i.e. we are trying to solve the optimization problem:
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min
x
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{ ||x-X0|| | F(x)=0 } |
| (11) |
So, the system we need to solve is:
where w is the tangent vector at point x. In Newton's method this system is solved using a linearization about X0. Taylor expansion about X0 gives:
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F(X0) + Fx(X0)(x-X0) + O(||x-X0||2) |
| | (14) |
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vT(x-X0) + O(||x-X0||2) . |
| | (15) |
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So when we discard the higher order terms we can see using (8) and (10) that the solution of this system is:
However, the null vector of Fx(X0) is not known, therefore we approximate it by V0 = vi, the tangent vector at xi. Geometrically this means
we are solving F(x)=0 in a hyperplane perpendicular to the previous tangent vector. In other words, the extra function g(x) in (2) becomes:
where Fx(Xk-1)Vk=0 for k=1,2,¼.
Thus, the Newton iteration we are doing is:
H(X,V) = |
æ ç
ç è
|
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ö ÷
÷ ø
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, Hx(X,V) = |
æ ç
ç è
|
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ö ÷
÷ ø
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|
| (20) |
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 xi+1 and the corresponding tangent vector vi+1, 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 Vk at each
iterate (19), so we only need to normalize the computed
tangent vectors.