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Pseudo-arclength continuation

One option for choosing g(x) is to select a hyperplane passing through X0 that is orthogonal to the vector vi:

g(x) = áx-X0,vi ñ .
(3)

So, the Newton iteration becomes:

Xk+1 = Xk - H-1x(Xk)H(Xk)
(4)


H(X) = æ
ç
ç
è
F(X)
0
ö
÷
÷
ø
,  Hx(X) = æ
ç
ç
è
Fx(X)
viT
ö
÷
÷
ø
 .
(5)

Then one can prove that the Newton iteration for (2) will converge to a point xi+1 on the curve from X0 provided that the stepsize h is sufficiently small and that the curve is regular (rank Fx(x)=n). Having found the new point xi+1 on the curve we need to compute the tangent vector at that point:

Fx(xi+1)vi+1 = 0 .
(6)

Furthermore the direction along the curve must be preserved: ávi,vi+1 ñ = 1, so we get the (n+1)-dimensional appended system

æ
ç
ç
è
Fx(xi+1)
viT
ö
÷
÷
ø
vi+1 = æ
ç
ç
è
0
1
ö
÷
÷
ø
.
(7)

Upon solving this system, vi+1 must be normalized.