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4. Conclusions 31 1. λ ⇐ λ0 , 2. λ[m + J] ⇐ µJ , 3. u, µJ ⇐ (u) − λ, where λ[m + J] denotes the set {λm+i }i∈J . A similar algorithm applies to the evaluation of the Jacobian of the left-hand side of this restricted continuation problem with respect to u and µJ . Specifically, suppose that access to the Jacobians of the functions and is stored in the data variable ∂u such that ∂u (u) = ∂u (u) ∂u (u) . 41) where IJ is the submatrix of the r × r identity matrix containing the columns indexed by J.

11) and K2 := {n 1 + 1, . . ,n s +n 2 } . , that the extended continuation problem constructed from ˜ and u˜ 0 := (u 0 )K2 is embedded in the extended continuation problem constructed from and u 0 . We illustrate this relationship in Fig. 1(b). In this case f 2 is clearly independent of v1 . 40 Chapter 3. Construction We say that the extended continuation problem Eˆ is embedded canonically in the extended ˆ continuation problem E and write Eˆ E if Eqs. 3) hold with L = {1, . . 1, the extended continuation problem L = {1, .