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An exact index-one eigenvector-following step is the Newton correction at a saddle

Statement

Let V∈C4(U,R) on an open U⊆Rd, and put F=∇V. At any accepted update of Finite-budget index-one eigenvector-following Newton step, the step is the unique vector p solving

DF(x)p=−F(x),DF(x)=∇2V(x).

Suppose x∗ is a critical point, its Hessian is nonsingular with exactly one negative eigenvalue, and the Hessian is Lipschitz on a neighborhood of x∗. Then for all initial points in a sufficiently small ball about x∗, every update that the algorithm attempts before it stops is the ordinary exact-Jacobian Newton update and satisfies the local quadratic estimate supplied by Updated-Jacobian Newton has local quadratic convergence at a simple root. A gradient or eigensolve stopping status creates no later update, and no global-start guarantee is asserted.

Facts & Assumptions

Given: The smooth potential, square gradient residual, accepted branch checks, nonsingular index-one critical point, and local Hessian-Lipschitz hypothesis in the Statement.

[F1]

Finite-budget index-one eigenvector-following Newton step: after its eigenpair and enclosure checks, it calls the square vector Newton correction with Jacobian H(x)=∇2V(x) and residual ∇V(x).

[F2]

Newton correction for a square vector residual: if the Jacobian is invertible, its correction is the unique p solving J(x)p=−F(x).

[F3]

Updated-Jacobian Newton has local quadratic convergence at a simple root: for a C2 square residual with a simple zero and locally Lipschitz Jacobian that is invertible at the zero, exact-Jacobian Newton updates from a sufficiently small ball remain local and satisfy a quadratic error bound; the finite method may stop first.

[F4]

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and The Hessian matrix and critical points of a scalar field fix the Jacobian, gradient, and Hessian conventions. Applying If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative to the first-partial components of V, and then their successive partials, supplies the continuous total derivatives of F=∇V through order two from V∈C4. A total derivative computes every directional derivative, and its matrix is the Jacobian identifies the matrix of D(∇V) with the gradient Jacobian, and The Hessian of a C2 scalar field is symmetric makes its entries agree with ∇2V. Critical points are precisely zeros of F.

[A1]

The Hessian at an index-one critical point is nonsingular: its one negative eigenvalue and all remaining positive eigenvalues are nonzero.

Proof

Given: The hypotheses of the Statement.

1.1F1F2F4

At an accepted update, [F1] supplies the exact Hessian Jacobian and gradient residual to the Newton-correction procedure, and its validated ordered enclosures put every Hessian eigenvalue at distance at least γ from zero. Thus the Hessian is invertible; [F2] makes the returned correction the unique solution of ∇2V(x)p=−∇V(x), so the selected step is exactly the vector Newton step.

1.2A1F3F4given

Let x∗ be the critical point. By [F4], F(x∗)=0 and DF(x∗)=∇2V(x∗) is invertible by [A1]. Since V∈C4, F is C2; the assumed local Hessian-Lipschitz bound is a local Lipschitz bound on DF. Thus the hypotheses of [F3] hold.

2.1F1F3step 1.1step 1.2∎

The algorithm's update, whenever it is attempted, is the exact correction from step 1.1. Applying [F3] to this recurrence gives the asserted local quadratic bound and invariant neighborhood for every attempted update; if the algorithm stops on a residual or certification branch, there is no later point to which that update claim applies. No conclusion outside the local hypotheses follows.

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