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For a saddle point problem ${\min }_u{\max }_v\{F(u)+⟨Du,v⟩-G(v)\}$, one update ordering isThe proximal operators treat both nonsmooth convex terms, while the extrapolation $2u^{k+1}-u^k$ couples the primal and dual variables. The primal objective is $F(u)+G^{\ast }(Du)$, with $G^{\ast }$ the convex conjugate. A dual-first ordering is obtained by exchanging roles; consistent indexing is needed when comparing the formulas.

For proper lower semicontinuous convex functions on finite-dimensional Hilbert spaces, a nonempty saddle point set, and constant positive steps satisfying the displayed inequality, the primal-dual hybrid gradient method with extrapolation parameter one converges to a saddle point. The strict condition controls the bilinear coupling by the operator norm. It does not remove the need for existence of a saddle point. The fixed-step theorem and assumptions are recalled in Malitsky and Pock, A first-order primal-dual algorithm with linesearch, Section 1.

For a Gaussian integral from $-1/\sqrt{2}$ to $e^{i\theta }$, a saddle point contribution dominates for $0\le \theta \le \pi /4$. The moving endpoint contribution in steepest descent dominates for $\pi /4<\theta <5\pi /6$. At $5\pi /6$ the two endpoints have equal exponential rates and both are needed; for larger angles the fixed endpoint dominates. Saddle connectivity switches at $\pi /2$, but that change is subdominant in the leading endpoint regime.

For a simple saddle point, a nearby pole coalesces with its contributing region when their separation is of order the inverse square root of the large parameter. A local quadratic phase coordinate changes the singular part to a Gaussian pole integral. Adding any residue crossed by the contour deformation gives a Faddeeva function rather than two separately singular approximations. The regular part of the transformed amplitude remains smaller by the ordinary saddle width.

A parabola is a plane curve whose points are equally distant from a focus and a directrix. In suitable coordinates it has equation $y=a{x}^2$, with $a\ne 0$. Its quadratic geometry can also describe a contour deformation through a saddle point.