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Nearly dependent constraints make the multiplier solve ill conditioned

Example

For $0 < ϵ \leq 1$ , take $M = I_{3}$ and $c_{ϵ} (q) = (q_{3}, q_{3} + ϵ q_{2}) .$ The Jacobian has full row rank for every fixed $ϵ > 0$ , but $A_{ϵ} = J_{ϵ} J_{ϵ}^{T} = (\begin{array}{cc} 1 & 1 \\ 1 & 1 + ϵ^{2} \end{array})$ has smallest eigenvalue $2 ϵ^{2} / (2 + ϵ^{2} + \sqrt{4 + ϵ^{4}}) \sim ϵ^{2} / 2$ . Thus the inverse multiplier bound in the RATTLE residual estimate is not uniform as the constraints approach dependence. At $ϵ = 0$ the rank hypothesis fails.

Facts & Assumptions

Given: The family $c_{ϵ}$ , $M = I_{3}$ , and the RATTLE multiplier matrix $A_{ϵ}$ in the Example.

[F1]

Regular holonomic phase space and constrained dynamics: regularity requires full row rank of $J$ and invertibility of $A = J M^{- 1} J^{T}$ .

[F2]

Scaled RATTLE residuals bound the finite solve trajectory error: on a fixed compact phase tube, residual-to-state constants depend in particular on uniform inverse bounds for the multiplier matrices; each fixed $ϵ > 0$ is regular, but these inverse bounds are not uniform as $ϵ ↓ 0$ .

Verification

1.1givenF1algebra

The two rows $(0, 0, 1)$ and $(0, ϵ, 1)$ of $J_{ϵ}$ are linearly independent exactly when $ϵ > 0$ . Their Gram matrix is the displayed $A_{ϵ}$ , whose trace is $2 + ϵ^{2}$ and determinant is $ϵ^{2}$ .

2.1step 1.1algebra

The quadratic formula gives its eigenvalues $(2 + ϵ^{2} \pm \sqrt{4 + ϵ^{4}}) / 2$ . Rationalizing the smaller one gives $2 ϵ^{2} / (2 + ϵ^{2} + \sqrt{4 + ϵ^{4}})$ , so $ϵ^{- 2} λ_{\min} (A_{ϵ}) \to 1 / 2$ and $∥ A_{ϵ}^{- 1} ∥_{2} \to \infty$ .

3.1F1F2step 2.1∎

Consequently no single residual-to-multiplier constant works for this family as $ϵ ↓ 0$ ; this is consistent with [F2], whose uniform estimates assume a fixed compact chart with uniformly bounded $A^{- 1}$ . At $ϵ = 0$ the two rows coincide, so [F1]'s full-rank hypothesis fails.

Depends on

Used by

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Sources

Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric Numerical Integration, 2nd ed., Chapter VII.1.4, pp. 245–247 (standard reference, not scraped)
Robert I. McLachlan, Klas Modin, Olivier Verdier, and Matt Wilkins, Geometric Generalisations of SHAKE and RATTLE, Foundations of Computational Mathematics 14 (2014), 339–370, §5.2 (standard reference, not scraped)