sdPathODEeulerExtendCoarsePath.m

Contents

sdPathODEeulerExtendCoarsePath.m#

This subroutine extends steepest descent contour approximation produced by sdPathODEeuler.m into the region of no return. This may be required to allocate a sufficiently large number of quadrature points.

[pArrayOut, hArrayOut, success] = ...
    sdPathODEeulerExtendCoarsePath(pIn, hIn, phaseCoeffs, SPs, ...
    baseStepSize, maxPts, pNewMax)

All paths \(h_\eta(p)\) are initially approximated in sdPathODEeuler.m, until \(p=P\), where the path has reached either another ball, or a region of no return. In the case of the latter, this routine extends this to an approximation to \(h(p)\) for \(p\leq P'\), where \(P'>P\) is now determined by the largest parameter value required by the underlying Gauss Laguerre rule. If there \(N\) points approximating the path \(h_\eta\) at the start of the routine, there will be \(N'>N\) points at the end of the routine. A thorough explanation can be found in [Gibbs et al., 2024], Section 2.6.2.

Inputs#

  • pIn : The coarse path parameter points \(p_1,\ldots,p_N\).

  • hIn : The coarse path points \(h_1\approx h_\eta(p_1),\ldots,h_N\approx h_\eta(p_N)\).

  • phaseCoeffs : The coefficients of the phase function \(g\).

  • SPs : An array \(\mathcal{P}_{\mathrm{stat}}:=\{\xi:g'(\xi)=0\}\)

  • baseStepSize : Scales the step in \(p\) of the Euler method.

  • maxPts : The algorithm terminates early when this number of points is reached.

  • pNewMax : The value \(P'\).

Outputs#

  • pArrayOut: The new values \(p_j\in[0,P']\) for \(j=1,\ldots,N'\).

  • hArrayOut : The new values \(h_j\approx h_\eta(p_j)\) for \(j=1,\ldots,N'\).

  • success : True if the algorithm did not terminate early, i.e. by reaching maxPts points.

Contents