Advanced use

Advanced use#

Adjustable parameters#

The optional inputs 'plot', 'plot graph' and 'infcontour' are just three of many adjustable parameters. In fact, all of the algorithm parameters defined in [Gibbs et al., 2024] can be easily modified, if the user wishes to do so. Here is a list of adjustable parameters - each can be adjusted by adding the optional input as a text string, followed by the new value.

Parameter

Meaning

Default

C_ball

Governs maximum number of oscillations across each non-oscillatory ball (and hence the ball radius)

\(2\pi\)

N_ball

Number of rays used when determining the ball radius

16

delta_ball

Governs when overlapping balls should be amalgamated

\(10^{-3}/(2\max(J-2,1))\), where \(J\) is the degree of the polynomial \(g\)

delta_ODE

Governs the local step size in the ODE solver for SD path tracing

\(10^{-1}\)

delta_coarse

Tolerance for the increment in the Newton iteration in the SD path tracing

\(10^{-2}\)

delta_fine

Tolerance for the increment in the Newton iteration in the quadrature

\(10^{-13}\)

delta_quad

Governs when the contribution from an integral on the quasi-SD deformation is computed

\(10^{-16}\)

inf quad rule

Determines which quadrature rule is used for the SD contours, from a choice of Gauss-Laguerre 'laguerre', or truncated Gauss-Legendre 'legendre'

'laguerre'

use mex

Determines if the MEX codes are to be used for higher performance. Should only be turned off if there is some issue with the MEX compiler, as described above.

true

Creating a log file#

There is also the option to record information about the algorithm at each step. This can be achieved by adding the optional input 'log'. A log file is then generated, and the name is based on the time and date that the file was created. For example:

PathFinder(-1, 1, @(x) x.^2, [1 -0.5 0.5 0 -1 0], 50, 10, 'log');

If you want to name it something different, use the option 'log name' followed by a text string argument with the desired name. For example:

PathFinder(-1, 1, @(x) x.^2, [1 -0.5 0.5 0 -1 0], 50, 10, 'log name','el_murad');

Hard-coded quadrature#

For efficiency, PathFinder uses hard-coded Gaussian quadrature to evaluate contour integrals. By default, these hard-coded values have been stored for up to 100 points each in the subroutines src/gauss_quadrature_rules/gausLagHC.m and src/gauss_quadrature_rules/gausLegHC.m, corresponding to Gauss-Laguerre and Gauss-Legendre respectively.

If more points are requested by PathFinder, these are generated from scratch using the Golub-Welsch algorithm, with code supplied by Dirk Laurie (22/6/1998); later edited by Walter Gautschi (4/4/2002).

If desired, you can increase the number of hard-coded points, overwriting the subroutines gausLagHC and gausLegHC. To do this:

  • Change the working directory to src/gauss_quadrature_rules

  • Open the script(s) hardCodeQuadratureGaulag and/or hardCodeQuadratureGauleg as required

  • Adjust the parameter maxNumHardCodedPts as required, and run the script.

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