induced-distributions

Computation of induced orthogonal polynomial distributions. See [1] for details.

Evaluating distribution functions

The following scripts evaluate distribution functions

• idist_jacobi.m : Induced distribution function for Jacobi weights
• idist_freud.m : Induced distribution function for Freud weights
• idist_hfreud.m : Induced distribution function for half-line Freud weights

Evaluating inverse distribution functions

The following scripts evaluate inverses of induced distribution functions

• idistinv_jacobi.m : Inverse induced distribution function for Jacobi weights
• idistinv_freud.m : Inverse induced distribution function for Freud weights
• idistinv_hfreud.m : Inverse induced distribution function for half-line Freud weights
• fidistinv_jacobi.m : Fast evaluation of idistinv_jacobi.m. Requires expensive one-time setup
• fidistinv_freud.m : Fast evaluation of idistinv_freud.m. Requires expensive one-time setup
• fidistinv_hfreud.m : Fast evaluation of idistinv_hfreud.m. Requires expensive one-time setup

Random sampling from mixtures of tensorial induced distributions

The script idist_mixture_sampling.m samples from an additive mixture of tensorial univariate induced distributions. The method only supports tensorial constructions where each of the univariate marginals are identical.

Demos

The following demo files illustrate usage of the methods above.

• demo_idist_jacobi.m : Illustrates basic usage of idist_jacobi.m, plots induced distribution. Produces part of Figure 1 in [1].
• demo_hfreud_error.m : Error plot for various quadrature sizes for half-line Freud weights. Showcases idist_hfreud.m. Produces part of Figure 5 in [1].
• demo_jacobi_error.m : Error plot for various quadrature sizes for Jacobi weights. Showcases idist_jacobi.m. Produces part of Figure 3 in [1].
• demo_medapprox_jacobi.m : Illustration of potential-theoretic induced distribution median versus exact median. Showcases idistinv_jacobi.m. Produces part of Figure 2 in [1].
• demo_intapprox_hfreud.m : Illustration of potential-theoretic induced distribution support interval versus exact support interval. Showcases idistinv_hfreud.m. Produces part of Figure 4 in [1].
• demo_equilibrium_measure.m : Empirical evidence for multivariate equilibrium measure conjecture. Showcases idist_mixture_sampling.m. Produces part of Figure 6 in [1].
• demo_fast_sampling_jacobi.m : Usage of idistinv_jacobi.m versus the faster fidistinv_jacobi.m
• demo_fast_sampling_hfreud.m : Usage of idistinv_hfreud.m versus the faster fidistinv_hfreud.m

References

[1] : "Computation of Induced Orthogonal Polynomial Distributions", Akil Narayan, arXiv:1704.08465 [math], http://arxiv.org/abs/1704.08465