Numerical Algorithm for Inverting the Pure Characteristics Model of Demand

I have provided an algorithm to invert market shares in pure characteristics model with arbitrary number of random coefficients by using first order information of direct mapping from structural parameters to market shares (Jacobian matrix). I have coded that algorithm, checked that it correctly calculates Jacobian and used it in series of MC-simulations. I find that this algorithm inverts market shares for small number of products exactly all the time if I give it a good guess and it successfully inverts market share from random starting point 100 percent of the time. Increasing number of products to 100 is possible, but the problem of zero market share during solving routine arises. I have provided algorithmic remedy for it so that I am able to invert market shares even for number of products as high as 100. I used the combination of good initial guess that is a solution to undisturbed problem and changing the delta_j for products that appeared to have zero market share during the solution routine. Combining these techniques allowed me to successfully invert market shares in all cases. I find that using sparse grids that have negative weights on some nodes makes the problem non-monotone and unstable. I also showed that using the quadrature integration rules is valid in this setting. I think this set of algorithms may be useful in empirical work and be incorporated into estimation procedures that estimate structural models with random coefficients based on pure characteristics model