Once the collocation points have been defined, we can translate these values into sets of model input parameters,. We then run the "black-box" model for each of these cases, and collect the resulting values for each of the uncertain output variables. With the output values, we can construct equations for the coefficients. This matrix of algebraic equations can then be solved using standard linear-algebra techniques. With the coefficients determined, we now have the complete PDFs defined for all of the variant model outputs. This allows us to perform a variety of analyses including error estimation, expected value, and variance analysis, discussed further below.