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Can you advise me how I can solve the following equation (equation of an ellipsoid) and find N solutions for that. Thanks.

$ (X-\mu)' \Sigma^{-1} (X-\mu) ={\chi}^2_{p,(1-\alpha)} $

where X is a vector of p variables , $\mu$ is a vector of p known parameters, $\Sigma$ is a p*p known matrix.

$\alpha$ is a known value between 0 and 1. The solutions would be infinite values for X (as variables). N number of solutions should be selected such that they represent the boundaries of the ellipsoid fairly good. Solutions would be points on an ellipsoid indicating a probability region ($\alpha$) of a multivariate normal distribution.

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    $\begingroup$ I'm interpreting this to mean you'd like to find points on an ellipsoid indicating a probability region reflecting the mean and covariance matrix of a multivariate normal. If so this may be of interest: level curves of multivariate normal. $\endgroup$ May 20 '15 at 15:32
  • $\begingroup$ Please clarify: what would a "solution" be? Since you don't mention anything about $\alpha$, and it is the only thing you don't mention, are you asking for a solution set of values of $\alpha$? Or by referring to $X$ as "variables" are you looking for values for its components? And since there will be an entire (infinite) manifold of solutions in most cases, by what criteria are your $N$ solutions intended to be selected from them? $\endgroup$
    – whuber
    May 20 '15 at 15:37
  • $\begingroup$ Thank you! The post is edited accordingly. Ellipse function in R does the job but I need to know how to find the N points without using Ellipse function, then I can compare the accuracy of the results I have found with those given by Ellipse function. $\endgroup$
    – Nile
    May 21 '15 at 10:14
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    $\begingroup$ @conjectures the link you have cited works only for bivariate distributions. Is it right? Here I need to deal with an 40 variate distribution $\endgroup$
    – Nile
    May 21 '15 at 10:24

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