Some stochastic processes Xt bring some challenges to Cholesky. Consider the stochastic process Xt with the covariance kernel of the form Cov(Xt, Xs) = exp{−|t − s|}. Let ti = i/n for i = 1,... ,n be...

Some stochastic processes Xt bring some challenges to Cholesky. Consider the stochastic process Xt with the covariance kernel of the form Cov(Xt, Xs) = exp{−|t − s|}. Let ti = i/n for i = 1,... ,n be points of evaluation of Xt for t ∈ [0, 1], and let A(n) be the covariance matrix, so (A(n))ij = exp{−|i − j |/n}. For various values of n, say 10, 20, 30,... , compute the Cholesky factors of A(n) and also the condition numbers κ = A(n)(A(n))−1using the p = 1 or p = ∞ norms. Does scaling with a parameter α in (A(n))ij = exp{−α|i − j |/n} make much of a difference?