Find the roots of the polynomial p(x) = 5×4 − 20x + 2 in the interval [0, 1] using binary search.
Consider the function f (x) = 41 x4 − x2 + 2x − 1. We want to minimize this function using Newton’s method. Verify that starting at a point close to 0 or 1 and using Newton’s method, one would opbtain iterates alternating between close neighborhoods of 0 and 1, and never converge. Apply Newton’s method to this problem with the Armijo-Goldstein condition, and backtracking, starting from point 0. Use μ = 0.5 and a backtracking ratio of 0.9. Experiment with other values of μ ∈ (0, 1) and the backtrcking ratio.
One of the most fundamental techniques of statistical analysis is the method of maxi- mum likelihood estimation. Given a sample set of independently drawn observations from a parametric distribution, the estimation problem is to determine the values of the distri- bution parameters that maximize the probability that the observed sample set comes from this distribution.
Consider, for example, the observations x1 = −0.24, x2 = 0.31, x3 = 0.23, x4 = −1.1, sampled from a normal distribution. If the mean of the distribution is known to be 0, what is the maximum likelihood estimate for the standard deviation σ? Construct the log-likelihood function and maximize it using the golden section search.
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