This post is just to collect a few recent entries on this topic under one umbrella.
1. This post set up the general problem
Numerically compute the following sum, for arbitrary \(x_i\) ,\[F = \log \left(e^{x_1} + e^{x_2} + ... + e^{x_N} \right).\] It also briefly discussed the major problem with doing this by brute force (overflow/underflow).
2. We then made the problem specific, whose answer could be analytically computed. \[F = \log \left(e^{-1000} + e^{-1001} + ... + e^{-1100} \right) = -999.54.\]
3. We briefly looked at numerically evaluating an even simpler model problem \[f(y) = \log(1 + e^y).\] While much simpler, this problem reflects all of the complexity in the original problem.
4. Equipped with the solution, we went back and solved our specific problem.
1. This post set up the general problem
Numerically compute the following sum, for arbitrary \(x_i\) ,\[F = \log \left(e^{x_1} + e^{x_2} + ... + e^{x_N} \right).\] It also briefly discussed the major problem with doing this by brute force (overflow/underflow).
2. We then made the problem specific, whose answer could be analytically computed. \[F = \log \left(e^{-1000} + e^{-1001} + ... + e^{-1100} \right) = -999.54.\]
3. We briefly looked at numerically evaluating an even simpler model problem \[f(y) = \log(1 + e^y).\] While much simpler, this problem reflects all of the complexity in the original problem.
4. Equipped with the solution, we went back and solved our specific problem.
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