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Fitting the result to 2 h n 2 one finds h ≃ 0.589 , but no exact formula for h has ever been found. With hexagonal cells, however, the exact solution of the so-called hard hexagon lattice gas model in 1980 showed that h ≃ 0.481 is the logarithm of the largest root of a degree 12 polynomial.

The most common are ones based on repetition or iteration, classic examples being Euclid's algorithm for GCD (page 915 ), Newton's method for FindRoot and the Gaussian elimination method for LinearSolve .

So, for example, if there are more parameters it becomes difficult to find continuous definitions that work for all complex values of these parameters. … And if one modifies the usual hypergeometric equation y''[x] f[y[x], y'[x]] by making f nonlinear then solutions typically become hard to find, and vary greatly in character with the form of f . … In Mathematica, however, functions like Root provide more convenient ways to access such results.

The main issue in evaluating those that exhibit regular oscillations at large x is to find their oscillation period with sufficient precision. … Any iterative procedure (such as FindRoot ) that yields a constant multiple more digits at each step will take about Log[n] steps to get n digits.

In the continued fraction for a randomly chosen number, the probability to find a term of size s is Log[2, (1 + 1/s)/(1 + 1/(s + 1))] , so that the probability of getting a 1 is about 41.50%, and the probability of getting a large term falls off like 1/s 2 . … Fairly large terms are sometimes seen quite early: in 5 1/3 term 19 is 3052, while in Root[10 + 8 # - # 3 &, 1] term 34 is 1,501,790.