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63 lines
1.5 KiB
Plaintext
63 lines
1.5 KiB
Plaintext
/*
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* Copyright (c) 1995 David I. Bell
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* Permission is granted to use, distribute, or modify this source,
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* provided that this copyright notice remains intact.
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*
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* Calculate the Nth Bernoulli number B(n).
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* This uses the following symbolic formula to calculate B(n):
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*
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* (b+1)^(n+1) - b^(n+1) = 0
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*
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* where b is a dummy value, and each power b^i gets replaced by B(i).
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* For example, for n = 3:
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* (b+1)^4 - b^4 = 0
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* b^4 + 4*b^3 + 6*b^2 + 4*b + 1 - b^4 = 0
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* 4*b^3 + 6*b^2 + 4*b + 1 = 0
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* 4*B(3) + 6*B(2) + 4*B(1) + 1 = 0
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* B(3) = -(6*B(2) + 4*B(1) + 1) / 4
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*
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* The combinatorial factors in the expansion of the above formula are
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* calculated interatively, and we use the fact that B(2i+1) = 0 if i > 0.
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* Since all previous B(n)'s are needed to calculate a particular B(n), all
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* values obtained are saved in an array for ease in repeated calculations.
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*/
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static Bnmax;
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static mat Bn[1001];
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define B(n)
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{
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local nn, np1, i, sum, mulval, divval, combval;
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if (!isint(n) || (n < 0))
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quit "Non-negative integer required for Bernoulli";
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if (n == 0)
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return 1;
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if (n == 1)
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return -1/2;
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if (isodd(n))
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return 0;
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if (n > 1000)
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quit "Very large Bernoulli";
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if (n <= Bnmax)
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return Bn[n];
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for (nn = Bnmax + 2; nn <= n; nn+=2) {
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np1 = nn + 1;
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mulval = np1;
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divval = 1;
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combval = 1;
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sum = 1 - np1 / 2;
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for (i = 2; i < np1; i+=2) {
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combval = combval * mulval-- / divval++;
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combval = combval * mulval-- / divval++;
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sum += combval * Bn[i];
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}
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Bn[nn] = -sum / np1;
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}
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Bnmax = n;
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return Bn[n];
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}
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