~mdw / mLib / blame
| Commit | Line | Data |
|---|---|---|
| 289651a7 MW |
1 | /* -*-c-*- |
| 2 | * | |
| 3 | * Compute elements of the Fibonacci sequence | |
| 4 | * | |
| 5 | * (c) 2024 Straylight/Edgeware | |
| 6 | */ | |
| 7 | ||
| 8 | /*----- Licensing notice --------------------------------------------------* | |
| 9 | * | |
| 10 | * This file is part of the mLib utilities library. | |
| 11 | * | |
| 12 | * mLib is free software: you can redistribute it and/or modify it under | |
| 13 | * the terms of the GNU Library General Public License as published by | |
| 14 | * the Free Software Foundation; either version 2 of the License, or (at | |
| 15 | * your option) any later version. | |
| 16 | * | |
| 17 | * mLib is distributed in the hope that it will be useful, but WITHOUT | |
| 18 | * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 19 | * FITNESS FOR A PARTICULAR PURPOSE. See the GNU Library General Public | |
| 20 | * License for more details. | |
| 21 | * | |
| 22 | * You should have received a copy of the GNU Library General Public | |
| 23 | * License along with mLib. If not, write to the Free Software | |
| 24 | * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, | |
| 25 | * USA. | |
| 26 | */ | |
| 27 | ||
| 28 | /*----- Header files ------------------------------------------------------*/ | |
| 29 | ||
| 30 | #include "example.h" | |
| 31 | ||
| 32 | /*----- Main code ---------------------------------------------------------*/ | |
| 33 | ||
| 34 | unsigned long recfib(unsigned n) | |
| 35 | { return (n <= 1 ? n : recfib(n - 1) + recfib(n - 2)); } | |
| 36 | ||
| 37 | unsigned long iterfib(unsigned n) | |
| 38 | { | |
| 39 | unsigned long u, v, t; | |
| 40 | ||
| 41 | for (u = 0, v = 1; n--; t = v, v = u, u += t); | |
| 42 | return (u); | |
| 43 | } | |
| 44 | ||
| 45 | unsigned long expfib(unsigned n) | |
| 46 | { | |
| 47 | unsigned long a, b, u, v, t; | |
| 48 | ||
| 49 | /* We work in %$\Q(\phi)$%, where %$\phi^2 = \phi + 1$%. I claim that | |
| 50 | * %$\phi^k = F_k \phi + F_{k-1} \pmod f(\phi))$%. Proof by induction: | |
| 51 | * note that * %$F_{-1} = F_1 - F_0 = 1$%, so %$\phi^0 = 1 = {}$% | |
| 52 | * %$F_0 \phi + F_{-1}$%; and %$\phi^{k+1} = F_k \phi^2 + {}$% | |
| 53 | * %$F_{k-1} \phi = F_k (\phi + 1) + F_{k-1} \phi = (F_k + {}$% | |
| 54 | * %$F_{k-1} \phi + F_k = F_{k+1} \phi + F_k$% as claimed. | |
| 55 | * | |
| 56 | * Now, notice that %$(a \phi + b) (c \phi + d) = a c \phi^2 + {}$% | |
| 57 | * $%(a d + b c) \phi + b d = a c (\phi + 1) + (a d + b c) \phi + {}$% | |
| 58 | * %$b d = (a c + a d + b c) \phi + (a c + b d)$%. In particular, | |
| 59 | * %$(u \phi + v)^2 \equiv (u^2 + 2 u v) \phi + (u^2 + v^2)$%. | |
| 60 | */ | |
| 61 | a = 0, b = 1; u = 1, v = 0; | |
| 62 | if (n) | |
| 63 | for (;;) { | |
| 64 | if (n%2) { t = a*u; a = t + a*v + b*u; b = t + b*v; } | |
| 65 | n /= 2; if (!n) break; | |
| 66 | t = u*u; u = t + 2*u*v; v = t + v*v; | |
| 67 | } | |
| 68 | return (a); | |
| 69 | } | |
| 70 | ||
| 71 | /*----- That's all, folks -------------------------------------------------*/ |