doc/ratseries/paper/binsplit.ps

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   2 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
   3 %%Title: binsplit.dvi
   4 %%Pages: 8
   5 %%PageOrder: Ascend
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   7 %%DocumentPaperSizes: a4
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  17 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
  18 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
  19 landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
  20 mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
  21 matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
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  66 B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
  67 p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
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1693 (series)2040 2310 y Fr(The)f(tec)n(hnique)e(presen)n(ted)g(here)i
1694 (applies)g(to)f(all)h(linearly)g(con)n(v)n(er-)2040 2393
1695 y(gen)n(t)f(sums)e(of)j(the)e(form)2602 2622 y Fo(S)g
1696 Fr(=)2785 2520 y Fm(1)2755 2545 y Fl(X)2758 2719 y Fk(n)p
1697 Fn(=0)2898 2573 y Fo(a)p Fr(\()p Fo(n)p Fr(\))p 2898
1698 2605 147 4 v 2902 2672 a Fo(b)p Fr(\()p Fo(n)p Fr(\))3065
1699 2573 y Fo(p)p Fr(\(0\))12 b Fj(\001)h(\001)g(\001)g Fo(p)p
1700 Fr(\()p Fo(n)p Fr(\))p 3065 2605 397 4 v 3067 2672 a
1701 Fo(q)s Fr(\(0\))f Fj(\001)h(\001)g(\001)g Fo(q)s Fr(\()p
1702 Fo(n)p Fr(\))2040 2867 y(where)25 b Fo(a)p Fr(\()p Fo(n)p
1703 Fr(\),)g Fo(b)p Fr(\()p Fo(n)p Fr(\),)h Fo(p)p Fr(\()p
1704 Fo(n)p Fr(\),)f Fo(q)s Fr(\()p Fo(n)p Fr(\))g(are)g(in)n(tegers)h(with)
1705 f Fo(O)r Fr(\(log)15 b Fo(n)p Fr(\))25 b(bits.)2040 2950
1706 y(The)j(most)g(often)h(used)e(case)i(is)g(that)f Fo(a)p
1707 Fr(\()p Fo(n)p Fr(\),)g Fo(b)p Fr(\()p Fo(n)p Fr(\),)h
1708 Fo(p)p Fr(\()p Fo(n)p Fr(\),)g Fo(q)s Fr(\()p Fo(n)p
1709 Fr(\))f(are)2040 3033 y(p)r(olynomials)e(in)g Fo(n)g
1710 Fr(with)g(in)n(teger)g(co)r(e\016cien)n(ts.)2040 3192
1711 y Fh(Algorithm:)2152 3352 y Fr(Giv)n(en)g(t)n(w)n(o)g(index)f(b)r
1712 (ounds)g Fo(n)3040 3360 y Fn(1)3101 3352 y Fr(and)h Fo(n)3297
1713 3360 y Fn(2)3332 3352 y Fr(,)g(consider)g(the)g(partial)2040
1714 3435 y(sum)2503 3623 y Fo(S)f Fr(=)2734 3546 y Fl(X)2656
1715 3720 y Fk(n)2694 3730 y Fg(1)2727 3720 y Fm(\024)p Fk(n<n)2899
1716 3730 y Fg(2)2955 3574 y Fo(a)p Fr(\()p Fo(n)p Fr(\))p
1717 2955 3606 147 4 v 2959 3673 a Fo(b)p Fr(\()p Fo(n)p Fr(\))3121
1718 3574 y Fo(p)p Fr(\()p Fo(n)3236 3582 y Fn(1)3271 3574
1719 y Fr(\))12 b Fj(\001)h(\001)h(\001)f Fo(p)p Fr(\()p Fo(n)p
1720 Fr(\))p 3121 3606 439 4 v 3123 3673 a Fo(q)s Fr(\()p
1721 Fo(n)3236 3681 y Fn(1)3271 3673 y Fr(\))f Fj(\001)h(\001)h(\001)f
1722 Fo(q)s Fr(\()p Fo(n)p Fr(\))2152 3873 y(It)35 b(is)i(not)e(computed)f
1723 (directly)-6 b(.)64 b(Instead,)38 b(w)n(e)e(compute)f(the)2040
1724 3956 y(in)n(tegers)j Fo(P)52 b Fr(=)41 b Fo(p)p Fr(\()p
1725 Fo(n)2653 3964 y Fn(1)2688 3956 y Fr(\))12 b Fj(\001)h(\001)g(\001)h
1726 Fo(p)p Fr(\()p Fo(n)2948 3964 y Fn(2)2999 3956 y Fj(\000)j
1727 Fr(1\),)41 b Fo(Q)g Fr(=)g Fo(q)s Fr(\()p Fo(n)3522 3964
1728 y Fn(1)3556 3956 y Fr(\))13 b Fj(\001)g(\001)g(\001)g
1729 Fo(q)s Fr(\()p Fo(n)3814 3964 y Fn(2)3866 3956 y Fj(\000)k
1730 Fr(1\),)2040 4039 y Fo(B)25 b Fr(=)c Fo(b)p Fr(\()p Fo(n)2313
1731 4047 y Fn(1)2348 4039 y Fr(\))13 b Fj(\001)g(\001)g(\001)g
1732 Fo(b)p Fr(\()p Fo(n)2602 4047 y Fn(2)2654 4039 y Fj(\000)k
1733 Fr(1\))23 b(and)g Fo(T)32 b Fr(=)21 b Fo(B)t(QS)t Fr(.)33
1734 b(If)24 b Fo(n)3476 4047 y Fn(2)3523 4039 y Fj(\000)12
1735 b Fo(n)3641 4047 y Fn(1)3697 4039 y Fo(<)21 b Fr(5,)j(these)2040
1736 4122 y(are)30 b(computed)f(directly)-6 b(.)46 b(If)30
1737 b Fo(n)2973 4130 y Fn(2)3028 4122 y Fj(\000)19 b Fo(n)3153
1738 4130 y Fn(1)3216 4122 y Fj(\025)28 b Fr(5,)k(they)d(are)h(computed)2040
1739 4205 y(using)24 b Fi(binary)j(splitting)p Fr(:)34 b(Cho)r(ose)26
1740 b(an)e(index)f Fo(n)3419 4213 y Fk(m)3502 4205 y Fr(in)h(the)g(middle)f
1741 (of)2040 4289 y Fo(n)2086 4297 y Fn(1)2141 4289 y Fr(and)c
1742 Fo(n)2330 4297 y Fn(2)2365 4289 y Fr(,)i(compute)d(the)i(comp)r(onen)n
1743 (ts)e Fo(P)3304 4298 y Fk(l)3329 4289 y Fr(,)j Fo(Q)3432
1744 4298 y Fk(l)3455 4289 y Fr(,)h Fo(B)3556 4298 y Fk(l)3580
1745 4289 y Fr(,)f Fo(T)3667 4298 y Fk(l)3710 4289 y Fr(b)r(elonging)2040
1746 4372 y(to)28 b(the)g(in)n(terv)l(al)g Fo(n)2597 4380
1747 y Fn(1)2657 4372 y Fj(\024)d Fo(n)h(<)f(n)2945 4380 y
1748 Fk(m)3004 4372 y Fr(,)k(compute)e(the)g(comp)r(onen)n(ts)h
1749 Fo(P)3977 4380 y Fk(r)4011 4372 y Fr(,)2040 4455 y Fo(Q)2101
1750 4463 y Fk(r)2135 4455 y Fr(,)g Fo(B)2242 4463 y Fk(r)2277
1751 4455 y Fr(,)g Fo(T)2371 4463 y Fk(r)2433 4455 y Fr(b)r(elonging)g(to)g
1752 (the)e(in)n(terv)l(al)i Fo(n)3337 4463 y Fk(m)3420 4455
1753 y Fj(\024)23 b Fo(n)i(<)e(n)3703 4463 y Fn(2)3738 4455
1754 y Fr(,)28 b(and)f(set)2040 4538 y Fo(P)32 b Fr(=)21 b
1755 Fo(P)2251 4547 y Fk(l)2275 4538 y Fo(P)2324 4546 y Fk(r)2359
1756 4538 y Fr(,)h Fo(Q)e Fr(=)i Fo(Q)2626 4547 y Fk(l)2649
1757 4538 y Fo(Q)2710 4546 y Fk(r)2744 4538 y Fr(,)g Fo(B)j
1758 Fr(=)c Fo(B)3009 4547 y Fk(l)3033 4538 y Fo(B)3091 4546
1759 y Fk(r)3146 4538 y Fr(and)f Fo(T)32 b Fr(=)21 b Fo(B)3506
1760 4546 y Fk(r)3541 4538 y Fo(Q)3602 4546 y Fk(r)3636 4538
1761 y Fo(T)3681 4547 y Fk(l)3711 4538 y Fr(+)6 b Fo(B)3835
1762 4547 y Fk(l)3859 4538 y Fo(P)3908 4547 y Fk(l)3932 4538
1763 y Fo(T)3977 4546 y Fk(r)4011 4538 y Fr(.)2152 4621 y(Finally)-6
1764 b(,)36 b(this)e(algorithm)g(is)g(applied)g(to)g Fo(n)3464
1765 4629 y Fn(1)3534 4621 y Fr(=)g(0)g(and)f Fo(n)3903 4629
1766 y Fn(2)3973 4621 y Fr(=)2040 4704 y Fo(n)2086 4712 y
1767 Fn(max)2224 4704 y Fr(=)21 b Fo(O)r Fr(\()p Fo(N)8 b
1768 Fr(\),)27 b(and)e(a)g(\014nal)g(\015oating-p)r(oin)n(t)h(division)g
1769 Fo(S)f Fr(=)3875 4673 y Fk(T)p 3849 4687 97 4 v 3849
1770 4729 a(B)r(Q)3981 4704 y Fr(is)2040 4799 y(p)r(erformed.)2040
1771 4958 y Fh(Complexit)n(y:)2152 5118 y Fr(The)k(bit)f(complexit)n(y)g(of)
1772 h(computing)f Fo(S)k Fr(with)d Fo(N)37 b Fr(bits)29 b(of)g(pre-)2040
1773 5201 y(cision)e(is)f Fo(O)r Fr(\(\(log)15 b Fo(N)8 b
1774 Fr(\))2662 5169 y Fn(2)2697 5201 y Fo(M)g Fr(\()p Fo(N)g
1775 Fr(\)\).)2040 5360 y Fh(Pro)r(of:)p eop
1776 %%Page: 2 2
1777 2 1 bop -40 -69 a Fr(Since)29 b(w)n(e)g(ha)n(v)n(e)g(assumed)f(the)g
1778 (series)i(to)f(b)r(e)g(linearly)h(con)n(v)n(er-)-152
1779 14 y(gen)n(t,)h(the)d Fo(n)p Fr(-th)h(term)f(is)i Fo(O)r
1780 Fr(\()p Fo(c)743 -17 y Fm(\000)p Fk(n)834 14 y Fr(\))f(with)g
1781 Fo(c)f(>)e Fr(1.)46 b(Hence)29 b(c)n(ho)r(osing)-152
1782 109 y Fo(n)-106 117 y Fn(max)32 109 y Fr(=)21 b Fo(N)193
1783 76 y Fn(log)12 b(2)p 193 92 121 4 v 194 135 a(log)g Fk(c)338
1784 109 y Fr(+)i Fo(O)r Fr(\(1\))25 b(will)g(ensure)g(that)f(the)g
1785 (round-o\013)f(error)i(is)-152 218 y Fo(<)h Fr(2)-28
1786 187 y Fm(\000)p Fk(N)78 218 y Fr(.)44 b(By)28 b(our)h(assumption)f
1787 (that)g Fo(a)p Fr(\()p Fo(n)p Fr(\),)i Fo(b)p Fr(\()p
1788 Fo(n)p Fr(\),)f Fo(p)p Fr(\()p Fo(n)p Fr(\),)h Fo(q)s
1789 Fr(\()p Fo(n)p Fr(\))e(are)-152 301 y(in)n(tegers)e(with)f
1790 Fo(O)r Fr(\(log)15 b Fo(n)p Fr(\))25 b(bits,)h(the)e(in)n(tegers)i
1791 Fo(P)11 b Fr(,)24 b Fo(Q)p Fr(,)h Fo(B)t Fr(,)g Fo(T)36
1792 b Fr(b)r(elong-)-152 384 y(ing)26 b(to)g(the)f(in)n(terv)l(al)g
1793 Fo(n)525 392 y Fn(1)581 384 y Fj(\024)c Fo(n)h(<)f(n)857
1794 392 y Fn(2)917 384 y Fr(all)27 b(ha)n(v)n(e)d Fo(O)r
1795 Fr(\(\()p Fo(n)1366 392 y Fn(2)1418 384 y Fj(\000)16
1796 b Fo(n)1540 392 y Fn(1)1575 384 y Fr(\))d(log)i Fo(n)1776
1797 392 y Fn(2)1811 384 y Fr(\))-152 467 y(bits.)-40 550
1798 y(The)29 b(algorithm's)g(recursion)g(depth)e(is)i Fo(d)c
1799 Fr(=)1318 517 y Fn(log)12 b Fk(n)1446 525 y Fg(max)p
1800 1318 533 236 4 v 1376 576 a Fn(log)f(2)1582 550 y Fr(+)19
1801 b Fo(O)r Fr(\(1\).)-152 650 y(A)n(t)36 b(recursion)i(depth)e
1802 Fo(k)j Fr(\(1)h Fj(\024)g Fo(k)i Fj(\024)e Fo(d)p Fr(\),)f(in)n(tegers)
1803 f(ha)n(ving)f(eac)n(h)-152 733 y Fo(O)r Fr(\()-52 701
1804 y Fk(n)-14 709 y Fg(max)p -51 716 146 4 v -12 762 a Fn(2)18
1805 748 y Ff(k)117 733 y Fr(log)15 b Fo(n)275 741 y Fn(max)392
1806 733 y Fr(\))25 b(bits)g(are)g(m)n(ultiplied.)33 b(Th)n(us,)26
1807 b(the)e(en)n(tire)h(com-)-152 816 y(putation)h(time)f
1808 Fo(t)g Fr(is)157 1042 y Fo(t)75 b Fr(=)438 940 y Fk(d)394
1809 965 y Fl(X)399 1139 y Fk(k)q Fn(=1)527 1042 y Fr(2)565
1810 1006 y Fk(k)q Fm(\000)p Fn(1)682 1042 y Fo(O)755 952
1811 y Fl(\020)805 1042 y Fo(M)900 952 y Fl(\020)959 993 y
1812 Fo(n)1005 1001 y Fn(max)p 959 1025 163 4 v 1003 1092
1813 a Fr(2)1041 1071 y Fk(k)1145 1042 y Fr(log)15 b Fo(n)1303
1814 1050 y Fn(max)1420 952 y Fl(\021)o(\021)260 1325 y Fr(=)438
1815 1224 y Fk(d)394 1249 y Fl(X)399 1423 y Fk(k)q Fn(=1)527
1816 1325 y Fo(O)g Fr(\()p Fo(M)21 b Fr(\()p Fo(n)801 1333
1817 y Fn(max)931 1325 y Fr(log)15 b Fo(n)1089 1333 y Fn(max)1206
1818 1325 y Fr(\))o(\))260 1521 y(=)74 b Fo(O)r Fr(\(log)16
1819 b Fo(n)643 1529 y Fn(max)760 1521 y Fo(M)8 b Fr(\()p
1820 Fo(n)918 1529 y Fn(max)1047 1521 y Fr(log)15 b Fo(n)1205
1821 1529 y Fn(max)1322 1521 y Fr(\)\))-152 1671 y(Because)27
1822 b(of)g Fo(n)276 1679 y Fn(max)414 1671 y Fr(=)21 b Fo(O)r
1823 Fr(\()627 1640 y Fk(N)p 595 1654 118 4 v 595 1696 a Fn(log)12
1824 b Fk(c)723 1671 y Fr(\))25 b(and)-122 1891 y Fo(M)-26
1825 1776 y Fl(\022)83 1842 y Fo(N)p 45 1874 146 4 v 45 1941
1826 a Fr(log)15 b Fo(c)213 1891 y Fr(log)372 1842 y Fo(N)p
1827 335 1874 V 335 1941 a Fr(log)g Fo(c)490 1776 y Fl(\023)626
1828 1891 y Fr(=)74 b Fo(O)833 1776 y Fl(\022)958 1842 y Fr(1)p
1829 904 1874 V 904 1941 a(log)15 b Fo(c)1060 1891 y(N)21
1830 b Fr(\(log)15 b Fo(N)8 b Fr(\))1383 1855 y Fn(2)1443
1831 1891 y Fr(log)15 b(log)g Fo(N)1736 1776 y Fl(\023)626
1832 2123 y Fr(=)74 b Fo(O)833 2008 y Fl(\022)958 2075 y Fr(1)p
1833 904 2106 V 904 2174 a(log)15 b Fo(c)1072 2123 y Fr(log)g
1834 Fo(N)22 b(M)8 b Fr(\()p Fo(N)g Fr(\))1478 2008 y Fl(\023)-152
1835 2323 y Fr(w)n(e)26 b(ha)n(v)n(e)356 2449 y Fo(t)21 b
1836 Fr(=)g Fo(O)559 2334 y Fl(\022)684 2401 y Fr(1)p 630
1837 2432 V 630 2500 a(log)15 b Fo(c)785 2449 y Fr(\(log)g
1838 Fo(N)8 b Fr(\))1026 2414 y Fn(2)1061 2449 y Fo(M)g Fr(\()p
1839 Fo(N)g Fr(\))1272 2334 y Fl(\023)-152 2625 y Fr(Considering)27
1840 b Fo(c)f Fr(as)g(constan)n(t,)h(this)e(is)h(the)g(desired)g(result.)
1841 -152 2766 y Fh(Chec)n(kp)r(oin)n(ting/P)n(arallelising:)-40
1842 2907 y Fr(A)f(c)n(hec)n(kp)r(oin)n(t)e(can)i(b)r(e)g(easily)h(done)e(b)
1843 n(y)g(storing)h(the)g(\(in)n(teger\))-152 2990 y(v)l(alues)i(of)g
1844 Fo(n)212 2998 y Fn(1)247 2990 y Fr(,)g Fo(n)341 2998
1845 y Fn(2)376 2990 y Fr(,)g Fo(P)11 b Fr(,)27 b Fo(Q)p Fr(,)f
1846 Fo(B)k Fr(and)d Fo(T)11 b Fr(.)36 b(Similarly)-6 b(,)27
1847 b(if)g Fo(m)f Fr(pro)r(cessors)-152 3073 y(are)e(a)n(v)l(ailable,)i
1848 (then)d(the)g(in)n(terv)l(al)g([0)p Fo(;)14 b(n)1028
1849 3081 y Fk(max)1156 3073 y Fr(])24 b(can)f(b)r(e)h(divided)f(in)n(to)
1850 -152 3156 y Fo(m)d Fr(pieces)g(of)h(length)f Fo(l)j Fr(=)e
1851 Fj(b)p Fo(n)670 3164 y Fk(max)797 3156 y Fo(=m)p Fj(c)p
1852 Fr(.)33 b(After)20 b(eac)n(h)g(pro)r(cessor)h Fo(i)f
1853 Fr(has)-152 3239 y(computed)d(the)h(sum)g(of)h(its)g(in)n(terv)l(al)f
1854 ([)p Fo(il)q(;)c Fr(\()p Fo(i)s Fr(+)s(1\))p Fo(l)q Fr(],)20
1855 b(the)e(partial)i(sums)-152 3322 y(are)35 b(com)n(bined)e(to)i(the)f
1856 (\014nal)g(result)h(using)g(the)f(rules)h(describ)r(ed)-152
1857 3405 y(ab)r(o)n(v)n(e.)-152 3547 y Fh(Note:)-40 3688
1858 y Fr(F)-6 b(or)22 b(the)f(sp)r(ecial)i(case)f Fo(a)p
1859 Fr(\()p Fo(n)p Fr(\))f(=)g Fo(b)p Fr(\()p Fo(n)p Fr(\))g(=)g(1,)i(the)e
1860 (binary)g(splitting)-152 3771 y(algorithm)e(has)g(already)g(b)r(een)f
1861 (do)r(cumen)n(ted)f(in)i([3)q(],)h(section)f(6,)i(and)-152
1862 3854 y([2)q(],)26 b(section)h(10.2.3.)-40 3937 y(Explicit)d
1863 (computation)e(of)i Fo(P)11 b Fr(,)23 b Fo(Q)p Fr(,)h
1864 Fo(B)t Fr(,)f Fo(T)34 b Fr(is)23 b(only)g(required)g(as)g(a)-152
1865 4020 y(recursion)28 b(base,)f(for)h Fo(n)537 4028 y Fn(2)590
1866 4020 y Fj(\000)17 b Fo(n)713 4028 y Fn(1)771 4020 y Fo(<)23
1867 b Fr(2,)k(but)f(a)n(v)n(oiding)h(recursions)h(for)-152
1868 4103 y Fo(n)-106 4111 y Fn(2)-54 4103 y Fj(\000)17 b
1869 Fo(n)69 4111 y Fn(1)125 4103 y Fo(<)k Fr(5)26 b(gains)h(some)f(p)r
1870 (ercen)n(t)f(of)h(execution)g(sp)r(eed.)-40 4186 y(The)34
1871 b(binary)f(splitting)i(algorithm)f(is)g(asymptotically)g(faster)-152
1872 4269 y(than)25 b(step-b)n(y-step)e(ev)l(aluation)i(of)h(the)f(sum)f({)h
1873 (whic)n(h)g(has)g(binary)-152 4352 y(complexit)n(y)36
1874 b Fo(O)r Fr(\()p Fo(N)409 4320 y Fn(2)445 4352 y Fr(\))h({)g(b)r
1875 (ecause)h(it)f(pushes)g(as)h(m)n(uc)n(h)d(m)n(ultipli-)-152
1876 4435 y(cation)d(w)n(ork)g(as)g(p)r(ossible)g(to)g(the)f(region)h(where)
1877 g(m)n(ultiplication)-152 4518 y(b)r(ecomes)i(e\016cien)n(t.)59
1878 b(If)34 b(the)g(m)n(ultiplication)g(w)n(ere)g(implemen)n(ted)-152
1879 4601 y(as)28 b(an)g Fo(M)8 b Fr(\()p Fo(N)g Fr(\))24
1880 b(=)g Fo(O)r Fr(\()p Fo(N)531 4569 y Fn(2)566 4601 y
1881 Fr(\))k(algorithm,)g(the)f(binary)g(splitting)h(algo-)-152
1882 4684 y(rithm)j(w)n(ould)i(pro)n(vide)e(no)h(sp)r(eedup)g(o)n(v)n(er)g
1883 (step-b)n(y-step)e(ev)l(alua-)-152 4767 y(tion.)-152
1884 4909 y Fh(Implemen)n(tation:)-40 5050 y Fr(In)24 b(the)f(follo)n(wing)k
1885 (w)n(e)e(presen)n(t)e(a)i(simpli\014ed)e(C++)h(implemen-)-152
1886 5133 y(tation)f(of)g(the)f(ab)r(o)n(v)n(e)g(algorithm)822
1887 5101 y Fn(1)857 5133 y Fr(.)33 b(The)22 b(initialisation)j(is)e(done)f
1888 (b)n(y)p -152 5171 797 4 v -65 5220 a Fg(1)-33 5245 y
1889 Fe(A)28 b(complete)d(implemen)n(tation)g(can)h(b)r(e)h(found)f(in)h
1890 (CLN)h([8].)45 b(The)27 b(im-)-152 5311 y(plemen)n(tation)21
1891 b(of)j(the)e(binary-splitting)g(metho)r(d)g(will)h(b)r(e)g(also)g(a)n
1892 (v)l(ailable)g(in)-152 5378 y Fd(LiDIA-1.4)2040 -69 y
1893 Fr(a)29 b(structure)g Fq(abpq)p 2601 -69 24 4 v 29 w(series)i
1894 Fr(con)n(taining)e(arra)n(ys)h Fq(a)p Fr(,)g Fq(b)p Fr(,)h
1895 Fq(p)e Fr(and)g Fq(q)g Fr(of)2040 14 y(m)n(ultiprecision)f(in)n(tegers)
1896 g(\()p Fq(bigint)p Fr(s\).)42 b(The)28 b(v)l(alues)f(of)i(the)e(arra)n
1897 (ys)2040 97 y(at)i(the)e(index)h Fo(n)h Fr(corresp)r(ond)f(to)h(the)f
1898 (v)l(alues)g(of)h(the)f(functions)h Fo(a)p Fr(,)2040
1899 180 y Fo(b)p Fr(,)23 b Fo(p)e Fr(and)g Fo(q)j Fr(at)e(the)f(in)n(teger)
1900 h(p)r(oin)n(t)g Fo(n)p Fr(.)33 b(The)22 b(\(partial\))g(results)h(of)f
1901 (the)2040 263 y(algorithm)k(are)g(stored)g(in)g(the)f
1902 Fq(abpq)p 3132 263 V 29 w(series)p 3395 263 V 30 w(result)i
1903 Fr(structure.)2040 413 y Fq(//)40 b(abpq_series)i(is)d(initialised)j
1904 (by)e(user)2040 496 y(struct)h({)e(bigint)i(*a,)f(*b,)g(*p,)f(*q;)2315
1905 579 y(})g(abpq_series;)2040 745 y(//)h(abpq_series_result)j(holds)d
1906 (the)g(partial)h(results)2040 828 y(struct)g({)e(bigint)i(P,)e(Q,)h(B,)
1907 g(T;)2315 911 y(})f(abpq_series_result;)2040 1077 y(//)h(binary)g
1908 (splitting)i(summation)f(for)f(abpq_series)2040 1160
1909 y(void)g(sum_abpq\(abpq_series_result)46 b(&)39 b(r,)2589
1910 1243 y(int)h(n1,)g(int)g(n2,)2589 1326 y(const)h(abpq_series)g(&)f
1911 (arg\))2040 1409 y({)2119 1492 y(//)f(check)i(the)f(length)g(of)g(the)g
1912 (summation)h(interval)2119 1575 y(switch)f(\(n2)g(-)g(n1\))2119
1913 1658 y({)2197 1741 y(case)g(0:)2275 1824 y(error_handler\("summation)45
1914 b(device",)2629 1907 y("sum_abpq::)c(n2-n1)g(should)f(be)g(>)f(0."\);)
1915 2275 1990 y(break;)2197 2156 y(case)h(1:)g(//)g(the)f(result)i(at)f
1916 (the)g(point)g(n1)2275 2239 y(r.P)g(=)g(arg.p[n1];)2275
1917 2322 y(r.Q)g(=)g(arg.q[n1];)2275 2405 y(r.B)g(=)g(arg.b[n1];)2275
1918 2488 y(r.T)g(=)g(arg.a[n1])h(*)f(arg.p[n1];)2275 2571
1919 y(break;)2197 2737 y(//)g(cases)g(2,)g(3,)f(4)h(left)g(out)g(for)g
1920 (simplicity)2197 2904 y(default:)h(//)f(the)g(general)g(case)2275
1921 3070 y(//)g(the)g(left)g(and)g(the)g(right)g(partial)h(sum)2275
1922 3153 y(abpq_series_result)j(L,)39 b(R;)2275 3319 y(//)h(find)g(the)g
1923 (middle)h(of)e(the)h(interval)2275 3402 y(int)g(nm)g(=)g(\(n1)f(+)h
1924 (n2\))g(/)f(2;)2275 3568 y(//)h(sum)g(left)g(side)2275
1925 3651 y(sum_abpq\(L,)i(n1,)e(nm,)g(arg\);)2275 3817 y(//)g(sum)g(right)g
1926 (side)2275 3900 y(sum_abpq\(R,)i(nm,)e(n2,)g(arg\);)2275
1927 4066 y(//)g(put)g(together)2275 4149 y(r.P)g(=)g(L.P)g(*)f(R.P;)2275
1928 4232 y(r.Q)h(=)g(L.Q)g(*)f(R.Q;)2275 4315 y(r.B)h(=)g(L.B)g(*)f(R.B;)
1929 2275 4398 y(r.T)h(=)g(R.B)g(*)f(R.Q)h(*)g(L.T)f(+)h(L.B)g(*)f(L.P)h(*)g
1930 (R.T;)2275 4481 y(break;)2119 4564 y(})2040 4647 y(})2152
1931 4796 y Fr(Note)30 b(that)f(the)g(m)n(ultiprecision)h(in)n(tegers)g
1932 (could)g(b)r(e)f(replaced)2040 4879 y(here)c(b)n(y)g(in)n(teger)g(p)r
1933 (olynomials,)h(or)g(b)n(y)e(an)n(y)h(other)g(ring)h(pro)n(viding)2040
1934 4962 y(the)d(op)r(erators)h(=)f(\(assignmen)n(t\),)g(+)g(\(addition\))g
1935 (and)f Fj(\003)i Fr(\(m)n(ultipli-)2040 5045 y(cation\).)42
1936 b(F)-6 b(or)28 b(example,)g(one)g(could)h(regard)f(a)h(biv)l(ariate)f
1937 (p)r(olyno-)2040 5129 y(mial)j(o)n(v)n(er)f(the)g(in)n(tegers)h(as)g(a)
1938 g(series)h(o)n(v)n(er)e(the)g(second)h(v)l(ariable,)2040
1939 5212 y(with)j(p)r(olynomials)f(o)n(v)n(er)h(the)e(\014rst)i(v)l
1940 (ariable)f(as)h(its)g(co)r(e\016cien)n(ts.)2040 5295
1941 y(This)25 b(w)n(ould)g(result)f(an)h(accelerated)h(algorithm)e(for)h
1942 (summing)e(bi-)2040 5378 y(v)l(ariate)j(\(and)f(th)n(us)g(m)n(ultiv)l
1943 (ariate\))g(p)r(olynomials.)p eop
1944 %%Page: 3 3
1945 3 2 bop -152 -69 a Fp(2.1)83 b(Example:)36 b(The)28 b(facto)n(rial)-152
1946 63 y Fr(This)36 b(is)g(the)e(most)h(classical)j(example)c(of)i(the)e
1947 (binary)h(splitting)-152 146 y(algorithm)26 b(and)g(w)n(as)g(probably)g
1948 (kno)n(wn)f(long)h(b)r(efore)h([2].)-40 229 y(Computation)j(of)i(the)e
1949 (factorial)i(is)f(b)r(est)f(done)h(using)f(the)g(bi-)-152
1950 312 y(nary)e(splitting)h(algorithm,)h(com)n(bined)d(with)i(a)g
1951 (reduction)f(of)h(the)-152 395 y(ev)n(en)18 b(factors)h(in)n(to)f(o)r
1952 (dd)g(factors)h(and)f(m)n(ultiplication)g(with)g(a)g(p)r(o)n(w)n(er)
1953 -152 478 y(of)27 b(2,)f(according)h(to)f(the)f(form)n(ula)37
1954 752 y Fo(n)p Fr(!)d(=)f(2)245 716 y Fk(n)p Fm(\000)p
1955 Fk(\033)367 726 y Fg(2)400 716 y Fn(\()p Fk(n)p Fn(\))507
1956 752 y Fj(\001)549 675 y Fl(Y)546 849 y Fk(k)q Fm(\025)p
1957 Fn(1)671 587 y Fl(0)671 736 y(@)940 675 y(Y)767 831 y
1958 Ff(n)p 753 840 65 4 v 753 884 a Fg(2)781 872 y Ff(k)828
1959 851 y Fk(<)p Fn(2)p Fk(m)p Fn(+1)p Fm(\024)1146 831 y
1960 Ff(n)p 1096 840 138 4 v 1096 884 a Fg(2)1124 872 y Ff(k)q
1961 Fc(\000)p Fg(1)1243 752 y Fr(\(2)p Fo(m)c Fr(+)g(1\))1541
1962 587 y Fl(1)1541 736 y(A)1614 604 y Fk(k)-152 1021 y Fr(and)26
1963 b(where)g(the)f(pro)r(ducts)341 1198 y Fo(P)11 b Fr(\()p
1964 Fo(n)477 1206 y Fn(1)511 1198 y Fo(;)i(n)591 1206 y Fn(2)626
1965 1198 y Fr(\))21 b(=)851 1121 y Fl(Y)759 1296 y Fk(n)797
1966 1306 y Fg(1)829 1296 y Fk(<m)p Fm(\024)p Fk(n)1017 1306
1967 y Fg(2)1050 1198 y Fr(\(2)p Fo(m)c Fr(+)g(1\))-152 1443
1968 y(are)35 b(ev)l(aluated)f(according)h(to)f(the)g(binary)f(splitting)i
1969 (algorithm:)-152 1534 y Fo(P)11 b Fr(\()p Fo(n)-16 1542
1970 y Fn(1)19 1534 y Fo(;)i(n)99 1542 y Fn(2)134 1534 y Fr(\))44
1971 b(=)f Fo(P)11 b Fr(\()p Fo(n)447 1542 y Fn(1)482 1534
1972 y Fo(;)i(n)562 1542 y Fk(m)621 1534 y Fr(\))p Fo(P)e
1973 Fr(\()p Fo(n)787 1542 y Fk(m)845 1534 y Fo(;)i(n)925
1974 1542 y Fn(2)960 1534 y Fr(\))39 b(with)h Fo(n)1264 1542
1975 y Fk(m)1367 1534 y Fr(=)1470 1469 y Fl(\004)1520 1501
1976 y Fk(n)1558 1511 y Fg(1)1590 1501 y Fn(+)p Fk(n)1675
1977 1511 y Fg(2)p 1520 1517 189 4 v 1598 1560 a Fn(2)1718
1978 1469 y Fl(\005)1796 1534 y Fr(if)-152 1617 y Fo(n)-106
1979 1625 y Fn(2)-54 1617 y Fj(\000)17 b Fo(n)69 1625 y Fn(1)125
1980 1617 y Fj(\025)k Fr(5.)-152 1804 y Fp(2.2)83 b(Example:)36
1981 b(Elementa)n(ry)28 b(functions)f(at)h(rational)g(p)r(oints)-152
1982 1935 y Fr(The)35 b(binary)f(splitting)h(algorithm)f(can)h(b)r(e)f
1983 (applied)g(to)h(the)f(fast)-152 2018 y(computation)28
1984 b(of)g(the)g(elemen)n(tary)f(functions)h(at)g(rational)i(p)r(oin)n(ts)
1985 -152 2101 y Fo(x)e Fr(=)19 2071 y Fk(u)p 19 2085 37 4
1986 v 21 2127 a(v)65 2101 y Fr(,)k(simply)d(b)n(y)g(using)h(the)g(p)r(o)n
1987 (w)n(er)g(series.)49 b(W)-6 b(e)30 b(presen)n(t)f(ho)n(w)-152
1988 2184 y(this)24 b(can)f(b)r(e)g(done)g(for)h(exp)o(\()p
1989 Fo(x)p Fr(\),)f(ln\()p Fo(x)p Fr(\),)g(sin)q(\()p Fo(x)p
1990 Fr(\),)g(cos)q(\()p Fo(x)p Fr(\),)g(arctan\()p Fo(x)p
1991 Fr(\),)-152 2267 y(sinh\()p Fo(x)p Fr(\))33 b(and)g(cosh)q(\()p
1992 Fo(x)p Fr(\).)57 b(The)34 b(calculation)i(of)e(other)g(elemen)n(tary)
1993 -152 2350 y(functions)27 b(is)h(similar)f(\(or)g(it)g(can)g(b)r(e)g
1994 (reduced)f(to)h(the)f(calculation)-152 2433 y(of)h(these)f
1995 (functions\).)-152 2620 y Fp(2.2.1)82 b Fr(exp\()p Fo(x)p
1996 Fr(\))26 b Fp(fo)n(r)i(rational)g Fo(x)-152 2751 y Fr(This)41
1997 b(is)f(a)f(direct)h(application)h(of)f(the)f(ab)r(o)n(v)n(e)h
1998 (algorithm)g(with)-152 2834 y Fo(a)p Fr(\()p Fo(n)p Fr(\))45
1999 b(=)g(1,)g Fo(b)p Fr(\()p Fo(n)p Fr(\))g(=)g(1,)f Fo(p)p
2000 Fr(\(0\))h(=)g Fo(q)s Fr(\(0\))g(=)g(1,)g(and)39 b Fo(p)p
2001 Fr(\()p Fo(n)p Fr(\))45 b(=)g Fo(u)p Fr(,)-152 2917 y
2002 Fo(q)s Fr(\()p Fo(n)p Fr(\))36 b(=)f Fo(nv)i Fr(for)e
2003 Fo(n)h(>)f Fr(0.)61 b(Because)35 b(the)f(series)h(is)g(not)f(only)g
2004 (lin-)-152 3000 y(early)g(con)n(v)n(ergen)n(t)e({)h(exp\()p
2005 Fo(x)p Fr(\))f(is)h(an)g(en)n(tire)g(function)g({,)i
2006 Fo(n)1631 3008 y Fn(max)1781 3000 y Fr(=)-152 3092 y
2007 Fo(O)r Fr(\()107 3061 y Fk(N)p -51 3075 371 4 v -51 3125
2008 a Fn(log)12 b Fk(N)5 b Fn(+log)260 3105 y Fg(1)p 238
2009 3114 71 4 v 238 3148 a Fc(j)p Ff(x)p Fc(j)329 3092 y
2010 Fr(\),)26 b(hence)f(the)h(bit)f(complexit)n(y)g(is)387
2011 3379 y Fo(O)460 3239 y Fl( )638 3330 y Fr(\(log)15 b
2012 Fo(N)8 b Fr(\))879 3298 y Fn(2)p 536 3362 480 4 v 536
2013 3435 a Fr(log)15 b Fo(N)25 b Fr(+)17 b(log)954 3404 y
2014 Fn(1)p 933 3418 73 4 v 933 3460 a Fm(j)p Fk(x)p Fm(j)1025
2015 3379 y Fo(M)8 b Fr(\()p Fo(N)g Fr(\))1236 3239 y Fl(!)-152
2016 3618 y Fr(Considering)27 b Fo(x)e Fr(as)i(constan)n(t,)f(this)g(is)g
2017 Fo(O)r Fr(\(log)15 b Fo(N)26 b(M)8 b Fr(\()p Fo(N)g Fr(\)\).)-152
2018 3805 y Fp(2.2.2)82 b Fr(exp\()p Fo(x)p Fr(\))26 b Fp(fo)n(r)i(real)f
2019 Fo(x)-152 3936 y Fr(This)j(can)e(b)r(e)h(computed)e(using)i(the)f
2020 (addition)h(theorem)f(for)h(exp,)-152 4019 y(b)n(y)20
2021 b(a)i(tric)n(k)f(due)f(to)i(Bren)n(t)f([3])h(\(see)f(also)i([2)q(],)f
2022 (section)g(10.2,)i(exercise)-152 4102 y(8\).)35 b(W)-6
2023 b(rite)568 4329 y Fo(x)21 b Fr(=)g Fo(x)758 4337 y Fn(0)809
2024 4329 y Fr(+)916 4227 y Fm(1)886 4252 y Fl(X)891 4427
2025 y Fk(k)q Fn(=0)1029 4281 y Fo(u)1073 4290 y Fk(k)p 1029
2026 4312 83 4 v 1032 4380 a Fo(v)1069 4389 y Fk(k)-152 4594
2027 y Fr(with)26 b Fo(x)67 4602 y Fn(0)127 4594 y Fr(in)n(teger,)g
2028 Fo(v)439 4603 y Fk(k)499 4594 y Fr(=)21 b(2)618 4562
2029 y Fn(2)648 4541 y Ff(k)715 4594 y Fr(and)k Fj(j)p Fo(u)929
2030 4603 y Fk(k)968 4594 y Fj(j)d Fo(<)f Fr(2)1130 4562 y
2031 Fn(2)1160 4541 y Ff(k)q Fc(\000)p Fg(1)1274 4594 y Fr(,)26
2032 b(and)f(compute)299 4783 y(exp\()p Fo(x)p Fr(\))20 b(=)h(exp\()p
2033 Fo(x)814 4791 y Fn(0)848 4783 y Fr(\))c Fj(\001)936 4706
2034 y Fl(Y)933 4880 y Fk(k)q Fm(\025)p Fn(0)1058 4783 y Fr(exp)1188
2035 4693 y Fl(\020)1248 4734 y Fo(u)1292 4743 y Fk(k)p 1248
2036 4766 V 1251 4833 a Fo(v)1288 4842 y Fk(k)1340 4693 y
2037 Fl(\021)-40 5028 y Fr(This)27 b(algorithm)f(has)g(bit)f(complexit)n(y)
2038 -31 5272 y Fo(O)43 5132 y Fl( )109 5165 y Fk(O)r Fn(\(log)11
2039 b Fk(N)5 b Fn(\))168 5195 y Fl(X)173 5369 y Fk(k)q Fn(=0)409
2040 5223 y Fr(\(log)15 b Fo(N)8 b Fr(\))650 5191 y Fn(2)p
2041 371 5255 352 4 v 371 5322 a Fr(log)15 b Fo(N)25 b Fr(+)17
2042 b(2)684 5301 y Fk(k)733 5272 y Fo(M)8 b Fr(\()p Fo(N)g
2043 Fr(\))944 5132 y Fl(!)1031 5272 y Fr(=)21 b Fo(O)r Fr(\(\(log)15
2044 b Fo(N)8 b Fr(\))1443 5236 y Fn(2)1478 5272 y Fo(M)g
2045 Fr(\()p Fo(N)g Fr(\)\))2040 -69 y Fp(2.2.3)110 b Fr(ln\()p
2046 Fo(x)p Fr(\))26 b Fp(fo)n(r)h(rational)i Fo(x)2040 63
2047 y Fr(F)-6 b(or)28 b(rational)g Fj(j)p Fo(x)18 b Fj(\000)g
2048 Fr(1)p Fj(j)25 b Fo(<)f Fr(1,)29 b(the)e(binary)g(splitting)h
2049 (algorithm)f(can)2040 146 y(also)g(b)r(e)f(applied)g(directly)g(to)f
2050 (the)h(p)r(o)n(w)n(er)g(series)h(for)g(ln\()p Fo(x)p
2051 Fr(\).)34 b(W)-6 b(rite)2040 229 y Fo(x)q Fj(\000)q Fr(1)21
2052 b(=)2296 198 y Fk(u)p 2296 212 37 4 v 2298 254 a(v)2361
2053 229 y Fr(and)c(compute)g(the)g(series)i(with)f Fo(a)p
2054 Fr(\()p Fo(n)p Fr(\))j(=)g(1,)f Fo(b)p Fr(\()p Fo(n)p
2055 Fr(\))h(=)g Fo(n)q Fr(+)q(1,)2040 312 y Fo(q)s Fr(\()p
2056 Fo(n)p Fr(\))g(=)h Fo(v)s Fr(,)j Fo(p)p Fr(\(0\))c(=)h
2057 Fo(u)p Fr(,)k(and)f Fo(p)p Fr(\()p Fo(n)p Fr(\))c(=)g
2058 Fj(\000)p Fo(u)k Fr(for)i Fo(n)22 b(>)f Fr(0.)2152 395
2059 y(This)27 b(algorithm)f(has)f(bit)h(complexit)n(y)e Fo(O)r
2060 Fr(\(\(log)16 b Fo(N)8 b Fr(\))3656 363 y Fn(2)3691 395
2061 y Fo(M)g Fr(\()p Fo(N)g Fr(\)\).)2040 582 y Fp(2.2.4)82
2062 b Fr(ln\()p Fo(x)p Fr(\))27 b Fp(fo)n(r)g(real)h Fo(x)2040
2063 714 y Fr(This)e(can)g(b)r(e)g(computed)e(using)i(the)f(\\in)n(v)n
2064 (erse")h(Bren)n(t)g(tric)n(k:)2152 797 y(Start)g(with)g
2065 Fo(y)d Fr(:=)f(0.)2152 880 y(As)31 b(long)h(as)f Fo(x)f
2066 Fj(6)p Fr(=)g(1)h(within)h(the)e(actual)i(precision,)h(c)n(ho)r(ose)f
2067 Fo(k)2040 971 y Fr(maximal)23 b(with)h Fj(j)p Fo(x)14
2068 b Fj(\000)g Fr(1)p Fj(j)22 b Fo(<)f Fr(2)2878 939 y Fm(\000)p
2069 Fk(k)2964 971 y Fr(.)34 b(Put)24 b Fo(z)g Fr(=)d(2)3347
2070 939 y Fm(\000)p Fn(2)p Fk(k)3477 906 y Fl(\002)3512 971
2071 y Fr(2)3550 939 y Fn(2)p Fk(k)3619 971 y Fr(\()p Fo(x)16
2072 b Fj(\000)h Fr(1\))3854 906 y Fl(\003)3889 971 y Fr(,)24
2073 b(i.e.)2040 1062 y(let)h Fo(z)j Fr(con)n(tain)d(the)f(\014rst)h
2074 Fo(k)i Fr(signi\014can)n(t)e(bits)g(of)h Fo(x)15 b Fj(\000)g
2075 Fr(1.)34 b Fo(z)28 b Fr(is)d(a)h(go)r(o)r(d)2040 1145
2076 y(appro)n(ximation)20 b(for)i(ln\()p Fo(x)p Fr(\).)32
2077 b(Set)20 b Fo(y)k Fr(:=)d Fo(y)10 b Fr(+)d Fo(z)22 b
2078 Fr(and)f Fo(x)g Fr(:=)g Fo(x)7 b Fj(\001)g Fr(exp)o(\()p
2079 Fj(\000)p Fo(z)s Fr(\).)2152 1228 y(Since)23 b Fo(x)11
2080 b Fj(\001)g Fr(exp\()p Fo(y)s Fr(\))22 b(is)h(an)g(in)n(v)l(arian)n(t)f
2081 (of)i(the)e(algorithm,)i(the)f(\014nal)2040 1311 y Fo(y)28
2082 b Fr(is)e(the)g(desired)g(v)l(alue)f(ln\()p Fo(x)p Fr(\).)2152
2083 1394 y(This)i(algorithm)f(has)f(bit)h(complexit)n(y)2161
2084 1644 y Fo(O)2235 1504 y Fl( )2300 1538 y Fk(O)r Fn(\(log)11
2085 b Fk(N)5 b Fn(\))2360 1568 y Fl(X)2365 1742 y Fk(k)q
2086 Fn(=0)2601 1595 y Fr(\(log)14 b Fo(N)8 b Fr(\))2841 1564
2087 y Fn(2)p 2563 1627 352 4 v 2563 1695 a Fr(log)15 b Fo(N)25
2088 b Fr(+)17 b(2)2876 1674 y Fk(k)2924 1644 y Fo(M)8 b Fr(\()p
2089 Fo(N)g Fr(\))3135 1504 y Fl(!)3223 1644 y Fr(=)21 b Fo(O)r
2090 Fr(\(\(log)15 b Fo(N)8 b Fr(\))3635 1608 y Fn(2)3670
2091 1644 y Fo(M)g Fr(\()p Fo(N)g Fr(\)\))2040 1919 y Fp(2.2.5)110
2092 b Fr(sin\()p Fo(x)p Fr(\))p Fp(,)26 b Fr(cos)q(\()p Fo(x)p
2093 Fr(\))h Fp(fo)n(r)g(rational)h Fo(x)2040 2050 y Fr(These)35
2094 b(are)g(direct)g(applications)h(of)f(the)f(binary)g(splitting)i(algo-)
2095 2040 2133 y(rithm:)63 b(F)-6 b(or)40 b(sin)q(\()p Fo(x)p
2096 Fr(\),)j(put)d Fo(a)p Fr(\()p Fo(n)p Fr(\))45 b(=)h(1,)e
2097 Fo(b)p Fr(\()p Fo(n)p Fr(\))i(=)f(1,)g Fo(p)p Fr(\(0\))h(=)f
2098 Fo(u)p Fr(,)2040 2216 y Fo(q)s Fr(\(0\))h(=)f Fo(v)s
2099 Fr(,)e(and)d Fo(p)p Fr(\()p Fo(n)p Fr(\))45 b(=)g Fj(\000)p
2100 Fo(u)2993 2185 y Fn(2)3028 2216 y Fr(,)e Fo(q)s Fr(\()p
2101 Fo(n)p Fr(\))j(=)f(\(2)p Fo(n)p Fr(\)\(2)p Fo(n)28 b
2102 Fr(+)e(1\))p Fo(v)3866 2185 y Fn(2)3941 2216 y Fr(for)2040
2103 2299 y Fo(n)41 b(>)g Fr(0.)70 b(F)-6 b(or)37 b(cos)q(\()p
2104 Fo(x)p Fr(\),)j(put)c Fo(a)p Fr(\()p Fo(n)p Fr(\))41
2105 b(=)g(1,)f Fo(b)p Fr(\()p Fo(n)p Fr(\))h(=)g(1,)g Fo(p)p
2106 Fr(\(0\))f(=)h(1,)2040 2382 y Fo(q)s Fr(\(0\))22 b(=)f(1,)k(and)g
2107 Fo(p)p Fr(\()p Fo(n)p Fr(\))c(=)g Fj(\000)p Fo(u)2862
2108 2351 y Fn(2)2897 2382 y Fr(,)k Fo(q)s Fr(\()p Fo(n)p
2109 Fr(\))d(=)f(\(2)p Fo(n)16 b Fj(\000)g Fr(1\)\(2)p Fo(n)p
2110 Fr(\))p Fo(v)3647 2351 y Fn(2)3707 2382 y Fr(for)26 b
2111 Fo(n)c(>)f Fr(0.)2040 2465 y(Of)39 b(course,)j(when)c(b)r(oth)g(sin\()p
2112 Fo(x)p Fr(\))g(and)g(cos)q(\()p Fo(x)p Fr(\))f(are)i(needed,)i(one)2040
2113 2548 y(should)25 b(only)g(compute)f(sin\()p Fo(x)p Fr(\))g(this)h(w)n
2114 (a)n(y)-6 b(,)25 b(and)g(then)f(set)h(cos)q(\()p Fo(x)p
2115 Fr(\))c(=)2040 2651 y Fj(\006)2100 2582 y Fl(p)p 2183
2116 2582 365 4 v 69 x Fr(1)c Fj(\000)g Fr(sin\()p Fo(x)p
2117 Fr(\))2513 2630 y Fn(2)2547 2651 y Fr(.)35 b(This)26
2118 b(is)g(a)g(20\045)h(sp)r(eedup)e(at)h(least.)2152 2743
2119 y(The)18 b(bit)e(complexit)n(y)g(of)i(these)f(algorithms)h(is)g
2120 Fo(O)r Fr(\(log)d Fo(N)8 b(M)g Fr(\()p Fo(N)g Fr(\)\).)2040
2121 2930 y Fp(2.2.6)110 b Fr(sin\()p Fo(x)p Fr(\))p Fp(,)26
2122 b Fr(cos)q(\()p Fo(x)p Fr(\))h Fp(fo)n(r)g(real)h Fo(x)2040
2123 3062 y Fr(T)-6 b(o)37 b(compute)f(cos\()p Fo(x)p Fr(\))24
2124 b(+)g Fo(i)13 b Fr(sin)q(\()p Fo(x)p Fr(\))38 b(=)i(exp)o(\()p
2125 Fo(ix)p Fr(\))c(for)i(real)f Fo(x)p Fr(,)i(again)2040
2126 3145 y(the)30 b(addition)g(theorems)g(and)g(Bren)n(t's)h(tric)n(k)f
2127 (can)g(b)r(e)g(used.)48 b(The)2040 3228 y(resulting)26
2128 b(algorithm)g(has)g(bit)g(complexit)n(y)e Fo(O)r Fr(\(\(log)16
2129 b Fo(N)8 b Fr(\))3685 3196 y Fn(2)3719 3228 y Fo(M)g
2130 Fr(\()p Fo(N)g Fr(\)\).)2040 3415 y Fp(2.2.7)110 b Fr(arctan\()p
2131 Fo(x)p Fr(\))27 b Fp(fo)n(r)g(rational)h Fo(x)2040 3547
2132 y Fr(F)-6 b(or)30 b(rational)g Fj(j)p Fo(x)p Fj(j)e Fo(<)f
2133 Fr(1,)k(the)e(fastest)i(w)n(a)n(y)e(to)h(compute)e(arctan\()p
2134 Fo(x)p Fr(\))2040 3630 y(with)19 b(bit)g(complexit)n(y)e
2135 Fo(O)r Fr(\(\(log)e Fo(N)8 b Fr(\))3035 3598 y Fn(2)3070
2136 3630 y Fo(M)g Fr(\()p Fo(N)g Fr(\)\))19 b(is)h(to)e(apply)h(the)f
2137 (binary)2040 3713 y(splitting)i(algorithm)f(directly)h(to)f(the)g(p)r
2138 (o)n(w)n(er)g(series)i(for)f(arctan\()p Fo(x)p Fr(\).)2040
2139 3796 y(Put)37 b Fo(a)p Fr(\()p Fo(n)p Fr(\))k(=)f(1,)i
2140 Fo(b)p Fr(\()p Fo(n)p Fr(\))e(=)h(2)p Fo(n)26 b Fr(+)e(1,)41
2141 b Fo(q)s Fr(\()p Fo(n)p Fr(\))g(=)g(1,)g Fo(p)p Fr(\(0\))f(=)h
2142 Fo(x)c Fr(and)2040 3879 y Fo(p)p Fr(\()p Fo(n)p Fr(\))21
2143 b(=)g Fj(\000)p Fo(x)2391 3847 y Fn(2)2451 3879 y Fr(for)26
2144 b Fo(n)c(>)f Fr(0.)2040 4066 y Fp(2.2.8)110 b Fr(arctan\()p
2145 Fo(x)p Fr(\))27 b Fp(fo)n(r)g(real)h Fo(x)2040 4197 y
2146 Fr(This)22 b(again)g(can)f(b)r(e)g(computed)f(using)h(the)g(\\in)n(v)n
2147 (erse")h(Bren)n(t)f(tric)n(k:)2152 4280 y(Start)g(out)g(with)g
2148 Fo(z)k Fr(:=)2908 4250 y Fn(1)p 2819 4264 209 4 v 2819
2149 4273 a Fj(p)p 2883 4273 145 4 v 55 x Fn(1+)p Fk(x)2995
2150 4314 y Fg(2)3045 4280 y Fr(+)8 b Fo(i)3235 4250 y Fk(x)p
2151 3149 4264 209 4 v 3149 4273 a Fj(p)p 3213 4273 145 4
2152 v 55 x Fn(1+)p Fk(x)3325 4314 y Fg(2)3388 4280 y Fr(and)21
2153 b Fo(')g Fr(:=)h(0.)33 b(During)2040 4409 y(the)26 b(algorithm)g
2154 Fo(z)j Fr(will)e(b)r(e)f(a)h(complex)e(n)n(um)n(b)r(er)f(with)i
2155 Fj(j)p Fo(z)s Fj(j)d Fr(=)f(1)k(and)2040 4492 y(Re\()p
2156 Fo(z)s Fr(\))21 b Fo(>)g Fr(0.)2152 4575 y(As)j(long)h(as)g(Im)n(\()p
2157 Fo(z)s Fr(\))c Fj(6)p Fr(=)g(0)k(within)f(the)g(actual)h(precision,)h
2158 (c)n(ho)r(ose)2040 4671 y Fo(k)31 b Fr(maximal)c(with)i
2159 Fj(j)13 b Fr(Im)n(\()p Fo(z)s Fr(\))p Fj(j)26 b Fo(<)g
2160 Fr(2)3000 4639 y Fm(\000)p Fk(k)3087 4671 y Fr(.)42 b(Put)28
2161 b Fo(\013)f Fr(=)e(2)3502 4639 y Fm(\000)p Fn(2)p Fk(k)3632
2162 4605 y Fl(\002)3667 4671 y Fr(2)3705 4639 y Fn(2)p Fk(k)3786
2163 4671 y Fr(Im)o(\()p Fo(z)s Fr(\))3977 4605 y Fl(\003)4011
2164 4671 y Fr(,)2040 4766 y(i.e.)36 b(let)27 b Fo(\013)f
2165 Fr(con)n(tain)g(the)g(\014rst)g Fo(k)i Fr(signi\014can)n(t)f(bits)f(of)
2166 g(Im)o(\()p Fo(z)s Fr(\).)35 b Fo(\013)27 b Fr(is)f(a)2040
2167 4849 y(go)r(o)r(d)j(appro)n(ximation)e(for)h(arcsin)q(\(Im)o(\()p
2168 Fo(z)s Fr(\)\).)39 b(Set)27 b Fo(')e Fr(:=)g Fo(')18
2169 b Fr(+)g Fo(\013)28 b Fr(and)2040 4932 y Fo(z)c Fr(:=)e
2170 Fo(z)e Fj(\001)d Fr(exp\()p Fj(\000)p Fo(i\013)p Fr(\).)2152
2171 5015 y(Since)j Fo(z)10 b Fj(\001)c Fr(exp\()p Fo(i')p
2172 Fr(\))20 b(is)h(an)f(in)n(v)l(arian)n(t)g(of)i(the)e(algorithm,)i(the)e
2173 (\014nal)2040 5098 y Fo(')26 b Fr(is)g(the)f(desired)h(v)l(alue)g
2174 (arcsin)3091 5067 y Fk(x)p 3004 5081 209 4 v 3004 5091
2175 a Fj(p)p 3068 5091 145 4 v 55 x Fn(1+)p Fk(x)3180 5132
2176 y Fg(2)3222 5098 y Fr(.)p eop
2177 %%Page: 4 4
2178 4 3 bop -40 -69 a Fr(This)27 b(algorithm)f(has)g(bit)f(complexit)n(y)
2179 -31 164 y Fo(O)43 24 y Fl( )109 57 y Fk(O)r Fn(\(log)11
2180 b Fk(N)5 b Fn(\))168 87 y Fl(X)173 262 y Fk(k)q Fn(=0)409
2181 115 y Fr(\(log)15 b Fo(N)8 b Fr(\))650 83 y Fn(2)p 371
2182 147 352 4 v 371 215 a Fr(log)15 b Fo(N)25 b Fr(+)17 b(2)684
2183 193 y Fk(k)733 164 y Fo(M)8 b Fr(\()p Fo(N)g Fr(\))944
2184 24 y Fl(!)1031 164 y Fr(=)21 b Fo(O)r Fr(\(\(log)15 b
2185 Fo(N)8 b Fr(\))1443 128 y Fn(2)1478 164 y Fo(M)g Fr(\()p
2186 Fo(N)g Fr(\)\))-152 436 y Fp(2.2.9)82 b Fr(sinh\()p Fo(x)p
2187 Fr(\))p Fp(,)27 b Fr(cosh\()p Fo(x)p Fr(\))g Fp(fo)n(r)g(rational)h
2188 (and)g(real)f Fo(x)-152 567 y Fr(These)f(can)e(b)r(e)h(computed)e(b)n
2189 (y)g(similar)j(algorithms)f(as)g(sin\()p Fo(x)p Fr(\))f(and)-152
2190 650 y(cos)q(\()p Fo(x)p Fr(\))c(ab)r(o)n(v)n(e,)h(with)g(the)f(same)g
2191 (asymptotic)f(bit)h(complexit)n(y)-6 b(.)32 b(The)-152
2192 733 y(standard)26 b(computation,)f(using)g(exp)o(\()p
2193 Fo(x)p Fr(\))g(and)g(its)h(recipro)r(cal)h(\(cal-)-152
2194 816 y(culated)37 b(b)n(y)f(the)h(Newton)f(metho)r(d\))g(results)h(also)
2195 i(to)e(the)f(same)-152 899 y(complexit)n(y)25 b(and)g(w)n(orks)h
2196 (equally)g(w)n(ell)h(in)e(practice.)-40 982 y(The)20
2197 b(bit)f(complexit)n(y)e(of)j(these)f(algorithms)h(is)g
2198 Fo(O)r Fr(\(log)15 b Fo(N)d(M)c Fr(\()p Fo(N)g Fr(\)\))-152
2199 1065 y(for)27 b(rational)g Fo(x)e Fr(and)g Fo(O)r Fr(\(\(log)16
2200 b Fo(N)8 b Fr(\))802 1034 y Fn(2)836 1065 y Fo(M)g Fr(\()p
2201 Fo(N)g Fr(\)\))26 b(for)h(real)f Fo(x)p Fr(.)-152 1250
2202 y Fp(2.3)83 b(Example:)32 b(Hyp)r(ergeometric)19 b(functions)f(at)h
2203 (rational)g(p)r(oints)-152 1381 y Fr(The)26 b(binary)e(splitting)i
2204 (algorithm)f(is)h(w)n(ell)g(suited)f(for)g(the)g(ev)l(alua-)-152
2205 1464 y(tion)h(of)h(a)f(h)n(yp)r(ergeometric)f(series)140
2206 1676 y Fo(F)213 1561 y Fl(\022)311 1632 y Fo(a)352 1640
2207 y Fn(1)387 1632 y Fo(;)75 b(:)13 b(:)g(:)g(;)75 b(a)722
2208 1640 y Fk(r)315 1716 y Fo(b)348 1724 y Fn(1)383 1716
2209 y Fo(;)k(:)13 b(:)g(:)g(;)80 b(b)719 1724 y Fk(s)794
2210 1608 y Fl(\014)794 1657 y(\014)822 1676 y Fo(x)866 1561
2211 y Fl(\023)947 1676 y Fr(=)1058 1574 y Fm(1)1029 1599
2212 y Fl(X)1031 1774 y Fk(n)p Fn(=0)1171 1628 y Fo(a)p 1212
2213 1565 43 4 v -32 x Fk(n)1212 1641 y Fn(1)1267 1628 y Fj(\001)13
2214 b(\001)g(\001)h Fo(a)p 1411 1565 V -32 x Fk(n)1411 1641
2215 y(r)p 1171 1659 282 4 v 1179 1733 a Fo(b)p 1212 1676
2216 43 4 v -26 x Fk(n)1212 1752 y Fn(1)1267 1733 y Fj(\001)f(\001)g(\001)h
2217 Fo(b)p 1403 1676 V -26 x Fk(n)1403 1742 y(s)1463 1676
2218 y Fo(x)1507 1640 y Fk(n)-152 1900 y Fr(with)30 b(rational)i(co)r
2219 (e\016cien)n(ts)f Fo(a)752 1908 y Fn(1)786 1900 y Fr(,)g(...,)h
2220 Fo(a)995 1908 y Fk(r)1030 1900 y Fr(,)f Fo(b)1115 1908
2221 y Fn(1)1150 1900 y Fr(,)g(...,)h Fo(b)1351 1908 y Fk(s)1414
2222 1900 y Fr(at)e(a)g(rational)-152 1983 y(p)r(oin)n(t)36
2223 b Fo(x)g Fr(in)g(the)f(in)n(terior)i(of)g(the)f(circle)h(of)g(con)n(v)n
2224 (ergence.)66 b(Just)-152 2079 y(put)38 b Fo(a)p Fr(\()p
2225 Fo(n)p Fr(\))k(=)g(1,)g Fo(b)p Fr(\()p Fo(n)p Fr(\))g(=)g(1,)g
2226 Fo(p)p Fr(\(0\))g(=)g Fo(q)s Fr(\(0\))g(=)g(1,)g(and)1611
2227 2043 y Fk(p)p Fn(\()p Fk(n)p Fn(\))p 1611 2062 119 4
2228 v 1612 2104 a Fk(q)r Fn(\()p Fk(n)p Fn(\))1781 2079 y
2229 Fr(=)-142 2162 y Fn(\()p Fk(a)-84 2172 y Fg(1)-52 2162
2230 y Fn(+)p Fk(n)p Fm(\000)p Fn(1\))p Fm(\001\001\001)p
2231 Fn(\()p Fk(a)250 2170 y Ff(r)283 2162 y Fn(+)p Fk(n)p
2232 Fm(\000)p Fn(1\))p Fk(x)p -142 2182 648 4 v -117 2224
2233 a Fn(\()p Fk(b)-66 2234 y Fg(1)-34 2224 y Fn(+)p Fk(n)p
2234 Fm(\000)p Fn(1\))p Fm(\001\001\001)p Fn(\()p Fk(b)261
2235 2232 y Ff(s)294 2224 y Fn(+)p Fk(n)p Fm(\000)p Fn(1\))543
2236 2199 y Fr(for)28 b Fo(n)d(>)g Fr(0.)40 b(The)28 b(ev)l(aluation)g(can)g
2237 (th)n(us)f(b)r(e)-152 2309 y(done)k(with)f(bit)h(complexit)n(y)e
2238 Fo(O)r Fr(\(\(log)15 b Fo(N)8 b Fr(\))1067 2277 y Fn(2)1102
2239 2309 y Fo(M)g Fr(\()p Fo(N)g Fr(\)\))31 b(for)g Fo(r)g
2240 Fr(=)e Fo(s)h Fr(and)-152 2392 y Fo(O)r Fr(\(log)16 b
2241 Fo(N)25 b(M)8 b Fr(\()p Fo(N)g Fr(\)\))26 b(for)h Fo(r)c(<)e(s)p
2242 Fr(.)-152 2576 y Fp(2.4)83 b(Example:)36 b Fo(\031)-152
2243 2707 y Fr(The)26 b(Raman)n(ujan)f(series)i(for)f Fo(\031)260
2244 2871 y Fr(1)p 256 2902 47 4 v 256 2970 a Fo(\031)334
2245 2919 y Fr(=)465 2871 y(12)p 425 2902 157 4 v 425 2971
2246 a Fo(C)485 2950 y Fn(3)p Fk(=)p Fn(2)634 2818 y Fm(1)605
2247 2842 y Fl(X)607 3017 y Fk(n)p Fn(=0)748 2870 y Fr(\()p
2248 Fj(\000)p Fr(1\))906 2839 y Fk(n)948 2870 y Fr(\(6)p
2249 Fo(n)p Fr(\)!\()p Fo(A)17 b Fr(+)g Fo(nB)t Fr(\))p 748
2250 2902 686 4 v 890 2970 a(\(3)p Fo(n)p Fr(\)!)p Fo(n)p
2251 Fr(!)1122 2948 y Fn(3)1158 2970 y Fo(C)1218 2948 y Fn(3)p
2252 Fk(n)-152 3143 y Fr(with)36 b Fo(A)h Fr(=)g(13591409,)k
2253 Fo(B)g Fr(=)c(545140134,)k Fo(C)h Fr(=)37 b(640320)h(found)-152
2254 3226 y(b)n(y)i(the)h(Ch)n(udno)n(vsky's)623 3194 y Fn(2)699
2255 3226 y Fr(and)g(whic)n(h)g(is)h(used)e(b)n(y)h(the)f
2256 Fb(LiDIA)-152 3309 y Fr([16)q(,)32 b(15)q(,)g(13)q(])g(and)f(the)h(P)n
2257 (ari)g([7)q(])g(system)f(to)h(compute)e Fo(\031)s Fr(,)j(is)g(usu-)-152
2258 3392 y(ally)i(written)f(as)h(an)e(algorithm)i(of)f(bit)g(complexit)n(y)
2259 f Fo(O)r Fr(\()p Fo(N)1638 3360 y Fn(2)1673 3392 y Fr(\).)59
2260 b(It)-152 3475 y(is,)34 b(ho)n(w)n(ev)n(er,)f(p)r(ossible)g(to)f(apply)
2261 f(binary)g(splitting)h(to)f(the)h(sum.)-152 3558 y(Put)j
2262 Fo(a)p Fr(\()p Fo(n)p Fr(\))h(=)g Fo(A)23 b Fr(+)g Fo(nB)t
2263 Fr(,)38 b Fo(b)p Fr(\()p Fo(n)p Fr(\))e(=)g(1,)i Fo(p)p
2264 Fr(\(0\))e(=)h(1,)g Fo(q)s Fr(\(0\))g(=)f(1,)i(and)-152
2265 3641 y Fo(p)p Fr(\()p Fo(n)p Fr(\))j(=)g Fj(\000)p Fr(\(6)p
2266 Fo(n)26 b Fj(\000)e Fr(5\)\(2)p Fo(n)i Fj(\000)f Fr(1\)\(6)p
2267 Fo(n)h Fj(\000)f Fr(1\),)41 b Fo(q)s Fr(\()p Fo(n)p Fr(\))g(=)h
2268 Fo(n)1467 3609 y Fn(3)1501 3641 y Fo(C)1561 3609 y Fn(3)1596
2269 3641 y Fo(=)p Fr(24)d(for)-152 3724 y Fo(n)30 b(>)f Fr(0.)50
2270 b(This)31 b(reduces)g(the)f(complexit)n(y)f(to)i Fo(O)r
2271 Fr(\(\(log)15 b Fo(N)8 b Fr(\))1543 3692 y Fn(2)1578
2272 3724 y Fo(M)g Fr(\()p Fo(N)g Fr(\)\).)-152 3807 y(Although)32
2273 b(this)g(is)h(theoretically)h(slo)n(w)n(er)f(than)f(Bren)n(t-Salamin's)
2274 -152 3890 y(quadratically)41 b(con)n(v)n(ergen)n(t)f(iteration,)46
2275 b(whic)n(h)41 b(has)g(a)f(bit)h(com-)-152 3973 y(plexit)n(y)34
2276 b(of)h Fo(O)r Fr(\(log)15 b Fo(N)32 b(M)8 b Fr(\()p Fo(N)g
2277 Fr(\)\),)37 b(in)d(practice)h(the)f(binary)g(splitted)-152
2278 4056 y(Raman)n(ujan)d(sum)g(is)i(three)f(times)f(faster)j(than)d(Bren)n
2279 (t-Salamin,)-152 4139 y(at)g(least)g(in)f(the)g(range)h(from)f
2280 Fo(N)37 b Fr(=)29 b(1000)i(bits)g(to)f Fo(N)37 b Fr(=)29
2281 b(1000000)-152 4222 y(bits.)-152 4407 y Fp(2.5)83 b(Example:)36
2282 b(Catalan's)28 b(constant)f Fo(G)-152 4538 y Fr(A)f(linearly)g(con)n(v)
2283 n(ergen)n(t)f(sum)g(for)h(Catalan's)i(constan)n(t)521
2284 4750 y Fo(G)21 b Fr(:=)734 4648 y Fm(1)705 4673 y Fl(X)707
2285 4847 y Fk(n)p Fn(=0)903 4701 y Fr(\()p Fj(\000)p Fr(1\))1061
2286 4669 y Fk(n)p 847 4733 312 4 v 847 4801 a Fr(\(2)p Fo(n)d
2287 Fr(+)f(1\))1124 4779 y Fn(2)-152 4978 y Fr(is)27 b(giv)n(en)e(in)h
2288 ([2],)h(p.)34 b(386:)140 5194 y Fo(G)21 b Fr(=)312 5145
2289 y(3)p 312 5177 39 4 v 312 5244 a(8)403 5092 y Fm(1)374
2290 5117 y Fl(X)376 5291 y Fk(n)p Fn(=0)725 5145 y Fr(1)p
2291 516 5177 457 4 v 516 5190 a Fl(\000)554 5221 y Fn(2)p
2292 Fk(n)570 5280 y(n)623 5190 y Fl(\001)661 5255 y Fr(\(2)p
2293 Fo(n)d Fr(+)f(1\))938 5234 y Fn(2)999 5194 y Fr(+)1086
2294 5145 y Fo(\031)p 1086 5177 47 4 v 1090 5244 a Fr(8)1156
2295 5194 y(log)r(\(2)g(+)1417 5128 y Fj(p)p 1481 5128 39
2296 4 v 66 x Fr(3\))p -152 5304 797 4 v -65 5353 a Fg(2)-33
2297 5378 y Fe(A)23 b(sp)r(ecial)e(case)g(of)i([2)o(],)f(form)n(ula)g
2298 (\(5.5.18\),)g(with)g(N=163.)2152 -69 y Fr(The)c(series)g(is)g(summed)c
2299 (using)k(binary)e(splitting,)k(putting)d Fo(a)p Fr(\()p
2300 Fo(n)p Fr(\))k(=)2040 14 y(1,)44 b Fo(b)p Fr(\()p Fo(n)p
2301 Fr(\))h(=)f(2)p Fo(n)27 b Fr(+)g(1,)44 b Fo(p)p Fr(\(0\))g(=)g(1,)g
2302 Fo(q)s Fr(\(0\))h(=)g(1,)f(and)39 b Fo(p)p Fr(\()p Fo(n)p
2303 Fr(\))44 b(=)h Fo(n)p Fr(,)2040 97 y Fo(q)s Fr(\()p Fo(n)p
2304 Fr(\))40 b(=)g(2\(2)p Fo(n)26 b Fr(+)e(1\))37 b(for)h
2305 Fo(n)j(>)e Fr(0.)69 b(Th)n(us)37 b Fo(G)g Fr(can)g(b)r(e)g(computed)
2306 2040 180 y(with)26 b(bit)f(complexit)n(y)g Fo(O)r Fr(\(\(log)15
2307 b Fo(N)8 b Fr(\))3056 149 y Fn(2)3091 180 y Fo(M)g Fr(\()p
2308 Fo(N)g Fr(\)\).)2040 368 y Fp(2.6)82 b(Example:)37 b(The)27
2309 b(Gamma)h(function)f(at)h(rational)g(p)r(oints)2040 499
2310 y Fr(F)-6 b(or)31 b(ev)l(aluating)f(\000\()p Fo(s)p Fr(\))g(for)i
2311 (rational)f Fo(s)p Fr(,)h(w)n(e)f(\014rst)f(reduce)g
2312 Fo(s)g Fr(to)h(the)2040 582 y(range)38 b(1)i Fj(\024)g
2313 Fo(s)g Fj(\024)g Fr(2)e(b)n(y)e(the)g(form)n(ula)h(\000\()p
2314 Fo(s)25 b Fr(+)f(1\))40 b(=)g Fo(s)p Fr(\000\()p Fo(s)p
2315 Fr(\).)68 b(T)-6 b(o)2040 665 y(compute)28 b(\000\()p
2316 Fo(s)p Fr(\))g(with)h(a)g(precision)h(of)g Fo(N)37 b
2317 Fr(bits,)30 b(c)n(ho)r(ose)g(a)f(p)r(ositiv)n(e)2040
2318 758 y(in)n(teger)22 b Fo(x)f Fr(with)h Fo(xe)2606 726
2319 y Fm(\000)p Fk(x)2713 758 y Fo(<)f Fr(2)2832 726 y Fm(\000)p
2320 Fk(N)2939 758 y Fr(.)33 b(P)n(artial)23 b(in)n(tegration)f(lets)g(us)g
2321 (write)2075 1036 y(\000\()p Fo(s)p Fr(\))74 b(=)2428
2322 925 y Fl(Z)2511 946 y Fm(1)2474 1114 y Fn(0)2589 1036
2323 y Fo(e)2625 1000 y Fm(\000)p Fk(t)2700 1036 y Fo(t)2728
2324 1000 y Fk(s)p Fm(\000)p Fn(1)2840 1036 y Fo(dt)2293 1278
2325 y Fr(=)h Fo(x)2472 1242 y Fk(s)2505 1278 y Fo(e)2541
2326 1242 y Fm(\000)p Fk(x)2687 1177 y Fm(1)2657 1202 y Fl(X)2659
2327 1376 y Fk(n)p Fn(=0)3064 1230 y Fo(x)3108 1198 y Fk(n)p
2328 2800 1261 615 4 v 2800 1329 a Fo(s)p Fr(\()p Fo(s)16
2329 b Fr(+)h(1\))c Fj(\001)g(\001)g(\001)h Fr(\()p Fo(s)i
2330 Fr(+)h Fo(n)p Fr(\))3441 1278 y(+)3518 1167 y Fl(Z)3601
2331 1188 y Fm(1)3564 1356 y Fk(x)3679 1278 y Fo(e)3715 1242
2332 y Fm(\000)p Fk(t)3791 1278 y Fo(t)3819 1242 y Fk(s)p
2333 Fm(\000)p Fn(1)3930 1278 y Fo(dt)2040 1575 y Fr(The)36
2334 b(last)g(in)n(tegral)g(is)g Fo(<)h(xe)2916 1543 y Fm(\000)p
2335 Fk(x)3039 1575 y Fo(<)g Fr(2)3174 1543 y Fm(\000)p Fk(N)3280
2336 1575 y Fr(.)63 b(The)36 b(series)g(is)g(ev)l(alu-)2040
2337 1658 y(ated)h(as)g(a)g(h)n(yp)r(ergeometric)g(function)g(\(see)g(ab)r
2338 (o)n(v)n(e\);)42 b(the)37 b(n)n(um-)2040 1741 y(b)r(er)32
2339 b(of)h(terms)e(to)h(b)r(e)g(summed)d(up)i(is)i Fo(O)r
2340 Fr(\()p Fo(N)8 b Fr(\),)34 b(since)f Fo(x)e Fr(=)h Fo(O)r
2341 Fr(\()p Fo(N)8 b Fr(\).)2040 1824 y(Th)n(us)23 b(the)g(en)n(tire)g
2342 (computation)g(can)g(b)r(e)g(done)g(with)h(bit)f(complex-)2040
2343 1907 y(it)n(y)i Fo(O)r Fr(\(\(log)16 b Fo(N)8 b Fr(\))2487
2344 1875 y Fn(2)2522 1907 y Fo(M)g Fr(\()p Fo(N)g Fr(\)\).)2040
2345 2067 y Fh(Note:)2152 2226 y Fr(This)27 b(result)e(is)i(already)f(men)n
2346 (tioned)e(in)i([4)q(].)2152 2309 y(E.)e(Karatsuba)f([9)q(])g(extends)f
2347 (this)h(result)g(to)h(\000\()p Fo(s)p Fr(\))e(for)i(algebraic)2040
2348 2392 y Fo(s)p Fr(.)2152 2475 y(F)-6 b(or)35 b(\000\()p
2349 Fo(s)p Fr(\))g(there)f(is)i(no)f(c)n(hec)n(kp)r(oin)n(ting)f(p)r
2350 (ossible)i(b)r(ecause)g(of)2040 2558 y(the)25 b(dep)r(endency)g(on)g
2351 Fo(x)g Fr(in)h(the)f(binary)g(splitting.)2040 2746 y
2352 Fp(2.7)82 b(Example:)37 b(The)27 b(Riemann)h(Zeta)g(value)g
2353 Fo(\020)5 b Fr(\(3\))2040 2877 y(Recen)n(tly)-6 b(,)31
2354 b(Doron)g(Zeilb)r(erger's)i(metho)r(d)d(of)h(\\creativ)n(e)h(telescop-)
2355 2040 2960 y(ing")d(has)g(b)r(een)f(applied)h(to)g(Riemann's)e(zeta)i
2356 (function)g(at)g Fo(s)c Fr(=)h(3)2040 3043 y(\(see)g([1)q(]\),)g(whic)n
2357 (h)g(is)g(also)h(kno)n(wn)e(as)h Fi(Ap)n(\023)-37 b(ery's)29
2358 b(c)l(onstant)p Fr(:)2324 3276 y Fo(\020)5 b Fr(\(3\))22
2359 b(=)2574 3228 y(1)p 2574 3259 39 4 v 2574 3327 a(2)2665
2360 3174 y Fm(1)2635 3199 y Fl(X)2637 3374 y Fk(n)p Fn(=1)2778
2361 3227 y Fr(\()p Fj(\000)p Fr(1\))2936 3195 y Fk(n)p Fm(\000)p
2362 Fn(1)3056 3227 y Fr(\(205)p Fo(n)3246 3195 y Fn(2)3300
2363 3227 y Fj(\000)16 b Fr(160)p Fo(n)j Fr(+)e(32\))p 2778
2364 3259 961 4 v 3128 3349 a Fo(n)3174 3328 y Fn(5)3209 3284
2365 y Fl(\000)3247 3316 y Fn(2)p Fk(n)3262 3375 y(n)3316
2366 3284 y Fl(\001)3354 3301 y Fn(5)2152 3517 y Fr(This)22
2367 b(sum)e(consists)j(of)f(three)f(h)n(yp)r(ergeometric)g(series.)34
2368 b(Binary)2040 3600 y(splitting)k(can)g(also)h(b)r(e)f(applied)g
2369 (directly)-6 b(,)40 b(b)n(y)d(putting)g Fo(a)p Fr(\()p
2370 Fo(n)p Fr(\))k(=)2040 3683 y(205)p Fo(n)2200 3651 y Fn(2)2260
2371 3683 y Fr(+)23 b(250)p Fo(n)i Fr(+)e(77,)39 b Fo(b)p
2372 Fr(\()p Fo(n)p Fr(\))e(=)g(1,)h Fo(p)p Fr(\(0\))f(=)h(1,)g
2373 Fo(p)p Fr(\()p Fo(n)p Fr(\))f(=)g Fj(\000)p Fo(n)3871
2374 3651 y Fn(5)3941 3683 y Fr(for)2040 3766 y Fo(n)24 b(>)f
2375 Fr(0,)28 b(and)e Fo(q)s Fr(\()p Fo(n)p Fr(\))e(=)f(32\(2)p
2376 Fo(n)c Fr(+)f(1\))3035 3734 y Fn(5)3070 3766 y Fr(.)38
2377 b(Th)n(us)27 b(the)f(bit)h(complexit)n(y)f(of)2040 3849
2378 y(computing)f Fo(\020)5 b Fr(\(3\))26 b(is)g Fo(O)r Fr(\(\(log)15
2379 b Fo(N)8 b Fr(\))2991 3817 y Fn(2)3026 3849 y Fo(M)g
2380 Fr(\()p Fo(N)g Fr(\)\).)2040 4008 y Fh(Note:)2152 4168
2381 y Fr(Using)18 b(this)g(the)g(authors)g(w)n(ere)h(able)f(to)h(establish)
2382 g(a)f(new)g(record)2040 4251 y(in)29 b(the)f(calculation)i(of)f
2383 Fo(\020)5 b Fr(\(3\))28 b(b)n(y)g(computing)f(1,000,000)32
2384 b(decimals)2040 4334 y([17)q(].)45 b(The)29 b(computation)g(to)r(ok)g
2385 (8)g(hours)g(on)g(a)g(Hewlett)h(P)n(ac)n(k)l(ard)2040
2386 4417 y(9000/712)i(mac)n(hine.)41 b(After)29 b(distributing)f(on)g(a)h
2387 (cluster)g(of)g(4)g(HP)2040 4500 y(9000/712)e(mac)n(hines)d(the)f(same)
2388 h(computation)f(required)h(only)g(2.5)2040 4583 y(hours.)66
2389 b(The)37 b(half)g(hour)f(w)n(as)h(necessary)g(for)g(reading)g(the)f
2390 (par-)2040 4666 y(tial)28 b(results)f(from)f(disk)h(and)f(for)h(recom)n
2391 (bining)g(them.)36 b(Again,)27 b(w)n(e)2040 4749 y(ha)n(v)n(e)c(used)g
2392 (binary-splitting)g(for)h(recom)n(bining:)32 b(the)23
2393 b(4)h(partial)g(re-)2040 4832 y(sult)36 b(pro)r(duced)e(2)i(results)g
2394 (whic)n(h)g(w)n(ere)g(com)n(bined)e(to)i(the)f(\014nal)2040
2395 4915 y(1,000,000)29 b(decimals)d(v)l(alue)f(of)i Fo(\020)5
2396 b Fr(\(3\).)2152 4998 y(This)40 b(example)e(sho)n(ws)i(the)f(imp)r
2397 (ortance)g(of)h(c)n(hec)n(kp)r(oin)n(ting.)2040 5081
2398 y(Ev)n(en)30 b(if)h(a)g(mac)n(hine)f(crashes)h(through)f(the)g
2399 (calculation,)k(the)c(re-)2040 5164 y(sults)35 b(of)g(the)e(other)i
2400 (mac)n(hines)e(are)i(still)g(usable.)61 b(Additionally)-6
2401 b(,)2040 5247 y(b)r(eing)27 b(able)h(to)f(parallelise)i(the)d
2402 (computation)h(reduced)f(the)h(com-)2040 5330 y(puting)e(time)g
2403 (dramatically)-6 b(.)p eop
2404 %%Page: 5 5
2405 5 4 bop -152 -69 a Fp(3)83 b(Evaluation)28 b(of)g(linea)n(rly)g
2406 (convergent)f(series)g(of)h(sums)-152 88 y Fr(The)23
2407 b(tec)n(hnique)f(presen)n(ted)g(in)h(the)f(previous)h(section)g(also)h
2408 (applies)-152 171 y(to)i(all)h(linearly)f(con)n(v)n(ergen)n(t)f(sums)g
2409 (of)i(the)e(form)31 400 y Fo(U)30 b Fr(=)223 298 y Fm(1)194
2410 323 y Fl(X)196 497 y Fk(n)p Fn(=0)336 351 y Fo(a)p Fr(\()p
2411 Fo(n)p Fr(\))p 336 383 147 4 v 340 450 a Fo(b)p Fr(\()p
2412 Fo(n)p Fr(\))506 285 y Fl(\022)580 351 y Fo(c)p Fr(\(0\))p
2413 577 383 138 4 v 577 450 a Fo(d)p Fr(\(0\))742 400 y(+)17
2414 b Fj(\001)c(\001)g(\001)k Fr(+)1015 351 y Fo(c)p Fr(\()p
2415 Fo(n)p Fr(\))p 1012 383 146 4 v 1012 450 a Fo(d)p Fr(\()p
2416 Fo(n)p Fr(\))1168 285 y Fl(\023)1252 351 y Fo(p)p Fr(\(0\))12
2417 b Fj(\001)h(\001)h(\001)f Fo(p)p Fr(\()p Fo(n)p Fr(\))p
2418 1252 383 397 4 v 1254 450 a Fo(q)s Fr(\(0\))f Fj(\001)h(\001)h(\001)f
2419 Fo(q)s Fr(\()p Fo(n)p Fr(\))-152 644 y(where)35 b Fo(a)p
2420 Fr(\()p Fo(n)p Fr(\),)i Fo(b)p Fr(\()p Fo(n)p Fr(\),)g
2421 Fo(c)p Fr(\()p Fo(n)p Fr(\),)g Fo(d)p Fr(\()p Fo(n)p
2422 Fr(\),)g Fo(p)p Fr(\()p Fo(n)p Fr(\),)f Fo(q)s Fr(\()p
2423 Fo(n)p Fr(\))e(are)h(in)n(tegers)h(with)-152 727 y Fo(O)r
2424 Fr(\(log)16 b Fo(n)p Fr(\))27 b(bits.)38 b(The)27 b(most)f(often)i
2425 (used)e(case)i(is)f(again)h(that)f Fo(a)p Fr(\()p Fo(n)p
2426 Fr(\),)-152 810 y Fo(b)p Fr(\()p Fo(n)p Fr(\),)22 b Fo(c)p
2427 Fr(\()p Fo(n)p Fr(\),)f Fo(d)p Fr(\()p Fo(n)p Fr(\),)g
2428 Fo(p)p Fr(\()p Fo(n)p Fr(\),)g Fo(q)s Fr(\()p Fo(n)p
2429 Fr(\))f(are)h(p)r(olynomials)f(in)g Fo(n)g Fr(with)g(in)n(teger)-152
2430 893 y(co)r(e\016cien)n(ts.)-152 1052 y Fh(Algorithm:)-40
2431 1211 y Fr(Giv)n(en)29 b(t)n(w)n(o)g(index)g(b)r(ounds)f
2432 Fo(n)861 1219 y Fn(1)896 1211 y Fr(and)h Fo(n)1095 1219
2433 y Fn(2)1130 1211 y Fr(,)h(consider)f(the)g(partial)-152
2434 1294 y(sums)311 1407 y Fo(S)c Fr(=)542 1330 y Fl(X)464
2435 1505 y Fk(n)502 1515 y Fg(1)535 1505 y Fm(\024)p Fk(n<n)707
2436 1515 y Fg(2)763 1358 y Fo(a)p Fr(\()p Fo(n)p Fr(\))p
2437 763 1390 147 4 v 767 1458 a Fo(b)p Fr(\()p Fo(n)p Fr(\))929
2438 1358 y Fo(p)p Fr(\()p Fo(n)1044 1366 y Fn(1)1079 1358
2439 y Fr(\))12 b Fj(\001)i(\001)f(\001)g Fo(p)p Fr(\()p Fo(n)p
2440 Fr(\))p 929 1390 439 4 v 931 1458 a Fo(q)s Fr(\()p Fo(n)1044
2441 1466 y Fn(1)1079 1458 y Fr(\))f Fj(\001)i(\001)f(\001)g
2442 Fo(q)s Fr(\()p Fo(n)p Fr(\))-152 1628 y(and)-89 1828
2443 y Fo(U)29 b Fr(=)151 1752 y Fl(X)73 1926 y Fk(n)111 1936
2444 y Fg(1)144 1926 y Fm(\024)p Fk(n<n)316 1936 y Fg(2)372
2445 1779 y Fo(a)p Fr(\()p Fo(n)p Fr(\))p 372 1811 147 4 v
2446 376 1879 a Fo(b)p Fr(\()p Fo(n)p Fr(\))541 1713 y Fl(\022)616
2447 1779 y Fo(c)p Fr(\()p Fo(n)725 1787 y Fn(1)760 1779 y
2448 Fr(\))p 612 1811 181 4 v 612 1879 a Fo(d)p Fr(\()p Fo(n)728
2449 1887 y Fn(1)763 1879 y Fr(\))820 1828 y(+)17 b Fj(\001)c(\001)g(\001)k
2450 Fr(+)1093 1779 y Fo(c)p Fr(\()p Fo(n)p Fr(\))p 1090 1811
2451 146 4 v 1090 1879 a Fo(d)p Fr(\()p Fo(n)p Fr(\))1246
2452 1713 y Fl(\023)1330 1779 y Fo(p)p Fr(\()p Fo(n)1445 1787
2453 y Fn(1)1479 1779 y Fr(\))c Fj(\001)g(\001)g(\001)g Fo(p)p
2454 Fr(\()p Fo(n)p Fr(\))p 1330 1811 439 4 v 1332 1879 a
2455 Fo(q)s Fr(\()p Fo(n)1445 1887 y Fn(1)1479 1879 y Fr(\))g
2456 Fj(\001)g(\001)g(\001)g Fo(q)s Fr(\()p Fo(n)p Fr(\))-40
2457 2083 y(As)k(ab)r(o)n(v)n(e,)i(w)n(e)f(compute)e(the)h(in)n(tegers)h
2458 Fo(P)32 b Fr(=)21 b Fo(p)p Fr(\()p Fo(n)1382 2091 y Fn(1)1416
2459 2083 y Fr(\))13 b Fj(\001)g(\001)g(\001)g Fo(p)p Fr(\()p
2460 Fo(n)1676 2091 y Fn(2)1727 2083 y Fj(\000)k Fr(1\),)-152
2461 2166 y Fo(Q)24 b Fr(=)g Fo(q)s Fr(\()p Fo(n)130 2174
2462 y Fn(1)165 2166 y Fr(\))12 b Fj(\001)i(\001)f(\001)g
2463 Fo(q)s Fr(\()p Fo(n)423 2174 y Fn(2)475 2166 y Fj(\000)j
2464 Fr(1\))q(,)28 b Fo(B)g Fr(=)c Fo(b)p Fr(\()p Fo(n)948
2465 2174 y Fn(1)982 2166 y Fr(\))13 b Fj(\001)g(\001)g(\001)g
2466 Fo(b)p Fr(\()p Fo(n)1236 2174 y Fn(2)1288 2166 y Fj(\000)k
2467 Fr(1\),)28 b Fo(T)35 b Fr(=)24 b Fo(B)t(QS)t Fr(,)-152
2468 2286 y Fo(D)g Fr(=)d Fo(d)p Fr(\()p Fo(n)132 2294 y Fn(1)167
2469 2286 y Fr(\))12 b Fj(\001)i(\001)f(\001)g Fo(d)p Fr(\()p
2470 Fo(n)428 2294 y Fn(2)479 2286 y Fj(\000)k Fr(1\),)j Fo(C)26
2471 b Fr(=)21 b Fo(D)905 2196 y Fl(\020)967 2250 y Fk(c)p
2472 Fn(\()p Fk(n)1057 2260 y Fg(1)1090 2250 y Fn(\))p 965
2473 2269 151 4 v 965 2312 a Fk(d)p Fn(\()p Fk(n)1059 2322
2474 y Fg(1)1092 2312 y Fn(\))1143 2286 y Fr(+)16 b Fj(\001)e(\001)f(\001)k
2475 Fr(+)1415 2250 y Fk(c)p Fn(\()p Fk(n)1505 2260 y Fg(2)1538
2476 2250 y Fm(\000)p Fn(1\))p 1413 2269 230 4 v 1413 2312
2477 a Fk(d)p Fn(\()p Fk(n)1507 2322 y Fg(2)1540 2312 y Fm(\000)p
2478 Fn(1\))1652 2196 y Fl(\021)1719 2286 y Fr(and)-152 2403
2479 y Fo(V)63 b Fr(=)45 b Fo(D)r(B)t(QU)8 b Fr(.)78 b(If)40
2480 b Fo(n)545 2411 y Fn(2)607 2403 y Fj(\000)27 b Fo(n)740
2481 2411 y Fn(1)820 2403 y Fo(<)45 b Fr(4,)g(these)40 b(are)h(computed)d
2482 (di-)-152 2486 y(rectly)-6 b(.)70 b(If)38 b Fo(n)258
2483 2494 y Fn(2)318 2486 y Fj(\000)25 b Fo(n)449 2494 y Fn(1)525
2484 2486 y Fj(\025)41 b Fr(4,)g(they)c(are)h(computed)e(using)i
2485 Fi(binary)-152 2569 y(splitting)p Fr(:)47 b(Cho)r(ose)33
2486 b(an)f(index)f Fo(n)827 2577 y Fk(m)917 2569 y Fr(in)h(the)f(middle)g
2487 (of)h Fo(n)1548 2577 y Fn(1)1583 2569 y Fr(and)g Fo(n)1785
2488 2577 y Fn(2)1820 2569 y Fr(,)-152 2652 y(compute)27 b(the)h(comp)r
2489 (onen)n(ts)f Fo(P)771 2661 y Fk(l)796 2652 y Fr(,)i Fo(Q)907
2490 2661 y Fk(l)930 2652 y Fr(,)g Fo(B)1038 2661 y Fk(l)1063
2491 2652 y Fr(,)g Fo(T)1158 2661 y Fk(l)1182 2652 y Fr(,)g
2492 Fo(D)1295 2661 y Fk(l)1319 2652 y Fr(,)g Fo(C)1424 2661
2493 y Fk(l)1448 2652 y Fr(,)g Fo(V)1543 2661 y Fk(l)1596
2494 2652 y Fr(b)r(elong-)-152 2735 y(ing)40 b(to)f(the)g(in)n(terv)l(al)g
2495 Fo(n)580 2743 y Fn(1)659 2735 y Fj(\024)k Fo(n)h(<)g(n)1002
2496 2743 y Fk(m)1061 2735 y Fr(,)f(compute)38 b(the)h(comp)r(o-)-152
2497 2818 y(nen)n(ts)29 b Fo(P)104 2826 y Fk(r)138 2818 y
2498 Fr(,)h Fo(Q)250 2826 y Fk(r)284 2818 y Fr(,)g Fo(B)393
2499 2826 y Fk(r)428 2818 y Fr(,)f Fo(T)523 2826 y Fk(r)558
2500 2818 y Fr(,)h Fo(D)672 2826 y Fk(r)707 2818 y Fr(,)f
2501 Fo(C)812 2826 y Fk(r)847 2818 y Fr(,)h Fo(V)943 2826
2502 y Fk(r)1006 2818 y Fr(b)r(elonging)g(to)f(the)f(in)n(terv)l(al)-152
2503 2901 y Fo(n)-106 2909 y Fk(m)-16 2901 y Fj(\024)i Fo(n)h(<)f(n)287
2504 2909 y Fn(2)322 2901 y Fr(,)j(and)e(set)g Fo(P)41 b Fr(=)30
2505 b Fo(P)885 2910 y Fk(l)909 2901 y Fo(P)958 2909 y Fk(r)993
2506 2901 y Fr(,)j Fo(Q)d Fr(=)g Fo(Q)1289 2910 y Fk(l)1313
2507 2901 y Fo(Q)1374 2909 y Fk(r)1408 2901 y Fr(,)j Fo(B)h
2508 Fr(=)c Fo(B)1702 2910 y Fk(l)1727 2901 y Fo(B)1785 2909
2509 y Fk(r)1820 2901 y Fr(,)-152 2984 y Fo(T)36 b Fr(=)26
2510 b Fo(B)73 2992 y Fk(r)108 2984 y Fo(Q)169 2992 y Fk(r)203
2511 2984 y Fo(T)248 2993 y Fk(l)291 2984 y Fr(+)18 b Fo(B)427
2512 2993 y Fk(l)451 2984 y Fo(P)500 2993 y Fk(l)525 2984
2513 y Fo(T)570 2992 y Fk(r)604 2984 y Fr(,)29 b Fo(D)f Fr(=)e
2514 Fo(D)894 2993 y Fk(l)918 2984 y Fo(D)981 2992 y Fk(r)1016
2515 2984 y Fr(,)j Fo(C)i Fr(=)26 b Fo(C)1293 2993 y Fk(l)1317
2516 2984 y Fo(D)1380 2992 y Fk(r)1433 2984 y Fr(+)19 b Fo(C)1567
2517 2992 y Fk(r)1601 2984 y Fo(D)1664 2993 y Fk(l)1717 2984
2518 y Fr(and)-152 3067 y Fo(V)39 b Fr(=)21 b Fo(D)76 3075
2519 y Fk(r)111 3067 y Fo(B)169 3075 y Fk(r)203 3067 y Fo(Q)264
2520 3075 y Fk(r)299 3067 y Fo(V)344 3076 y Fk(l)385 3067
2521 y Fr(+)16 b Fo(D)524 3075 y Fk(r)559 3067 y Fo(C)614
2522 3076 y Fk(l)638 3067 y Fo(B)696 3076 y Fk(l)721 3067
2523 y Fo(P)770 3076 y Fk(l)794 3067 y Fo(T)839 3075 y Fk(r)890
2524 3067 y Fr(+)h Fo(D)1030 3076 y Fk(l)1055 3067 y Fo(B)1113
2525 3076 y Fk(l)1137 3067 y Fo(P)1186 3076 y Fk(l)1210 3067
2526 y Fo(V)1255 3075 y Fk(r)1289 3067 y Fr(.)-40 3150 y(Finally)-6
2527 b(,)37 b(this)c(algorithm)h(is)h(applied)e(to)h Fo(n)1272
2528 3158 y Fn(1)1342 3150 y Fr(=)g(0)g(and)g Fo(n)1712 3158
2529 y Fn(2)1781 3150 y Fr(=)-152 3233 y Fo(n)-106 3241 y
2530 Fn(max)48 3233 y Fr(=)i Fo(O)r Fr(\()p Fo(N)8 b Fr(\),)38
2531 b(and)c(\014nal)h(\015oating-p)r(oin)n(t)g(divisions)g
2532 Fo(S)41 b Fr(=)1760 3202 y Fk(T)p 1734 3216 97 4 v 1734
2533 3258 a(B)r(Q)-152 3341 y Fr(and)26 b Fo(U)j Fr(=)220
2534 3310 y Fk(V)p 170 3324 149 4 v 170 3366 a(D)r(B)r(Q)354
2535 3341 y Fr(are)d(p)r(erformed.)-152 3512 y Fh(Complexit)n(y:)-40
2536 3670 y Fr(The)k(bit)g(complexit)n(y)e(of)i(computing)f
2537 Fo(S)34 b Fr(and)29 b Fo(U)38 b Fr(with)30 b Fo(N)38
2538 b Fr(bits)-152 3753 y(of)27 b(precision)f(is)h Fo(O)r
2539 Fr(\(\(log)15 b Fo(N)8 b Fr(\))665 3722 y Fn(2)700 3753
2540 y Fo(M)g Fr(\()p Fo(N)g Fr(\)\).)-152 3912 y Fh(Pro)r(of:)-40
2541 4071 y Fr(By)21 b(our)f(assumption)g(that)g Fo(a)p Fr(\()p
2542 Fo(n)p Fr(\),)h Fo(b)p Fr(\()p Fo(n)p Fr(\),)h Fo(c)p
2543 Fr(\()p Fo(n)p Fr(\),)g Fo(d)p Fr(\()p Fo(n)p Fr(\),)g
2544 Fo(p)p Fr(\()p Fo(n)p Fr(\),)f Fo(q)s Fr(\()p Fo(n)p
2545 Fr(\))-152 4154 y(are)37 b(in)n(tegers)g(with)f Fo(O)r
2546 Fr(\(log)15 b Fo(n)p Fr(\))36 b(bits,)j(the)d(in)n(tegers)h
2547 Fo(P)11 b Fr(,)38 b Fo(Q)p Fr(,)g Fo(B)t Fr(,)h Fo(T)11
2548 b Fr(,)-152 4237 y Fo(D)r Fr(,)37 b Fo(C)5 b Fr(,)37
2549 b Fo(V)51 b Fr(b)r(elonging)35 b(to)g(the)e(in)n(terv)l(al)h
2550 Fo(n)1116 4245 y Fn(1)1187 4237 y Fj(\024)h Fo(n)h(<)f(n)1505
2551 4245 y Fn(2)1574 4237 y Fr(all)g(ha)n(v)n(e)-152 4320
2552 y Fo(O)r Fr(\(\()p Fo(n)14 4328 y Fn(2)69 4320 y Fj(\000)19
2553 b Fo(n)194 4328 y Fn(1)229 4320 y Fr(\))13 b(log)i Fo(n)430
2554 4328 y Fn(2)465 4320 y Fr(\))29 b(bits.)46 b(The)30 b(rest)f(of)h(the)f
2555 (pro)r(of)i(is)f(as)g(in)f(the)-152 4403 y(previous)d(section.)-152
2556 4562 y Fh(Chec)n(kp)r(oin)n(ting/P)n(arallelising:)-40
2557 4721 y Fr(A)f(c)n(hec)n(kp)r(oin)n(t)e(can)i(b)r(e)g(easily)h(done)e(b)
2558 n(y)g(storing)h(the)g(\(in)n(teger\))-152 4804 y(v)l(alues)20
2559 b(of)g Fo(n)198 4812 y Fn(1)233 4804 y Fr(,)h Fo(n)321
2560 4812 y Fn(2)356 4804 y Fr(,)g Fo(P)11 b Fr(,)20 b Fo(Q)p
2561 Fr(,)h Fo(B)t Fr(,)f Fo(T)30 b Fr(and)19 b(additionally)h
2562 Fo(D)r Fr(,)i Fo(C)5 b Fr(,)21 b Fo(V)c Fr(.)32 b(Simi-)-152
2563 4887 y(larly)-6 b(,)20 b(if)f Fo(m)e Fr(pro)r(cessors)i(are)g(a)n(v)l
2564 (ailable,)h(then)d(the)h(in)n(terv)l(al)f([0)p Fo(;)d(n)1692
2565 4895 y Fk(max)1820 4887 y Fr(])-152 4970 y(can)29 b(b)r(e)g(divided)f
2566 (in)n(to)h Fo(m)g Fr(pieces)g(of)h(length)f Fo(l)f Fr(=)e
2567 Fj(b)p Fo(n)1401 4978 y Fk(max)1529 4970 y Fo(=m)p Fj(c)p
2568 Fr(.)44 b(Af-)-152 5053 y(ter)34 b(eac)n(h)g(pro)r(cessor)h
2569 Fo(i)f Fr(has)g(computed)e(the)h(sum)f(of)j(its)f(in)n(terv)l(al)-152
2570 5136 y([)p Fo(il)q(;)14 b Fr(\()p Fo(i)9 b Fr(+)g(1\))p
2571 Fo(l)q Fr(],)23 b(the)e(partial)h(sums)f(are)h(com)n(bined)e(to)i(the)f
2572 (\014nal)g(result)-152 5219 y(using)26 b(the)g(rules)g(describ)r(ed)f
2573 (ab)r(o)n(v)n(e.)-152 5378 y Fh(Implemen)n(tation:)2152
2574 -69 y Fr(The)d(C++)f(implemen)n(tation)f(of)i(the)f(ab)r(o)n(v)n(e)h
2575 (algorithm)g(is)g(v)n(ery)2040 14 y(similar)30 b(to)g(the)f(previous)g
2576 (one.)46 b(The)29 b(initialisation)j(is)e(done)f(no)n(w)2040
2577 97 y(b)n(y)g(a)i(structure)f Fq(abpqcd)p 2793 97 24 4
2578 v 29 w(series)i Fr(con)n(taining)f(arra)n(ys)f Fq(a)p
2579 Fr(,)i Fq(b)p Fr(,)g Fq(p)p Fr(,)g Fq(q)p Fr(,)2040 180
2580 y Fq(c)26 b Fr(and)g Fq(d)g Fr(of)g(m)n(ultiprecision)g(in)n(tegers.)36
2581 b(The)26 b(v)l(alues)g(of)h(the)e(arra)n(ys)2040 263
2582 y(at)k(the)e(index)h Fo(n)h Fr(corresp)r(ond)f(to)h(the)f(v)l(alues)g
2583 (of)h(the)f(functions)h Fo(a)p Fr(,)2040 346 y Fo(b)p
2584 Fr(,)d Fo(p)p Fr(,)g Fo(q)s Fr(,)g Fo(c)g Fr(and)g Fo(d)f
2585 Fr(at)h(the)g(in)n(teger)g(p)r(oin)n(t)g Fo(n)p Fr(.)35
2586 b(The)26 b(\(partial\))h(results)2040 429 y(of)38 b(the)e(algorithm)i
2587 (are)f(stored)h(in)e(the)h Fq(abpqcd)p 3509 429 V 30
2588 w(series)p 3773 429 V 29 w(result)2040 512 y Fr(structure,)26
2589 b(whic)n(h)f(no)n(w)h(con)n(tains)h(3)f(new)g(elemen)n(ts)f(\()p
2590 Fq(C)p Fr(,)h Fq(D)f Fr(and)h Fq(V)p Fr(\).)2040 639
2591 y Fq(//)40 b(abpqcd_series)i(is)e(initialised)h(by)f(user)2040
2592 722 y(struct)h({)e(bigint)i(*a,)f(*b,)g(*p,)f(*q,)h(*c,)g(*d;)2315
2593 805 y(})f(abpqcd_series;)2040 971 y(//)h(abpqcd_series_result)k(holds)c
2594 (the)g(partial)h(results)2040 1054 y(struct)g({)e(bigint)i(P,)e(Q,)h
2595 (B,)g(T,)f(C,)h(D,)g(V;)2315 1137 y(})f(abpqcd_series_result;)2040
2596 1303 y(void)h(sum_abpqcd\(abpqcd_series_res)q(ult)46
2597 b(&)39 b(r,)2668 1386 y(int)h(n1,)g(int)f(n2,)2668 1469
2598 y(const)h(abpqcd_series)i(&)e(arg\))2040 1552 y({)2119
2599 1635 y(switch)g(\(n2)g(-)g(n1\))2119 1718 y({)2197 1801
2600 y(case)g(0:)2275 1884 y(error_handler\("summation)45
2601 b(device",)2511 1967 y("sum_abpqcd::)d(n2-n1)e(should)h(be)f(>)f
2602 (0."\);)2275 2050 y(break;)2197 2217 y(case)h(1:)g(//)g(the)f(result)i
2603 (at)f(the)g(point)g(n1)2275 2300 y(r.P)g(=)g(arg.p[n1];)2275
2604 2383 y(r.Q)g(=)g(arg.q[n1];)2275 2466 y(r.B)g(=)g(arg.b[n1];)2275
2605 2549 y(r.T)g(=)g(arg.a[n1])h(*)f(arg.p[n1];)2275 2632
2606 y(r.D)g(=)g(arg.d[n1];)2275 2715 y(r.C)g(=)g(arg.c[n1];)2275
2607 2798 y(r.V)g(=)g(arg.a[n1])h(*)f(arg.c[n1])h(*)e(arg.p[n1];)2275
2608 2881 y(break;)2197 3047 y(//)h(cases)g(2,)g(3,)f(4)h(left)g(out)g(for)g
2609 (simplicity)2197 3213 y(default:)h(//)f(general)h(case)2275
2610 3379 y(//)f(the)g(left)g(and)g(the)g(right)g(partial)h(sum)2275
2611 3462 y(abpqcd_series_result)j(L,)c(R;)2275 3628 y(//)g(find)g(the)g
2612 (middle)h(of)e(the)h(interval)2275 3711 y(int)g(nm)g(=)g(\(n1)f(+)h
2613 (n2\))g(/)f(2;)2275 3877 y(//)h(sum)g(left)g(side)2275
2614 3960 y(sum_abpqcd\(L,)j(n1,)d(nm,)f(arg\);)2275 4126
2615 y(//)h(sum)g(right)g(side)2275 4209 y(sum_abpqcd\(R,)j(nm,)d(n2,)f
2616 (arg\);)2275 4375 y(//)h(put)g(together)2275 4458 y(r.P)g(=)g(L.P)g(*)f
2617 (R.P;)2275 4541 y(r.Q)h(=)g(R.Q)g(*)f(L.Q;)2275 4624
2618 y(r.B)h(=)g(L.B)g(*)f(R.B;)2275 4707 y(bigint)i(tmp)f(=)f(L.B)h(*)g
2619 (L.P)g(*)f(R.T;)2275 4790 y(r.T)h(=)g(R.B)g(*)f(R.Q)h(*)g(L.T)f(+)h
2620 (tmp;)2275 4873 y(r.D)g(=)g(L.D)g(*)f(R.D;)2275 4956
2621 y(r.C)h(=)g(L.C)g(*)f(R.D)h(+)g(R.C)f(*)h(L.D;)2275 5039
2622 y(r.V)g(=)g(R.D)g(*)f(\(R.B)h(*)g(R.Q)g(*)f(L.V)h(+)g(L.C)f(*)h(tmp\))
2623 2511 5122 y(+)f(L.D)h(*)g(L.B)g(*)f(L.P)h(*)f(R.V;)2275
2624 5205 y(break;)2119 5288 y(})2040 5371 y(})p eop
2625 %%Page: 6 6
2626 6 5 bop -152 -69 a Fp(3.1)83 b(Example:)36 b(Euler's)27
2627 b(constant)g Fo(C)-152 63 y Fh(Theorem:)-40 222 y Fr(Let)32
2628 b Fo(f)8 b Fr(\()p Fo(x)p Fr(\))32 b(=)376 162 y Fl(P)464
2629 182 y Fm(1)464 249 y Fk(n)p Fn(=0)612 191 y Fk(x)647
2630 171 y Ff(n)p 606 205 89 4 v 606 249 a Fk(n)p Fn(!)662
2631 235 y Fg(2)736 222 y Fr(and)g Fo(g)s Fr(\()p Fo(x)p Fr(\))e(=)1158
2632 162 y Fl(P)1245 182 y Fm(1)1245 249 y Fk(n)p Fn(=0)1377
2633 222 y Fo(H)1441 230 y Fk(n)1500 191 y(x)1535 171 y Ff(n)p
2634 1493 205 V 1493 249 a Fk(n)p Fn(!)1549 235 y Fg(2)1592
2635 222 y Fr(.)53 b(Then)-152 331 y(for)27 b Fo(x)21 b Fj(!)g(1)p
2636 Fr(,)264 295 y Fk(g)r Fn(\()p Fk(x)p Fn(\))p 262 314
2637 119 4 v 262 357 a Fk(f)6 b Fn(\()p Fk(x)p Fn(\))412 331
2638 y Fr(=)503 301 y Fn(1)p 503 315 31 4 v 503 357 a(2)556
2639 331 y Fr(log)15 b Fo(x)i Fr(+)g Fo(C)22 b Fr(+)17 b Fo(O)1033
2640 266 y Fl(\000)1071 331 y Fo(e)1107 299 y Fm(\000)p Fn(4)1185
2641 263 y Fm(p)p 1235 263 35 3 v 36 x Fk(x)1274 266 y Fl(\001)1312
2642 331 y Fr(.)-152 506 y Fh(Pro)r(of:)-40 665 y Fr(The)27
2643 b(Laplace)h(metho)r(d)e(for)h(asymptotic)g(ev)l(aluation)g(of)g(exp)r
2644 (o-)-152 748 y(nen)n(tially)f(gro)n(wing)h(sums)e(and)g(in)n(tegrals)i
2645 (yields)257 942 y Fo(f)8 b Fr(\()p Fo(x)p Fr(\))21 b(=)g
2646 Fo(e)544 906 y Fn(2)574 870 y Fm(p)p 624 870 V 36 x Fk(x)663
2647 942 y Fo(x)707 906 y Fm(\000)765 887 y Fg(1)p 765 896
2648 29 4 v 765 927 a(4)872 894 y Fr(1)p 817 925 150 4 v 817
2649 995 a(2)855 941 y Fj(p)p 919 941 47 4 v 54 x Fo(\031)976
2650 942 y Fr(\(1)c(+)g Fo(O)r Fr(\()p Fo(x)1272 906 y Fm(\000)1330
2651 887 y Fg(1)p 1330 896 29 4 v 1330 927 a(4)1372 942 y
2652 Fr(\)\))-152 1148 y(and)-42 1327 y Fo(g)s Fr(\()p Fo(x)p
2653 Fr(\))i(=)i Fo(e)238 1291 y Fn(2)268 1255 y Fm(p)p 318
2654 1255 35 3 v 36 x Fk(x)357 1327 y Fo(x)401 1291 y Fm(\000)459
2655 1272 y Fg(1)p 459 1281 29 4 v 459 1313 a(4)566 1279 y
2656 Fr(1)p 511 1310 150 4 v 511 1380 a(2)549 1326 y Fj(p)p
2657 614 1326 47 4 v 614 1380 a Fo(\031)683 1237 y Fl(\020)742
2658 1279 y Fr(1)p 742 1310 39 4 v 742 1378 a(2)804 1327 y(log)15
2659 b Fo(x)h Fr(+)h Fo(C)22 b Fr(+)17 b Fo(O)r Fr(\(log)e
2660 Fo(x)i Fj(\001)g Fo(x)1552 1291 y Fm(\000)1610 1272 y
2661 Fg(1)p 1610 1281 29 4 v 1610 1313 a(4)1652 1327 y Fr(\))1682
2662 1237 y Fl(\021)-152 1555 y Fr(On)36 b(the)f(other)h(hand,)j
2663 Fo(h)p Fr(\()p Fo(x)p Fr(\))f(:=)884 1518 y Fk(g)r Fn(\()p
2664 Fk(x)p Fn(\))p 882 1538 119 4 v 882 1580 a Fk(f)6 b Fn(\()p
2665 Fk(x)p Fn(\))1046 1555 y Fr(satis\014es)38 b(the)d(di\013eren)n(tial)
2666 -152 1651 y(equation)53 1808 y Fo(xf)8 b Fr(\()p Fo(x)p
2667 Fr(\))16 b Fj(\001)h Fo(h)344 1772 y Fm(00)385 1808 y
2668 Fr(\()p Fo(x)p Fr(\))g(+)f(\(2)p Fo(xf)739 1772 y Fm(0)762
2669 1808 y Fr(\()p Fo(x)p Fr(\))g(+)h Fo(f)8 b Fr(\()p Fo(x)p
2670 Fr(\)\))17 b Fj(\001)g Fo(h)1237 1772 y Fm(0)1260 1808
2671 y Fr(\()p Fo(x)p Fr(\))j(=)h Fo(f)1510 1772 y Fm(0)1533
2672 1808 y Fr(\()p Fo(x)p Fr(\))-152 1966 y(hence)-152 2161
2673 y Fo(h)p Fr(\()p Fo(x)p Fr(\))g(=)108 2113 y(1)p 108
2674 2144 39 4 v 108 2212 a(2)169 2161 y(log)15 b Fo(x)p Fr(+)p
2675 Fo(C)5 b Fr(+)p Fo(c)538 2169 y Fn(2)585 2050 y Fl(Z)668
2676 2071 y Fm(1)631 2239 y Fk(x)834 2113 y Fr(1)p 756 2144
2677 196 4 v 756 2212 a Fo(tf)j Fr(\()p Fo(t)p Fr(\))917 2190
2678 y Fn(2)961 2161 y Fo(dt)21 b Fr(=)1141 2113 y(1)p 1141
2679 2144 39 4 v 1141 2212 a(2)1202 2161 y(log)15 b Fo(x)p
2680 Fr(+)p Fo(C)5 b Fr(+)p Fo(O)r Fr(\()p Fo(e)1664 2125
2681 y Fm(\000)p Fn(4)1742 2089 y Fm(p)p 1792 2089 35 3 v
2682 36 x Fk(x)1830 2161 y Fr(\))-152 2375 y Fh(Algorithm:)-40
2683 2535 y Fr(T)-6 b(o)26 b(compute)f Fo(C)31 b Fr(with)26
2684 b(a)g(precision)g(of)h Fo(N)34 b Fr(bits,)26 b(set)486
2685 2744 y Fo(x)21 b Fr(=)632 2654 y Fl(l)676 2744 y Fr(\()p
2686 Fo(N)k Fr(+)17 b(2\))964 2696 y(log)e(2)p 964 2727 151
2687 4 v 1020 2795 a(4)1125 2654 y Fl(m)1168 2664 y Fn(2)-152
2688 2939 y Fr(and)36 b(ev)l(aluate)h(the)e(series)j(for)f
2689 Fo(g)s Fr(\()p Fo(x)p Fr(\))d(and)i Fo(f)8 b Fr(\()p
2690 Fo(x)p Fr(\))36 b(sim)n(ultaneously)-6 b(,)-152 3022
2691 y(using)27 b(the)g(binary-splitting)g(algorithm,)h(with)f
2692 Fo(a)p Fr(\()p Fo(n)p Fr(\))c(=)g(1,)28 b Fo(b)p Fr(\()p
2693 Fo(n)p Fr(\))23 b(=)-152 3105 y(1,)f Fo(c)p Fr(\()p Fo(n)p
2694 Fr(\))g(=)f(1,)g Fo(d)p Fr(\()p Fo(n)p Fr(\))h(=)f Fo(n)5
2695 b Fr(+)g(1,)22 b Fo(p)p Fr(\()p Fo(n)p Fr(\))f(=)g Fo(x)p
2696 Fr(,)g Fo(q)s Fr(\()p Fo(n)p Fr(\))g(=)h(\()p Fo(n)5
2697 b Fr(+)g(1\))1490 3073 y Fn(2)1525 3105 y Fr(.)33 b(Let)20
2698 b Fo(\013)h Fr(=)-152 3188 y(3)p Fo(:)p Fr(591121477)c
2699 Fo(:)d(:)f(:)31 b Fr(b)r(e)f(the)h(solution)g(of)h(the)e(equation)h
2700 Fj(\000)p Fo(\013)12 b Fr(log)j Fo(\013)21 b Fr(+)-152
2701 3271 y Fo(\013)f Fr(+)f(1)29 b(=)e(0.)46 b(Then)29 b
2702 Fo(\013)508 3217 y Fj(p)p 573 3217 44 4 v 573 3271 a
2703 Fo(x)19 b Fj(\000)795 3241 y Fn(1)p 725 3255 171 4 v
2704 725 3297 a(4)11 b(log)h Fk(\013)919 3271 y Fr(log)1031
2705 3217 y Fj(p)p 1095 3217 44 4 v 54 x Fo(x)19 b Fr(+)g
2706 Fo(O)r Fr(\(1\))30 b(terms)f(of)h(the)-152 3380 y(series)d(su\016ce)f
2707 (for)h(the)e(relativ)n(e)h(error)g(to)g(b)r(e)g(b)r(ounded)e(b)n(y)h(2)
2708 1653 3349 y Fm(\000)p Fk(N)1759 3380 y Fr(.)-152 3540
2709 y Fh(Complexit)n(y:)-40 3699 y Fr(The)18 b(bit)f(complexit)n(y)f(of)h
2710 (this)h(algorithm)f(is)h Fo(O)r Fr(\(\(log)d Fo(N)8 b
2711 Fr(\))1564 3667 y Fn(2)1599 3699 y Fo(M)g Fr(\()p Fo(N)g
2712 Fr(\)\).)-152 3859 y Fh(Note:)-40 4018 y Fr(This)23 b(algorithm)f(w)n
2713 (as)g(\014rst)g(men)n(tioned)e(in)i([5].)34 b(It)21 b(is)h(b)n(y)f(far)
2714 h(the)-152 4101 y(fastest)27 b(kno)n(wn)e(algorithm)h(for)h(computing)e
2715 (Euler's)h(constan)n(t.)-40 4184 y(F)-6 b(or)33 b(Euler's)g(constan)n
2716 (t)g(there)f(is)h(no)g(c)n(hec)n(kp)r(oin)n(ting)f(p)r(ossible)-152
2717 4267 y(b)r(ecause)27 b(of)f(the)f(dep)r(endency)f(on)i
2718 Fo(x)f Fr(in)h(the)f(binary)g(splitting.)-152 4463 y
2719 Fp(4)83 b(Computational)27 b(results)-152 4620 y Fr(In)22
2720 b(this)g(section)g(w)n(e)h(presen)n(t)f(some)f(computational)h(results)
2721 h(of)g(our)-152 4703 y(CLN)f(and)g Fb(LiDIA)g Fr(implemen)n(tation)f
2722 (of)i(the)e(algorithms)i(presen)n(ted)-152 4786 y(in)j(this)g(note.)35
2723 b(W)-6 b(e)26 b(use)g(the)f(o\016cial)i(v)n(ersion)g(\(1.3\))f(and)g
2724 (an)g(exp)r(er-)-152 4869 y(imen)n(tal)33 b(v)n(ersion)f(\(1.4a\))i(of)
2725 g Fb(LiDIA)p Fr(.)e(W)-6 b(e)33 b(ha)n(v)n(e)f(tak)n(en)g(adv)l(an)n
2726 (tage)-152 4952 y(of)26 b Fb(LiDIA)p Fr('s)g(abilit)n(y)f(to)h(replace)
2727 g(its)g(k)n(ernel)f(\(m)n(ultiprecision)g(arith-)-152
2728 5035 y(metic)17 b(and)h(memory)e(managemen)n(t\))g([16)q(,)i(15)q(,)g
2729 (13],)i(so)f(w)n(e)f(w)n(ere)g(able)-152 5118 y(to)26
2730 b(use)g(in)f(b)r(oth)h(cases)h(CLN's)f(fast)h(in)n(teger)f(arithmetic)f
2731 (routines.)p 2040 -64 2022 4 v 2038 19 4 84 v 2085 -6
2732 a(D)p 2239 19 V 142 w(exp)o(\(1\))p 2545 19 V 90 w(log)r(\(2\))p
2733 2848 19 V 106 w Fo(\031)p 3151 19 V 259 w(C)p 3454 19
2734 V 248 w(G)p 3757 19 V 243 w(\020)5 b Fr(\(3\))p 4060
2735 19 V 2040 23 2022 4 v 2040 38 V 2038 126 4 89 v 2085
2736 101 a(10)2161 70 y Fn(2)p 2239 126 V 2286 101 a Fr(0.0005)p
2737 2545 126 V 94 w(0.0020)p 2848 126 V 92 w(0.0014)p 3151
2738 126 V 92 w(0.0309)p 3454 126 V 92 w(0.0179)p 3757 126
2739 V 92 w(0.0027)p 4060 126 V 2040 130 2022 4 v 2038 218
2740 4 89 v 2085 194 a(10)2161 162 y Fn(3)p 2239 218 V 2286
2741 194 a Fr(0.0069)p 2545 218 V 94 w(0.0474)p 2848 218 V
2742 92 w(0.0141)p 3151 218 V 92 w(0.8110)p 3454 218 V 92
2743 w(0.3580)p 3757 218 V 92 w(0.0696)p 4060 218 V 2040 222
2744 2022 4 v 2038 311 4 89 v 2085 286 a(10)2161 254 y Fn(4)p
2745 2239 311 V 2286 286 a Fr(0.2566)p 2545 311 V 94 w(1.9100)p
2746 2848 311 V 92 w(0.6750)p 3151 311 V 92 w(33.190)p 3454
2747 311 V 92 w(13.370)p 3757 311 V 92 w(2.5600)p 4060 311
2748 V 2040 314 2022 4 v 2038 403 4 89 v 2085 378 a(10)2161
2749 346 y Fn(5)p 2239 403 V 2286 378 a Fr(5.5549)p 2545 403
2750 V 94 w(45.640)p 2848 403 V 92 w(17.430)p 3151 403 V 92
2751 w(784.93)p 3454 403 V 92 w(340.33)p 3757 403 V 92 w(72.970)p
2752 4060 403 V 2040 406 2022 4 v 2040 549 a(Figure)35 b(1:)53
2753 b Fb(LiDIA-1.4a)34 b Fr(timings)h(of)g(computation)f(of)h(constan)n(ts)
2754 2040 632 y(using)26 b(binary-splitting)2040 955 y Fp(4.1)82
2755 b(Timings)2040 1087 y Fr(The)29 b(table)g(in)g(Figure)g(1)g(sho)n(ws)h
2756 (the)e(running)g(times)g(for)i(the)e(cal-)2040 1170 y(culation)37
2757 b(of)g(exp\(1\),)i(log)r(\(2\),)g Fo(\031)s Fr(,)g Fo(C)5
2758 b Fr(,)40 b Fo(G)c Fr(and)g Fo(\020)5 b Fr(\(3\))36 b(to)h(precision)
2759 2040 1253 y(100,)g(1000,)g(10000)e(and)e(100000)j(decimal)d(digits.)58
2760 b(The)34 b(timings)2040 1336 y(are)29 b(giv)n(en)e(in)h(seconds)h(and)f
2761 (they)f(denote)g(the)h Fi(r)l(e)l(al)h Fr(time)e(needed,)2040
2762 1419 y(i.e.)52 b(system)30 b(and)h(user)g(time.)50 b(The)31
2763 b(computation)g(w)n(as)h(done)f(on)2040 1502 y(an)26
2764 b(In)n(tel)f(P)n(en)n(tium)f(with)i(133Hz)h(and)e(32MB)i(of)g(RAM.)2152
2765 1585 y(The)20 b(second)f(table)h(\(Figure)g(2\))f(summarizes)g(the)f(p)
2766 r(erformance)2040 1668 y(of)25 b Fo(exp)p Fr(\()p Fo(x)p
2767 Fr(\))e(in)h(v)l(arious)h(Computer)f(Algebra)h(systems)3631
2768 1636 y Fn(3)3665 1668 y Fr(.)34 b(F)-6 b(or)25 b(a)f(fair)2040
2769 1751 y(comparison)h(of)h(the)f(algorithms,)h(b)r(oth)f(argumen)n(t)f
2770 (and)h(precision)2040 1834 y(are)d(c)n(hosen)g(in)f(suc)n(h)g(a)h(w)n
2771 (a)n(y)-6 b(,)22 b(that)g(system{sp)r(eci\014c)f(optimizations)2040
2772 1917 y(\(BCD)e(arithmetic)g(in)g(Maple,)i(FFT)f(m)n(ultiplication)f(in)
2773 g(CLN,)g(sp)r(e-)2040 2000 y(cial)j(exact)f(argumen)n(t)f(handling)h
2774 (in)g Fb(LiDIA)p Fr(\))f(do)h(not)g(w)n(ork.)33 b(W)-6
2775 b(e)21 b(use)2040 2096 y Fo(x)g Fr(=)g Fj(\000)2246 2034
2776 y(p)p 2310 2034 39 4 v 62 x Fr(2)j(and)f(precision)h(10)2913
2777 2064 y Fn(\()p Fk(i=)p Fn(3\))3050 2096 y Fr(,)h(with)e
2778 Fo(i)h Fr(running)f(from)g(4)h(to)g(15.)p 2106 2282 1862
2779 4 v 2104 2365 4 84 v 2322 2340 a(D)p 2424 2365 V 90 w(Maple)p
2780 2804 2365 V 173 w(P)n(ari)p 3222 2365 V 280 w Fb(LiDIA-1.3)p
2781 3624 2365 V 89 w Fr(CLN)p 3965 2365 V 2106 2368 1862
2782 4 v 2106 2383 V 2104 2466 4 84 v 2304 2441 a(21)p 2424
2783 2466 V 91 w(0.00090)p 2804 2466 V 130 w(0.00047)p 3222
2784 2466 V 170 w(0.00191)p 3624 2466 V 152 w(0.00075)p 3965
2785 2466 V 2106 2469 1862 4 v 2104 2552 4 84 v 2304 2527
2786 a(46)p 2424 2552 V 91 w(0.00250)p 2804 2552 V 130 w(0.00065)p
2787 3222 2552 V 170 w(0.00239)p 3624 2552 V 152 w(0.00109)p
2788 3965 2552 V 2106 2556 1862 4 v 2104 2639 4 84 v 2266
2789 2614 a(100)p 2424 2639 V 91 w(0.01000)p 2804 2639 V 130
2790 w(0.00160)p 3222 2639 V 170 w(0.00389)p 3624 2639 V 152
2791 w(0.00239)p 3965 2639 V 2106 2642 1862 4 v 2104 2725
2792 4 84 v 2266 2700 a(215)p 2424 2725 V 91 w(0.03100)p 2804
2793 2725 V 130 w(0.00530)p 3222 2725 V 170 w(0.00750)p 3624
2794 2725 V 152 w(0.00690)p 3965 2725 V 2106 2728 1862 4 v
2795 2104 2811 4 84 v 2266 2786 a(464)p 2424 2811 V 91 w(0.11000)p
2796 2804 2811 V 130 w(0.02500)p 3222 2811 V 170 w(0.02050)p
2797 3624 2811 V 152 w(0.02991)p 3965 2811 V 2106 2815 1862
2798 4 v 2104 2898 4 84 v 2227 2873 a(1000)p 2424 2898 V 92
2799 w(0.4000)p 2804 2898 V 168 w(0.2940)p 3222 2898 V 208
2800 w(0.0704)p 3624 2898 V 190 w(0.0861)p 3965 2898 V 2106
2801 2901 1862 4 v 2104 2984 4 84 v 2227 2959 a(2154)p 2424
2802 2984 V 92 w(1.7190)p 2804 2984 V 168 w(0.8980)p 3222
2803 2984 V 208 w(0.2990)p 3624 2984 V 190 w(0.2527)p 3965
2804 2984 V 2106 2987 1862 4 v 2104 3070 4 84 v 2227 3045
2805 a(4641)p 2424 3070 V 92 w(8.121)p 2804 3070 V 206 w(5.941)p
2806 3222 3070 V 246 w(1.510)p 3624 3070 V 228 w(0.906)p 3965
2807 3070 V 2106 3074 1862 4 v 2104 3157 4 84 v 2189 3132
2808 a(10000)p 2424 3157 V 92 w(39.340)p 2804 3157 V 168 w(39.776)p
2809 3222 3157 V 208 w(7.360)p 3624 3157 V 228 w(4.059)p 3965
2810 3157 V 2106 3160 1862 4 v 2104 3243 4 84 v 2189 3218
2811 a(21544)p 2424 3243 V 92 w(172.499)p 2804 3243 V 130
2812 w(280.207)p 3222 3243 V 170 w(39.900)p 3624 3243 V 190
2813 w(15.010)p 3965 3243 V 2106 3246 1862 4 v 2104 3329 4
2814 84 v 2189 3304 a(46415)p 2424 3329 V 92 w(868.841)p 2804
2815 3329 V 130 w(1972.184)p 3222 3329 V 132 w(129.000)p 3624
2816 3329 V 152 w(39.848)p 3965 3329 V 2106 3333 1862 4 v
2817 2104 3416 4 84 v 2151 3391 a(100000)p 2424 3416 V 92
2818 w(4873.829)p 2804 3416 V 92 w(21369.197)p 3222 3416 V
2819 94 w(437.000)p 3624 3416 V 152 w(106.990)p 3965 3416
2820 V 2106 3419 1862 4 v 2238 3579 a(Figure)i(2:)35 b(Timings)26
2821 b(of)g(computation)f(of)i(exp)o(\()p Fj(\000)3703 3517
2822 y(p)p 3767 3517 39 4 v 62 x Fr(2\))2152 3834 y(MapleV)39
2823 b(R3)g(is)g(the)f(slo)n(w)n(est)i(system)e(in)g(this)h(comparison.)2040
2824 3917 y(This)29 b(is)h(probably)e(due)g(to)h(the)f(BCD)i(arithmetic)e
2825 (it)h(uses.)44 b(Ho)n(w-)2040 4000 y(ev)n(er,)37 b(Maple)e(seems)g(to)f
2826 (ha)n(v)n(e)g(an)h(asymptotically)f(b)r(etter)h(algo-)2040
2827 4083 y(rithm)24 b(for)i Fo(exp)p Fr(\()p Fo(x)p Fr(\))c(for)k(n)n(um)n
2828 (b)r(ers)d(ha)n(ving)i(more)f(than)h(10000)i(deci-)2040
2829 4166 y(mals.)33 b(In)21 b(this)h(range)h(it)f(outp)r(erforms)g(P)n
2830 (ari-1.39.03,)k(whic)n(h)c(is)g(the)2040 4249 y(fastest)27
2831 b(system)e(in)g(the)h(0{200)h(decimals)f(range.)2152
2832 4332 y(The)18 b(comparison)f(indicating)h(the)f(strength)g(of)g
2833 (binary-splitting)2040 4415 y(is)28 b(b)r(et)n(w)n(een)e
2834 Fb(LiDIA-1.3)h Fr(and)g(CLN)g(itself.)40 b(Ha)n(ving)26
2835 b(the)h(same)f(k)n(er-)2040 4498 y(nel,)33 b(the)e(only)g(di\013erence)
2836 g(is)h(here)f(that)g Fb(LiDIA-1.3)g Fr(uses)h(Bren)n(t's)2040
2837 4581 y Fo(O)r Fr(\()2130 4527 y Fj(p)p 2195 4527 47 4
2838 v 2195 4581 a Fo(nM)8 b Fr(\()p Fo(n)p Fr(\)\))23 b(for)g(exp)o(\()p
2839 Fo(x)p Fr(\),)g(whereas)h(CLN)e(c)n(hanges)i(from)e(Bren)n(t's)2040
2840 4664 y(metho)r(d)j(to)g(a)h(binary-splitting)g(v)n(ersion)g(for)g
2841 (large)h(n)n(um)n(b)r(ers.)2152 4747 y(As)g(exp)r(ected)f(in)g(the)g
2842 (range)h(of)h(1000{100000)j(decimals)c(CLN)2040 4830
2843 y(outp)r(erforms)j Fb(LiDIA-1.3)g Fr(b)n(y)f(far.)49
2844 b(The)30 b(fact)h(that)f Fb(LiDIA-1.2.1)h Fr(is)2040
2845 4913 y(faster)d(in)f(the)f(range)h(of)h(200{1000)i(decimals)d(\(also)h
2846 (in)e(some)h(trig.)2040 4996 y(functions\))18 b(is)f(probably)g(due)f
2847 (to)i(a)f(b)r(etter)g(optimized)g Fo(O)r Fr(\()3708 4942
2848 y Fj(p)p 3772 4942 V 54 x Fo(n)q(M)8 b Fr(\()p Fo(n)p
2849 Fr(\)\))2040 5079 y(metho)r(d)25 b(for)h(exp)o(\()p Fo(x)p
2850 Fr(\).)p 2040 5124 797 4 v 2127 5173 a Fg(3)2159 5198
2851 y Fe(W)-5 b(e)15 b(do)h(not)g(list)g(the)g(timings)f(of)h
2852 Fd(LiDIA-1.4a)g Fe(since)f(these)g(are)h(comparable)2040
2853 5264 y(to)23 b(those)e(of)h(CLN.)p eop
2854 %%Page: 7 7
2855 7 6 bop -152 -69 a Fp(4.2)83 b(Distributed)27 b(computing)g(of)h
2856 Fo(\020)5 b Fr(\(3\))-152 63 y(Using)22 b(the)f(metho)r(d)g(describ)r
2857 (ed)g(in)h(2.7)h(the)e(authors)h(w)n(ere)g(the)f(\014rst)-152
2858 146 y(to)30 b(compute)f(1,000,000)k(decimals)c(of)i Fo(\020)5
2859 b Fr(\(3\))30 b([17)q(].)46 b(The)30 b(computa-)-152
2860 229 y(tion)d(to)r(ok)f(8)h(hours)f(on)g(a)g(Hewlett)h(P)n(ac)n(k)l(ard)
2861 f(9000/712)j(mac)n(hine.)-152 312 y(After)g(distributing)g(on)g(a)g
2862 (cluster)h(of)f(4)h(HP)e(9000/712)k(mac)n(hines)-152
2863 395 y(the)k(same)g(computation)f(required)g(only)h(2.5)h(hours.)66
2864 b(The)36 b(half)-152 478 y(hour)25 b(w)n(as)h(necessary)g(for)f
2865 (reading)h(the)f(partial)h(results)f(from)g(disk)-152
2866 561 y(and)37 b(for)g(recom)n(bining)g(them.)66 b(Again,)41
2867 b(w)n(e)c(ha)n(v)n(e)f(used)h(binary-)-152 644 y(splitting)e(for)g
2868 (recom)n(bining:)52 b(the)33 b(4)i(partial)g(result)g(pro)r(duced)e(2)
2869 -152 727 y(results)25 b(whic)n(h)f(w)n(ere)h(com)n(bined)e(to)h(the)g
2870 (\014nal)g(1,000,000)k(decimals)-152 810 y(v)l(alue)e(of)g
2871 Fo(\020)5 b Fr(\(3\).)-40 893 y(This)40 b(example)e(sho)n(ws)i(the)f
2872 (imp)r(ortance)g(of)h(c)n(hec)n(kp)r(oin)n(ting.)-152
2873 976 y(Ev)n(en)30 b(if)h(a)g(mac)n(hine)f(crashes)h(through)f(the)g
2874 (calculation,)k(the)c(re-)-152 1059 y(sults)35 b(of)g(the)f(other)g
2875 (mac)n(hines)f(are)i(still)g(usable.)61 b(Additionally)-6
2876 b(,)-152 1142 y(b)r(eing)27 b(able)h(to)f(parallelise)i(the)e
2877 (computation)f(reduced)g(the)h(com-)-152 1225 y(puting)f(time)e
2878 (dramatically)-6 b(.)-152 1413 y Fp(4.3)83 b(Euler's)26
2879 b(constant)h Fo(C)-152 1544 y Fr(W)-6 b(e)29 b(ha)n(v)n(e)g(implemen)n
2880 (ted)e(a)j(v)n(ersion)f(of)h(Bren)n(t's)g(and)f(McMillan's)-152
2881 1627 y(algorithm)j([5)q(])f(and)g(a)h(v)n(ersion)f(accelerated)i(b)n(y)
2882 d(binary-splitting)-152 1710 y(as)d(sho)n(wn)f(in)f(3.1.)-40
2883 1793 y(The)40 b(computation)e(of)i Fo(C)k Fr(w)n(as)c(done)e(t)n(wice)i
2884 (on)f(a)g(SP)-6 b(AR)n(C-)-152 1876 y(Ultra)36 b(mac)n(hine)e(with)h
2885 (167)h(MHz)f(and)g(256)h(MB)g(of)g(RAM.)f(The)-152 1959
2886 y(\014rst)23 b(computation)f(using)h(the)g(non-acellerated)g(v)n
2887 (ersion)g(required)-152 2042 y(160)j(hours.)34 b(The)24
2888 b(result)h(of)g(this)g(computation)e(w)n(as)j(then)d(v)n(eri\014ed)-152
2889 2125 y(b)n(y)i(the)g(binary)g(splitting)i(v)n(ersion)f(in)f(\(only\))h
2890 (14)g(hours.)-40 2208 y(The)h(\014rst)f(475006)j(partial)e(quotien)n
2891 (ts)f(of)i(the)e(con)n(tin)n(ued)f(frac-)-152 2291 y(tion)d(of)g
2892 Fo(C)k Fr(w)n(ere)c(computed)e(on)h(an)g(In)n(tel)g(P)n(en)n(tium)f
2893 (with)h(133)i(MHz)-152 2374 y(and)k(32)h(MB)g(of)f(RAM)g(in)g(3)g
2894 (hours)g(using)h(a)f(programm)f(b)n(y)g(H.)h(te)-152
2895 2457 y(Riele)33 b(based)g(on)g([6],)i(whic)n(h)e(w)n(as)h(translated)f
2896 (to)g Fb(LiDIA)f Fr(for)h(e\016-)-152 2540 y(ciency)j(reasons.)64
2897 b(Computing)35 b(the)g(475006th)i(con)n(v)n(ergen)n(t)e(pro-)-152
2898 2623 y(duced)25 b(the)g(follo)n(wing)k(impro)n(v)n(ed)24
2899 b(theorem:)-73 2756 y(If)i Fo(C)31 b Fr(is)26 b(a)g(rational)h(n)n(um)n
2900 (b)r(er,)d Fo(C)i Fr(=)21 b Fo(p=q)s Fr(,)26 b(then)f
2901 Fj(j)p Fo(q)s Fj(j)d Fo(>)f Fr(10)1574 2724 y Fn(244663)-40
2902 2889 y Fr(Details)33 b(of)e(this)h(computation)e(\(including)h
2903 (statistics)i(on)e(the)-152 2972 y(partial)c(quotien)n(ts\))e(can)h(b)r
2904 (e)f(found)h(in)f([14)r(].)-152 3167 y Fp(5)83 b(Conclusions)-152
2905 3324 y Fr(Although)17 b(p)r(o)n(w)n(erful,)j(the)d(binary)g(splitting)h
2906 (metho)r(d)e(has)i(not)f(b)r(een)-152 3407 y(widely)30
2907 b(used.)46 b(Esp)r(ecially)-6 b(,)32 b(no)e(information)g(existed)f(on)
2908 h(the)f(ap-)-152 3490 y(plicabilit)n(y)e(of)f(this)g(metho)r(d.)-40
2909 3573 y(In)c(this)h(note)g(w)n(e)g(presen)n(ted)f(a)g(generic)i
2910 (binary-splitting)e(sum-)-152 3656 y(mation)d(device)f(for)h(ev)l
2911 (aluating)g(t)n(w)n(o)g(t)n(yp)r(es)g(of)g(linearly)g(con)n(v)n(ergen)n
2912 (t)-152 3740 y(series.)35 b(F)-6 b(rom)22 b(this)i(w)n(e)f(deriv)n(ed)g
2913 (simple)g(and)g(computationally)g(ef-)-152 3823 y(\014cien)n(t)g
2914 (algorithms)g(for)h(the)e(ev)l(aluation)h(of)h(elemen)n(tary)d
2915 (functions)-152 3906 y(and)32 b(constan)n(ts.)52 b(These)32
2916 b(algorithms)h(w)n(ork)e(with)h Fi(exact)h Fr(ob)t(jects,)-152
2917 3989 y(making)22 b(them)f(suitable)h(for)h(use)g(within)f(Computer)f
2918 (Algebra)i(sys-)-152 4072 y(tems.)-40 4155 y(W)-6 b(e)36
2919 b(ha)n(v)n(e)g(sho)n(wn)h(that)f(the)g(practical)h(p)r(erformance)g(of)
2920 g(our)-152 4238 y(algorithms)e(is)f(sup)r(erior)g(to)g(curren)n(t)f
2921 (system)g(implemen)n(tations.)-152 4321 y(In)24 b(addition)g(to)g
2922 (existing)h(metho)r(ds,)f(our)g(algorithms)h(pro)n(vide)e(the)-152
2923 4404 y(p)r(ossibilit)n(y)d(of)g(c)n(hec)n(kp)r(oin)n(ting)e(and)h
2924 (parallelising.)34 b(These)20 b(features)-152 4487 y(can)33
2925 b(b)r(e)g(useful)g(for)h(h)n(uge)e(calculations,)37 b(suc)n(h)32
2926 b(as)h(those)g(done)g(in)-152 4570 y(analytic)27 b(n)n(um)n(b)r(er)c
2927 (theory)i(researc)n(h.)-152 4765 y Fp(6)83 b(Thanks)-152
2928 4922 y Fr(The)22 b(authors)h(w)n(ould)f(lik)n(e)g(to)g(thank)f(J\177)
2929 -38 b(org)23 b(Arndt,)f(for)h(p)r(oin)n(ting)f(us)-152
2930 5005 y(to)i(c)n(hapter)f(10)h(in)f([2)q(].)34 b(W)-6
2931 b(e)23 b(w)n(ould)h(also)g(lik)n(e)g(to)f(thank)g(Ric)n(hard)g(P)-6
2932 b(.)-152 5088 y(Bren)n(t)24 b(for)g(his)f(commen)n(ts)f(and)h(Hermann)f
2933 (te)h(Riele)h(for)g(pro)n(viding)-152 5171 y(us)35 b(his)g(program)g
2934 (for)g(the)g(con)n(tin)n(ued)f(fraction)i(computation)e(of)-152
2935 5254 y(Euler's)27 b(constan)n(t.)2040 -69 y Fp(References)2078
2936 88 y Fr([1])39 b Fa(Theodor)45 b(Amdeberhan,)j(and)c(Dor)n(on)j
2937 (Zeilber)n(ger)p Fr(.)2197 171 y(Acceleration)56 b(of)g(h)n(yp)r
2938 (ergeometric)f(series)h(via)f(the)g(WZ)2197 254 y(metho)r(d.)d(T)-6
2939 b(o)33 b(app)r(ear)g(in:)47 b(Electronic)33 b(Journal)g(of)g(Com)n(bi-)
2940 2197 337 y(natorics,)27 b(Wilf)g(F)-6 b(estsc)n(hrift)26
2941 b(V)-6 b(olume.)2078 480 y([2])39 b Fa(Jona)-5 b(than)25
2942 b(M.)g(Bor)-7 b(wein,)25 b(and)g(Peter)f(B.)h(Bor)-7
2943 b(wein)p Fr(.)29 b Fi(Pi)2197 563 y(and)f(the)g(A)n(GM)p
2944 Fr(.)35 b(Wiley)-6 b(,)26 b(1987.)2078 705 y([3])39 b
2945 Fa(Richard)48 b(P.)h(Brent)p Fr(.)95 b(The)46 b(complexit)n(y)f(of)i(m)
2946 n(ultiple-)2197 788 y(precision)i(arithmetic.)102 b Fi(Complexity)49
2947 b(of)f(Computational)2197 871 y(Pr)l(oblem)28 b(Solving)k
2948 Fr(\(1976\).)2078 1013 y([4])39 b Fa(Richard)21 b(P.)i(Brent)p
2949 Fr(.)h(F)-6 b(ast)20 b(m)n(ultiple-precision)g(ev)l(aluation)2197
2950 1096 y(of)f(elemen)n(tary)f(functions.)23 b Fi(Journal)e(of)g(the)h(A)n
2951 (CM)e Fh(23)f Fr(\(1976\),)2197 1179 y(242{251.)2078
2952 1322 y([5])39 b Fa(Richard)51 b(P.)h(Brent,)57 b(and)51
2953 b(Ed)n(win)h(M.)f(McMillan)p Fr(.)2197 1405 y(Some)37
2954 b(new)h(algorithms)h(for)g(high-precision)f(computation)2197
2955 1488 y(of)c(Euler's)g(constan)n(t.)57 b Fi(Mathematics)36
2956 b(of)f(Computation)g Fh(34)2197 1571 y Fr(\(1980\),)27
2957 b(305{312.)2078 1713 y([6])39 b Fa(Richard)i(P.)h(Brent,)j(Alf)d(v)-7
2958 b(an)42 b(der)f(Poor)-5 b(ten,)46 b(and)2197 1796 y(Hermann)40
2959 b(te)h(Riele)p Fr(.)71 b(A)37 b(comparativ)n(e)h(study)f(of)j(algo-)
2960 2197 1879 y(rithms)32 b(for)i(computing)e(con)n(tin)n(ued)g(fractions)j
2961 (of)f(algebraic)2197 1962 y(n)n(um)n(b)r(ers.)86 b(In)43
2962 b Fi(H.)g(Cohen)i(\(e)l(ditor\),)k(A)n(lgorithmic)44
2963 b(Num-)2197 2045 y(b)l(er)21 b(The)l(ory:)33 b(Se)l(c)l(ond)22
2964 b(International)f(Symp)l(osium,)h(ANTS-II)2197 2128 y
2965 Fr(\(1996\),)27 b(Springer)f(V)-6 b(erlag,)26 b(pp.)f(37{49.)2078
2966 2270 y([7])39 b Fa(Henri)47 b(Cohen,)53 b(C.)48 b(Ba)-5
2967 b(tut,)53 b(Dominique)48 b(Bernardi,)2197 2353 y(and)64
2968 b(M.)g(Olivier)p Fr(.)140 b(GP/P)-6 b(ARI)60 b(calculator)j({)f(V)-6
2969 b(er-)2197 2437 y(sion)49 b(1.39.03.)106 b(Av)l(ailable)49
2970 b(via)g(anon)n(ymous)e(FTP)j(from)2197 2520 y Fq
2971 (ftp://megrez.math.u-bordeaux.)q(fr)p Fr(,)32 b(1995.)2078
2972 2662 y([8])39 b Fa(Br)n(uno)136 b(Haible)p Fr(.)349 b(CLN,)131
2973 b(a)h(class)g(li-)2197 2745 y(brary)21 b(for)h(n)n(um)n(b)r(ers.)k(Av)l
2974 (ailable)c(via)f(anon)n(ymous)g(FTP)h(from)2197 2828
2975 y Fq(ftp://ftp.santafe.edu/pub/gnu)q(/cln.)q(tar.)q(gz)p
2976 Fr(,)32 b(1996.)2078 2970 y([9])39 b Fa(E.)33 b(A.)g(Kara)-5
2977 b(tsuba)p Fr(.)49 b(F)-6 b(ast)30 b(ev)l(aluation)h(of)g(transcenden)n
2978 (tal)2197 3053 y(functions.)k Fi(Pr)l(obl.)27 b(Per)l(e)l(dachi)i
2979 (Inform.)e Fh(27)f Fr(\(1991\),)h(76{99.)2040 3195 y([10])39
2980 b Fa(E.)32 b(A.)g(Kara)-5 b(tsuba)p Fr(.)45 b(On)29 b(fast)h
2981 (computation)f(of)h(transcen-)2197 3278 y(den)n(tal)i(functions.)54
2982 b Fi(Soviet)35 b(Math.)e(Dokl.)g Fh(43)g Fr(\(1991\),)i(693{)2197
2983 3362 y(694.)2040 3504 y([11])k Fa(E.)i(A.)g(Kara)-5 b(tsuba)p
2984 Fr(.)71 b(F)-6 b(ast)39 b(ev)l(aluation)f(of)h Fo(\020)5
2985 b Fr(\(3\).)72 b Fi(Pr)l(obl.)2197 3587 y(Per)l(e)l(dachi)29
2986 b(Inform.)d Fh(29)g Fr(\(1993\),)i(68{73.)2040 3729 y([12])39
2987 b Fa(E.)e(A.)g(Kara)-5 b(tsuba)p Fr(.)59 b(F)-6 b(ast)35
2988 b(calculation)g(of)g(the)f(Riemann)2197 3812 y(zeta)25
2989 b(function)g Fo(\020)5 b Fr(\()p Fo(s)p Fr(\))24 b(for)i(in)n(teger)f
2990 (v)l(alues)g(of)g(the)g(argumen)n(t)f Fo(s)p Fr(.)2197
2991 3895 y Fi(Pr)l(obl.)j(Inform.)g(T)-6 b(r)l(ansm.)28 b
2992 Fh(31)e Fr(\(1995\),)h(353{362.)2040 4037 y([13])39 b
2993 Fb(LiDIA)p Fa(-Gr)n(oup)p Fr(.)146 b Fb(LiDIA)63 b(1.3)g
2994 Fi({)f(a)g(libr)l(ary)h(for)f(c)l(om-)2197 4120 y(putational)76
2995 b(numb)l(er)h(the)l(ory)p Fr(.)191 b(T)-6 b(ec)n(hnisc)n(he)78
2996 b(Univ)n(er-)2197 4204 y(sit\177)-38 b(at)54 b(Darmstadt,)61
2997 b(1997.)120 b(Av)l(ailable)54 b(via)g(anon)n(ymous)2197
2998 4287 y(FTP)i(from)e Fq(ftp://ftp.informatik.tu-dar)q(mstad)q(t.de)2197
2999 4370 y(/pub/TI/systems/LiDIA)117 b Fr(or)c(via)g(WWW)g(from)2197
3000 4453 y Fq(http://www.informatik.tu-darm)q(stadt)q(.de/)q(TI/Li)q(DIA)p
3001 Fr(.)2040 4595 y([14])39 b Fa(Thomas)176 b(P)-7 b(ap)i(anik)n(ola)n(ou)
3002 p Fr(.)469 b(Homepage.)2197 4678 y Fq(http://www.math.u-bordeaux.fr)q
3003 (/~pap)q(anik)q Fr(.)2040 4820 y([15])39 b Fa(Thomas)45
3004 b(P)-7 b(ap)i(anik)n(ola)n(ou)p Fr(.)86 b Fi(Entwurf)43
3005 b(und)h(Entwicklung)2197 4903 y(einer)f(objektorientierten)i
3006 (Bibliothek)e(f)q(\177)-40 b(ur)42 b(algorithmische)2197
3007 4986 y(Zahlenthe)l(orie)p Fr(.)36 b(PhD)25 b(Thesis,)i(1997.)2040
3008 5129 y([16])39 b Fa(Thomas)54 b(P)-7 b(ap)i(anik)n(ola)n(ou,)63
3009 b(Ingrid)53 b(Biehl,)60 b(and)54 b(Jo-)2197 5212 y(hannes)21
3010 b(Buchmann)p Fr(.)h Fb(LiDIA)p Fr(:)d(a)h(library)f(for)i
3011 (computational)2197 5295 y(n)n(um)n(b)r(er)k(theory)-6
3012 b(.)39 b(SFB)28 b(124)g(rep)r(ort,)h(Univ)n(ersit\177)-38
3013 b(at)27 b(des)h(Saar-)2197 5378 y(landes,)e(1995.)p eop
3014 %%Page: 8 8
3015 8 7 bop -152 -69 a Fr([17])39 b Fa(Simon)195 b(Plouffe)p
3016 Fr(.)522 b(ISC:)190 b(In)n(v)n(erse)5 14 y(Sym)n(b)r(olic)21
3017 b(Calculator.)30 b(T)-6 b(ables)23 b(of)f(records)h(of)f(computation,)5
3018 97 y Fq(http://www.cecm.sfu.ca/proje)q(cts/I)q(SC/r)q(ecord)q(s2.h)q
3019 (tml)p Fr(.)-152 240 y([18])39 b Fa(Eugene)23 b(Salamin)p
3020 Fr(.)k(Computation)21 b(of)i Fo(\031)h Fr(using)d(arithmetic-)5
3021 323 y(geometric)50 b(mean.)104 b Fi(Mathematics)50 b(of)f(Computation)h
3022 Fh(30)5 406 y Fr(\(1976\),)27 b(565{570.)-152 549 y([19])39
3023 b Fa(Arnold)83 b(Sch)511 543 y(\177)508 549 y(onha)n(ge,)96
3024 b(and)82 b(V)n(olker)h(Strassen)p Fr(.)5 632 y(Sc)n(hnelle)38
3025 b(Multiplik)l(ation)g(gro\031er)h(Zahlen.)71 b Fi(Computing)39
3026 b Fh(7)5 715 y Fr(\(1971\),)27 b(281{292.)p eop
3027 %%Trailer
3028 end
3029 userdict /end-hook known{end-hook}if
3030 %%EOF