// catalanconst().
// General includes.
#include "cl_sysdep.h"
// Specification.
#include "cl_F_tran.h"
// Implementation.
#include "cln/lfloat.h"
#include "cl_LF_tran.h"
#include "cl_LF.h"
#include "cln/integer.h"
#include "cl_alloca.h"
namespace cln {
const cl_LF compute_catalanconst_ramanujan (uintC len)
{
// [Jonathan M. Borwein, Peter B. Borwein: Pi and the AGM.
// Wiley 1987. Section 11.3, exercise 16 g, p. 386]
// G = 3/8 * sum(n=0..infty, n!^2 / (2n+1)!*(2n+1))
// + pi/8 * log(2+sqrt(3)).
// Every summand gives 0.6 new decimal digits in precision.
// The sum is best evaluated using fixed-point arithmetic,
// so that the precision is reduced for the later summands.
var uintL actuallen = len + 2; // 2 Schutz-Digits
var sintL scale = intDsize*actuallen;
var cl_I sum = 0;
var cl_I n = 0;
var cl_I factor = ash(1,scale);
while (!zerop(factor)) {
sum = sum + truncate1(factor,2*n+1);
n = n+1;
factor = truncate1(factor*n,2*(2*n+1));
}
var cl_LF fsum = scale_float(cl_I_to_LF(sum,actuallen),-scale);
var cl_LF g =
scale_float(The(cl_LF)(3*fsum)
+ The(cl_LF)(pi(actuallen))
* ln(cl_I_to_LF(2,actuallen)+sqrt(cl_I_to_LF(3,actuallen))),
-3);
return shorten(g,len); // verkürzen und fertig
}
// Bit complexity (N := len): O(N^2).
const cl_LF compute_catalanconst_ramanujan_fast (uintC len)
{
// Same formula as above, using a binary splitting evaluation.
// See [Borwein, Borwein, section 10.2.3].
var uintL actuallen = len + 2; // 2 Schutz-Digits
// Evaluate a sum(0 <= n < N, a(n)/b(n) * (p(0)...p(n))/(q(0)...q(n)))
// with appropriate N, and
// a(n) = 1, b(n) = 2*n+1,
// p(n) = n for n>0, q(n) = 2*(2*n+1) for n>0.
var uintL N = (intDsize/2)*actuallen;
// 4^-N <= 2^(-intDsize*actuallen).
CL_ALLOCA_STACK;
var cl_I* bv = (cl_I*) cl_alloca(N*sizeof(cl_I));
var cl_I* pv = (cl_I*) cl_alloca(N*sizeof(cl_I));
var cl_I* qv = (cl_I*) cl_alloca(N*sizeof(cl_I));
var uintL n;
init1(cl_I, bv[0]) (1);
init1(cl_I, pv[0]) (1);
init1(cl_I, qv[0]) (1);
for (n = 1; n < N; n++) {
init1(cl_I, bv[n]) (2*n+1);
init1(cl_I, pv[n]) (n);
init1(cl_I, qv[n]) (2*(2*n+1));
}
var cl_pqb_series series;
series.bv = bv;
series.pv = pv; series.qv = qv; series.qsv = NULL;
var cl_LF fsum = eval_rational_series(N,series,actuallen);
for (n = 0; n < N; n++) {
bv[n].~cl_I();
pv[n].~cl_I();
qv[n].~cl_I();
}
var cl_LF g =
scale_float(The(cl_LF)(3*fsum)
+ The(cl_LF)(pi(actuallen))
* ln(cl_I_to_LF(2,actuallen)+sqrt(cl_I_to_LF(3,actuallen))),
-3);
return shorten(g,len); // verkürzen und fertig
}
// Bit complexity (N := len): O(log(N)^2*M(N)).
const cl_LF compute_catalanconst_expintegral1 (uintC len)
{
// We compute f(x) classically and g(x) using the partial sums of f(x).
var uintC actuallen = len+2; // 2 Schutz-Digits
var uintL x = (uintL)(0.693148*intDsize*actuallen)+1;
var uintL N = (uintL)(2.718281828*x);
var cl_LF fterm = cl_I_to_LF(1,actuallen);
var cl_LF fsum = fterm;
var cl_LF gterm = fterm;
var cl_LF gsum = gterm;
var uintL n;
// After n loops
// fterm = x^n/n!, fsum = 1 + x/1! + ... + x^n/n!,
// gterm = S_n*x^n/n!, gsum = S_0*x^0/0! + ... + S_n*x^n/n!.
for (n = 1; n < N; n++) {
fterm = The(cl_LF)(fterm*x)/n;
fsum = fsum + fterm;
gterm = The(cl_LF)(gterm*x)/n;
if (evenp(n))
gterm = gterm + fterm/square((cl_I)(2*n+1));
else
gterm = gterm - fterm/square((cl_I)(2*n+1));
gsum = gsum + gterm;
}
var cl_LF result = gsum/fsum;
return shorten(result,len); // verkürzen und fertig
}
// Bit complexity (N = len): O(N^2).
// Same algorithm as expintegral1, but using binary splitting to evaluate
// the sums.
const cl_LF compute_catalanconst_expintegral2 (uintC len)
{
var uintC actuallen = len+2; // 2 Schutz-Digits
var uintL x = (uintL)(0.693148*intDsize*actuallen)+1;
var uintL N = (uintL)(2.718281828*x);
CL_ALLOCA_STACK;
var cl_pqd_series_term* args = (cl_pqd_series_term*) cl_alloca(N*sizeof(cl_pqd_series_term));
var uintL n;
for (n = 0; n < N; n++) {
if (n==0) {
init1(cl_I, args[n].p) (1);
init1(cl_I, args[n].q) (1);
} else {
init1(cl_I, args[n].p) (x);
init1(cl_I, args[n].q) (n);
}
init1(cl_I, args[n].d) (evenp(n)
? square((cl_I)(2*n+1))
: -square((cl_I)(2*n+1)));
}
var cl_LF result = eval_pqd_series(N,args,actuallen);
for (n = 0; n < N; n++) {
args[n].p.~cl_I();
args[n].q.~cl_I();
args[n].d.~cl_I();
}
return shorten(result,len); // verkürzen und fertig
}
// Bit complexity (N = len): O(log(N)^2*M(N)).
// Using Cohen-Villegas-Zagier acceleration, but without binary splitting.
const cl_LF compute_catalanconst_cvz1 (uintC len)
{
var uintC actuallen = len+2; // 2 Schutz-Digits
var uintL N = (uintL)(0.39321985*intDsize*actuallen)+1;
#if 0
var cl_LF fterm = cl_I_to_LF(2*(cl_I)N*(cl_I)N,actuallen);
var cl_LF fsum = fterm;
var cl_LF gterm = fterm;
var cl_LF gsum = gterm;
var uintL n;
// After n loops
// fterm = (N+n)!N/(2n+2)!(N-n-1)!*2^(2n+2), fsum = ... + fterm,
// gterm = S_n*fterm, gsum = ... + gterm.
for (n = 1; n < N; n++) {
fterm = The(cl_LF)(fterm*(2*(cl_I)(N-n)*(cl_I)(N+n)))/((cl_I)(2*n+1)*(cl_I)(n+1));
fsum = fsum + fterm;
gterm = The(cl_LF)(gterm*(2*(cl_I)(N-n)*(cl_I)(N+n)))/((cl_I)(2*n+1)*(cl_I)(n+1));
if (evenp(n))
gterm = gterm + fterm/square((cl_I)(2*n+1));
else
gterm = gterm - fterm/square((cl_I)(2*n+1));
gsum = gsum + gterm;
}
var cl_LF result = gsum/(cl_I_to_LF(1,actuallen)+fsum);
#else
// Take advantage of the fact that fterm and fsum are integers.
var cl_I fterm = 2*(cl_I)N*(cl_I)N;
var cl_I fsum = fterm;
var cl_LF gterm = cl_I_to_LF(fterm,actuallen);
var cl_LF gsum = gterm;
var uintL n;
// After n loops
// fterm = (N+n)!N/(2n+2)!(N-n-1)!*2^(2n+2), fsum = ... + fterm,
// gterm = S_n*fterm, gsum = ... + gterm.
for (n = 1; n < N; n++) {
fterm = exquopos(fterm*(2*(cl_I)(N-n)*(cl_I)(N+n)),(cl_I)(2*n+1)*(cl_I)(n+1));
fsum = fsum + fterm;
gterm = The(cl_LF)(gterm*(2*(cl_I)(N-n)*(cl_I)(N+n)))/((cl_I)(2*n+1)*(cl_I)(n+1));
if (evenp(n))
gterm = gterm + cl_I_to_LF(fterm,actuallen)/square((cl_I)(2*n+1));
else
gterm = gterm - cl_I_to_LF(fterm,actuallen)/square((cl_I)(2*n+1));
gsum = gsum + gterm;
}
var cl_LF result = gsum/cl_I_to_LF(1+fsum,actuallen);
#endif
return shorten(result,len); // verkürzen und fertig
}
// Bit complexity (N = len): O(N^2).
// Using Cohen-Villegas-Zagier acceleration, with binary splitting.
const cl_LF compute_catalanconst_cvz2 (uintC len)
{
var uintC actuallen = len+2; // 2 Schutz-Digits
var uintL N = (uintL)(0.39321985*intDsize*actuallen)+1;
CL_ALLOCA_STACK;
var cl_pqd_series_term* args = (cl_pqd_series_term*) cl_alloca(N*sizeof(cl_pqd_series_term));
var uintL n;
for (n = 0; n < N; n++) {
init1(cl_I, args[n].p) (2*(cl_I)(N-n)*(cl_I)(N+n));
init1(cl_I, args[n].q) ((cl_I)(2*n+1)*(cl_I)(n+1));
init1(cl_I, args[n].d) (evenp(n)
? square((cl_I)(2*n+1))
: -square((cl_I)(2*n+1)));
}
var cl_pqd_series_result sums;
eval_pqd_series_aux(N,args,sums);
// Here we need U/(1+S) = V/D(Q+T).
var cl_LF result =
cl_I_to_LF(sums.V,actuallen) / The(cl_LF)(sums.D * cl_I_to_LF(sums.Q+sums.T,actuallen));
for (n = 0; n < N; n++) {
args[n].p.~cl_I();
args[n].q.~cl_I();
args[n].d.~cl_I();
}
return shorten(result,len); // verkürzen und fertig
}
// Bit complexity (N = len): O(log(N)^2*M(N)).
// Timings of the above algorithms, on an i486 33 MHz, running Linux.
// N ram ramfast exp1 exp2 cvz1 cvz2
// 10 0.055 0.068 0.32 0.91 0.076 0.11
// 25 0.17 0.26 0.95 3.78 0.23 0.43
// 50 0.43 0.73 2.81 11.5 0.63 1.36
// 100 1.32 2.24 8.82 34.1 1.90 4.48
// 250 6.60 10.4 48.7 127.5 10.3 20.8
// 500 24.0 30.9 186 329 38.4 58.6
// 1000 83.0 89.0 944 860 149 163
// 2500 446 352 6964 3096 1032 545
// 5000 1547 899
// asymp. N^2 FAST N^2 FAST N^2 FAST
// (FAST means O(log(N)^2*M(N)))
//
// The "exp1" and "exp2" algorithms are always about 10 to 15 times slower
// than the "ram" and "ramfast" algorithms.
// The break-even point between "ram" and "ramfast" is at about N = 1410.
const cl_LF compute_catalanconst (uintC len)
{
if (len >= 1410)
return compute_catalanconst_ramanujan_fast(len);
else
return compute_catalanconst_ramanujan(len);
}
// Bit complexity (N := len): O(log(N)^2*M(N)).
const cl_LF catalanconst (uintC len)
{
var uintC oldlen = TheLfloat(cl_LF_catalanconst)->len; // vorhandene Länge
if (len < oldlen)
return shorten(cl_LF_catalanconst,len);
if (len == oldlen)
return cl_LF_catalanconst;
// TheLfloat(cl_LF_catalanconst)->len um mindestens einen konstanten Faktor
// > 1 wachsen lassen, damit es nicht zu häufig nachberechnet wird:
var uintC newlen = len;
oldlen += floor(oldlen,2); // oldlen * 3/2
if (newlen < oldlen)
newlen = oldlen;
// gewünschte > vorhandene Länge -> muß nachberechnen:
cl_LF_catalanconst = compute_catalanconst(newlen);
return (len < newlen ? shorten(cl_LF_catalanconst,len) : cl_LF_catalanconst);
}
} // namespace cln
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