Science Forums 2

[modest]

Originally Posted by Don Blazys

This is exciting!

I agree

Using Qfwfq's derivative and newton's method (starting at 6.22) with perl at 100 digit accuracy I get:

6.220715628778645210593969670313416058685026190653 40698446526069757873158980184498130804275755047236 1

I'm out of my depth with this so this result should be verified. I'm also concerned and very confused that this converged after only 7 iterations. I wasn't expecting that at all. I used:

Algorithm:

use Math::BigFloat;

$n = 1;

my $x = Math::BigFloat->new(6.22 ,100);
my $half = Math::BigFloat->new(0.5 ,100);
my $one = Math::BigFloat->new(1 ,100);

while ($n < 10) {
    $xtohalf = $x->copy()->bpow($half);
    $logx = $x->copy()->blog();
    $loglogx = $logx->copy()->blog();
    $sinxtohalf = $xtohalf->copy()->bsin();
    $fx = $sinxtohalf->bsub($loglogx);
    $cosxtohalf = $xtohalf->copy()->bcos();
    $xtohalf2 = $xtohalf->copy()->bmul(2);
    $xlogx = $x->copy()->bmul($logx);
    $a = $cosxtohalf->bdiv($xtohalf2);
    $b = $one->copy()->bdiv($xlogx);
    $fpx = $a->bsub($b);
    $c = $fx->bdiv($fpx);
    $x->bsub($c);


print "iteration number $n: $x \n";
$n += 1;
    }

Output:

iteration number 1: 6.2207156179471774117867122228858042459075718269744372662836
23662266398210344786538184650666024140324
iteration number 2: 6.2207156287786452081126066015652927240261786788708637739507
49260445300617582198175750501665456398001
iteration number 3: 6.2207156287786452105939696703134160585548006057806747212612
23028733807513931628148855900138989610367
iteration number 4: 6.2207156287786452105939696703134160586850261906534069844652
60697578731589801486300349586510071804013
iteration number 5: 6.2207156287786452105939696703134160586850261906534069844652
60697578731589801844981308042757550472360
iteration number 6: 6.2207156287786452105939696703134160586850261906534069844652
60697578731589801844981308042757550472363
iteration number 7: 6.2207156287786452105939696703134160586850261906534069844652
60697578731589801844981308042757550472361
iteration number 8: 6.2207156287786452105939696703134160586850261906534069844652
60697578731589801844981308042757550472361
iteration number 9: 6.2207156287786452105939696703134160586850261906534069844652
60697578731589801844981308042757550472361

~modest