Re: Truncation Precision in PARI

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

* Peter Pein [2018-07-09 14:17]:
> that is not very accurate :(
> 
> I get:
> 
> precision(-1 - Euler + 3/2*log(2*Pi) + 6*zetahurwitz(-1, 1, der = 1), 100)
> %1 =
> 0.1870730725097797894509591576777666319578148029622159376465535484192711630046534855901322306210633101

Please check ??precision :
[...] If n is smaller than the precision of a t_REAL component of x,  it is
truncated, otherwise it is extended with zeros.
[...]

What you wrote first evaluates the expression to the default accuracy
(38 decimal digits) then pads the result with trailing zeros.

What you intended is achieved as follows:

1) Changing the global default accuracy
? \p100
  realprecision = 115 significant digits (100 digits displayed)

? -1 - Euler + 3/2*log(2*Pi) + 6*zetahurwitz(-1, 1, 1)
  time = 11 ms.
%2 = 0.1870730725097797894509591576777666319578148029622159376465535484192711630046534855901322306210633101

2) OR [usually better in programs] changing locally the accuracy in the
dynamic scope.

? localprec(100); z = -1 - Euler + 3/2*log(2*Pi) + 6*zetahurwitz(-1, 1, 1)
%1 = 0.18707307250977978945095915767776663196

This still prints results at the default accuracy, but precision(z) is 115
as requested and formatted printing indeed shows the expected digits:

? printf("%.100f", z)
0.1870730725097797894509591576777666319578148029622159376465535484192711630046534855901322306210633101


Cheers,

    K.B.

P.S. Note that your second solution using sumalt is not rigorous in general
(although rigorous error bounds can be given for specific cases), see ??sumalt

In this case, it gives an essentially correct answer in this case, but 2
decimal digits are wrong at \p115 :

? \p115
? sumalt(n=2,(-1)^n*(n-1)/(n+2)*zeta(n))
%3 = 0.1870730725097797894509591576777666319578148029622159376465535484192711630046534855901322306210633101211227714967803
                                        ^^ wrong

? -1 - Euler + 3/2*log(2*Pi) + 6*zetahurwitz(-1, 1, der = 1)
%4 = 0.1870730725097797894509591576777666319578148029622159376465535484192711630046534855901322306210633101211227714967866
                                        ^^ OK !

? \p120
   realprecision = 134 significant digits (120 digits displayed)
? sumalt(n=2,(-1)^n*(n-1)/(n+2)*zeta(n))
%4 = 0.187073072509779789450959157677766631957814802962215937646553548419271163004653485590132230621063310121122771496786571185

--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
`