Re: [Boost-users] limited accuracy for tanh-sinh integrator

21 Aug 2020

      *First problem - more precision*
Check the stopping condition of boost.math, it says the tolerance is
\sqrt{\epsilon}. So if you use 100-digits precision, then tolerance is
roughly 10^-50. For more precision, you have to use more digits such as
1000 digits by using BF= number >; . Full code:

#include<iostream>
#include
#include 

using namespace std;
using namespace boost::math::quadrature;
using namespace boost::multiprecision;
using BF= number >;

int main(){
auto f2 = [](BF x) { return x/BF(sqrt(x*x-BF("0.25"))); };
tanh_sinh<BF> integrator;

std::cout << std::setprecision(std::numeric_limits<BF>::max_digits10)
     << "Comp.: " << integrator.integrate(f2, BF("0.5"), sqrt(BF("1.25")))
<< std::endl;

    return 0;
};

Be patient, this run about 10 mins in my single core server.

*Second problem - error*
I didn't reproduce it. Just modify my code and everything is OK.

On Fri, Aug 21, 2020 at 9:18 AM Anirban Pal via Boost-users <
boost-users@lists.boost.org> wrote:
...
Thank you for your response. I’ve run into a few more issues with
accuracy. Here is my function, which has a singularity at 0.5 and I'm
integrating it from 0.5 to sqrt(1.25):
using BF = boost::multiprecision::cpp_bin_float_100;
auto f1 = [](BF x) { return x/BF(sqrt(x*x-BF("0.25"))); };
BF Q1 = integrator.integrate(f1, BF("0.5"), BF(sqrt(BF("1.25"))));
This gives me a value of Q1 which is close to the true value of 1 with an
error of O(10^-51). I am curious if this should be lower, like O(10^-100).
(I hope I’m not being greedy!)
Nevertheless, if I try a related function.
auto f2 = [](BF x) { return x/BF(sqrt(x*x-BF("0.25"))) - x; };
BF Q2 = integrator.integrate(f2, BF("0.5"), BF(sqrt(BF("1.25"))));
or an equivalent one:
auto f3 = [](BF x) { return (x - x*BF(sqrt(x*x-BF("0.25"))) )/BF(sqrt(x*x-BF("0.25"))); };
BF Q3 = integrator.integrate(f3, BF("0.5"), BF(sqrt(BF("1.25"))));
I get an error message:
boost_test5: /usr/local/include/boost/math/quadrature/tanh_sinh.hpp:196: boost::math::quadrature::tanh_sinh::integrate(F, Real, Real, Real, Real*, Real*, std::size_t*):: [with F = main(int, char**)::; Real = boost::multiprecision::number >; Policy = boost::math::policies::policy; result_type = boost::multiprecision::number >]: Assertion `position != a' failed.
Aborted (core dumped)
I'm not sure how I could improve my code. I tried using
boost::multiprecision::sqrt but i didn't notice any difference.
Thank you for your attention.
On Mon, 17 Aug 2020 at 04:47, Paul A. Bristow via Boost-users
boost-users@lists.boost.org http://mailto:boost-users@lists.boost.org
wrote:
...
...
-----Original Message-----
From: Boost-users  On Behalf Of
John Maddock via Boost-users
Sent: 16 August 2020 21:24
To: Anirban Pal via Boost-users 
Cc: John Maddock 
Subject: Re: [Boost-users] limited accuracy for tanh-sinh integrator
On 16/08/2020 21:15, Anirban Pal via Boost-users wrote:
...
Hello,
I’m using the tanh-sinh integrator to integrate a simple function f(x)
= 0.26*x from 3.0 to 4.0.
The exact result is 0.91. With the integrator I’m getting a result
accurate to only 10^-18 with cpp_bin_float_100 multiprecision.
You have the double precision constant 0.26 in your code, and since
this is an inexact binary value,
everything that depends on that constant is inherently limited to
double precision.  Try constructing it
as Real(26) / 100.
Everyone falls into this pit, and in my case, repeatedly ☹
You need to be ever vigilant not to use double precision items by mistake.
Using numeric_limits<>::max_digits10 is useful because it shows the
digits beyond double at about 17 decimal digits.  Being random should raise
suspicions.
If you are only using multiprecision types, then always using decimal
digit strings like "1.23456" is simple.
If you can use a fraction like Real(1)/1000 that also works fine.
You may also find the macro BOOST_MATH_TEST_VALUE  in
I:\boost\libs\math\test\test_value.hpp helpful if you want to make code
portable over different floating-point types from fundamental to
multiprecision?
Paul
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Anirban Pal
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Re: [Boost-users] limited accuracy for tanh-sinh integrator

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