On 24 Apr 2017, at 13:00, Ankang Gao <gaoak@pku.edu.cn> wrote:

Hi,

I think it's not exactly the case Spencer said. Firstly, EPS=1.E-12 is small enough, and in addition the polynomial's first derivative almost vanishes there, so the numerical error caused by giving value 1 when fabs(dz)<EPS is very small.

I use a much larger threshold value(EPSERR=3.E-3 compared with EPS=2.E-12) when swapping to the classical form of the Lagrange polynomial, since double-precision floating-point format is be challenged in the original evaluation method  if fabs(dz) is too small (like 1.E-11) but still larger than EPS. In the classical Lagrange polynomial, the challenging to double-precision format is avoided since no subtraction of to very close float numbers appears there.  

Hui's code snippet is amazing(at least for me) and very accurate in calculating 1/(a-b) when 'a' and 'b' are very close, but I'm afraid it is too costly since log10 and pow functions are called, and at the same time its performance is not very stable when fabs(a-b) is not small enough, for example when a=1.1, and b=1.+ 1.E-11: direct calculate of 1/(a-b) gives 10.0000000009999912009561739978 which is acceptable, while Hui's code gives -100000000000 which is wrong.

best regard 

Ankang

-----原始邮件-----
发件人:"Sherwin, Spencer J" <s.sherwin@imperial.ac.uk>
发送时间:2017-04-24 17:31:20 (星期一)
收件人: "Xu, Hui" <hui.xu@imperial.ac.uk>
抄送: nektar-users <nektar-users@imperial.ac.uk>, "Ankang Gao" <gaoak@pku.edu.cn>
主题: Re: [Nektar-users] 2 bugs and 1 question

Hi Hui,

Thanks for this code snippet which is very interesting. 

However I think the issue here is that we are using a threshold where we set the value to 1 when a points z is near to zj within a prescribed tolerance EPS. However depending on this value of EPS we are giving a value of 1 when it should be 0.999999xxx.   In the code from Ankang I believe he is swapping to the classical form of the Lagrange polynomial when we are within this tolerance (@Ankang can you confirm I am correct). 

So although we could swap to the arithmetic you suggest below, will we not be left with the issue that we still need to evaluate the function when z = zj.  Will the approach below allow for this case? If not we might prefer to swap to Anyang’s suggested modification. 

Cheers,

Spencer

On 22 Apr 2017, at 15:28, Xu, Hui <hui.xu@imperial.ac.uk> wrote:

Hi All,

From Ankang’s report about polylib.cpp, it was said that

“Problem happens when z is very close to zj but their distance is still greater than EPS, since at this situation the formula in hj contains subtraction of two very close (double) float numbers, which causes heavy loss of accuracy."

It seemed that the double-precision floating-point format  was being challenged.

Double precision only gives 15-17 significant decimal digits precision.

(See wikipedia)

Sign bit: 1  bit

Exponent: 11 bits

Significant precision: 53 bits (52 explicitly stored) -> 15-17 significant decimal digits 

However, we could have a more professional way to circumvent/reduce "subtractive cancellation".

See the example given below and it can be a way to do some modifications.

The codes:

#include <iostream>

#include <limits.h>

#include <float.h>

#include <iomanip>

#include <cmath>

using namespace std;

typedef std::numeric_limits< double > dbl;

double scale=1e16;

int main(){

double a    = 1.111111112355343;

double b    = 1.111111112344554;

cout<< setprecision(30)<< a << endl << setprecision(30) << b <<endl;

double expn = floor(log10(abs(a-b)));  // Assuming that the two numbers are very close to each other and  both are greater than 1. The difference is approaching to the limitation of significant digits.

double a0 = a*pow(10,-expn-1)-floor(a*pow(10,-expn-1));  

double b0 = b*pow(10,-expn-1)-floor(b*pow(10,-expn-1));

double a1 = round(a0*pow(10,15+expn+1));                     // get rid of rubbish

double b1 = round(b0*pow(10,15+expn+1));                     // get rid of rubbish

cout<< setprecision(30)<< a1 << endl << setprecision(30) << b1 <<endl;

cout<<"The difference between two number : "<<endl<< setprecision(20)<< a-b <<endl;

cout<<"Reference Value :"<<endl

    <<"                  92686996014.459171378255630735" <<endl;

cout<<"Calculations    : "<<endl;

cout<<"Direct Calc     : "<<setprecision(30)<<1/(a-b)<<endl;

cout<<"Optimised Calc  : "<<setprecision(30)<<1e15/(a1-b1)<<endl;

return 0;

}

The output:

1.11111111235534298913307793555

1.11111111234455406382437558932

55343

44554

The difference between two number : 

1.0788925308702346229e-11

Reference Value: 92686996014.459171378255630735

Calculations       : 

Direct Calc         : 92687637682.8190765380859375

Optimised Calc  : 92686996014.45916748046875

Hope it could be useful.

Cheers,

Hui

On 21 Apr 2017, at 19:47, Ankang Gao <gaoak@pku.edu.cn> wrote:

Hi,

I use the same Lagrange interpolation polynomial as Nektar::LibUtilities::GaussPoints::LagrangePoly or Nektar::LibUtilities::PolyEPoints::LagrangePoly. The Polylib.cpp file I modified is attached, and the differences between this file and the original one are shown in another attachment 'difference.txt' which is obtained by command "diff" in linux OS.

After this modification, the main numerical error of funtion PhysEvaluate is contributed by NewtonIterationForLocCoord, which is 2.e-6 maximum in my tests with 12 points each direction in a quad geometry.

I'm glad to see that my report is of use to improve nektar++. In my opinion, Nektar++ is a wonderful code, and it has become the main numerical tool in my research. I am looking forward to nektar++ getting more and more powerful.

Ankang

-----原始邮件-----
发件人:"Sherwin, Spencer J" <s.sherwin@imperial.ac.uk>
发送时间:2017-04-21 22:51:00 (星期五)
收件人: "Ankang Gao" <gaoak@pku.edu.cn>
抄送: nektar-users <nektar-users@imperial.ac.uk>
主题: Re: [Nektar-users] 2 bugs and 1 question

Hi Ankang,

Sorry it has taken rather long to get back to you. Thanks for your clear explanation which I think is very interesting and we should do something about this. Do you have a modified polylib.cpp we can include.

I am not sure if the papers that Dave forwarded offer a more robust way of evaluating the lagging polynomial near the quadrature points since that is the only other modification I can think we might consider.

I am cc’ing Mike so he can also perhaps comments on this point.

Cheers,
Spencer.

> On 14 Apr 2017, at 13:06, Ankang Gao <gaoak@pku.edu.cn> wrote:
>
> Hellow,
>
> I have found two bugs in the codes, and I listed them in the attachment.
>
> I have a question about why the backward differentiation formula starts after m_pressureCalls>2, in function
> void Nektar::Extrapolate::AccelerationBDF(Array< OneD, Array< OneD, NekDouble > > & array).
> Since when m_pressureCalls=2, we can already get a first order accuracy result.
>
>
> thanks
>
> Ankang
> <reportbug_question.pdf>_______________________________________________
> Nektar-users mailing list
> Nektar-users@imperial.ac.uk
> https://mailman.ic.ac.uk/mailman/listinfo/nektar-users

Spencer  Sherwin
McLaren Racing/Royal Academy of Engineering Research Chair,
Professor of Computational Fluid Mechanics,
Department of Aeronautics,
Imperial College London
South Kensington Campus
London SW7 2AZ

s.sherwin@imperial.ac.uk
+44 (0) 20 759 45052

<Polylib.cpp><difference.txt>_______________________________________________
Nektar-users mailing list
Nektar-users@imperial.ac.uk
https://mailman.ic.ac.uk/mailman/listinfo/nektar-users

Spencer  Sherwin

McLaren Racing/Royal Academy of Engineering Research Chair, 

Professor of Computational Fluid Mechanics,

Department of Aeronautics,

Imperial College London

South Kensington Campus

London SW7 2AZ

s.sherwin@imperial.ac.uk

+44 (0) 20 759 45052