[Scilab-users] Numerical precision

[Claus Futtrup]

Hi there

OK, so I went ahead and took some Fortran input ... yes, I do get the 
same output as Fortran, so it seems there's no calculation error per se 
in Scilab. It's just tiny differences in the input which generate large 
differences in output. It's scary.

If you compare the input (dz1 in scilab vs. dz1 in fortran, you get 
differences of around :

1->dz1s./dz1f
  ans  =

     0.9998331 + 0.0000339i
     0.9998132 + 0.0000391i
     0.9998197 + 0.0000391i


And if you compare input for dz2, you get similar tiny differences :

-1->dz2s./dz2f
  ans  =

     0.9999629 + 0.0000126i
     0.9999600 + 0.0000145i
     0.9999635 + 0.0000145i

The difference in magnitude is on the scale of less than 0.02%.

Here's a complete script, including comments which shows the results and 
explains. I only include the three datapoints around fs, which are of 
interest:

// dz_test.sce

// Scilab input/output for zm_star:

     dz1s  =  [-0.503483777422459-0.5757953650438843*%i
-0.5114164682962974-0.5883274915921053*%i
-0.5189524114871444-0.6013228498050527*%i]

     dz2s  =  [-0.9956247120533783-1.1585555511721832*%i
-1.019729681961281-1.1864195137402458*%i
-1.0436439978573127-1.2149095747738672*%i]

     mu  =  1.997015667744342;

     zm_star  =  (1-mu)*  dz1s  .*  dz2s  ./  (dz2s-mu*dz1s);

// =
// 67.582546 - 57.549843i
// 104.56363 + 0.9610187i
// 64.846465 + 47.016802i

// Fortran input:

     dz1f  =  [-0.50358736782056468-0.57587442109900699*%i
-0.51153502148422891-0.58841744553307240*%i
-0.51906952093260994-0.60141095056474114*%i]

     dz2f  =  [-0.99567617243878548-1.1585859673221321*%i
-1.0197876382178386-1.1864521484345998*%i
-1.0436997434456954-1.2149387979780446*%i]

     zm_star  =  (1-mu)*  dz1f  .*  dz2f  ./  (dz2f-mu*dz1f);

// Scilab output:

// =
// 66.547108 - 58.427427i
// 105.67638 - 0.8443203i
// 66.018415 + 47.279403i

// Fortran output:
// ( 66.547108467527437 , -58.427426595183157 )
// ( 105.67638116311093 ,-0.84432029182632362 )
// ( 66.018415098214774 , 47.279402998470466 )


As can be seen. Scilab gives the same output as Fortran, when the input 
is the same.

I'm just totally surprised how such small differences in input can 
generate such large differences in output.

Best regards,
Claus

On 21-08-2016 16:11, Claus Futtrup wrote:
>
> Dear Scilab Users
>
> I have a script, which imitates a Fortran script. I can see there's a 
> difference in calculation of about 1%, which is very strange to me. 
> Input parameters (dz1 and dz2 vectors, length 1200) to the equation 
> seem to agree within 0,1% ... so right now my theory is that the 
> precision slips away when I do the following calculation:
>
> Scilab: zm_star = (1-mu)* dz1 .* dz2 ./ (dz2-mu*dz1); // model free 
> mech. impedance
>
> Fortran: zm_star = (1-mu)*dz1*dz2/(dz2-mu*dz1)
>
> Do you see anything in the Scilab formulation, which should worry me / 
> which would give me such a high error?
>
> P.S. mu is a mass-ratio = 1.9970156677443420... and it's exactly the 
> same value in both Scilab and Fortran.
>
> Most of the calculations in the output vector are OK, but in 
> particular around the resonance frequency, I can list the following 
> three datapoints (zm_star(47:49)), to show what is worrying me:
>
> In Fortran, data no. 47-48-49 (near fs):
>
>  (  66.547108467527437     , -58.427426595183157     )
>  (  105.67638116311093     ,-0.84432029182632362     )
>  (  66.018415098214774     ,  47.279402998470466     )
>
> Whereas in Scilab I get:
>
> 67.58254632254881 - 57.549843258298814*%i
> 104.56362768103634 + 0.9610187273575014*%i
> 64.84646498264745 + 47.01680213507681*%i
>
> In particular the imaginary part is different at the middle data point 
> (near fs) where from Fortran the value is negative, whereas in Scilab 
> the value is positive. It seems that the calculation of zm_star 
> involves some math operations that could be critical to the precision.
>
> If you study the impedance magnitude (pythagoras...), the results from 
> Fortran are about 1% higher in value - IMHO that's non-negligible.
>
> Studying the Nyquist circle plot of the data, it seems to me that in 
> general the Fortran calculation is more correct.
>
> What is going wrong with the Scilab equation? Please let me know if 
> you have any ideas how to increase the precision of the calculation. 
> Thanks.
>
> P.S. I'm using Scilab 5.5.0 (64 bit, Windows 10). Could it be the 
> Intel Math Kernel that's doing this wrong?
>
> /Claus
>

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