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Metamaterial Edge and Bandpass Filters

jameszbw

https://www.jobs.ac.uk/job/CDG262/metamaterial-edge-and-bandpass-filters

Bangor University

Metamaterial Edge and Bandpass Filters

Bangor University – School of Computer Science and Electronic Engineering

Location:Bangor
Salary:£11,586 annual stipend
Hours:Full Time
Contract Type:Fixed-Term/Contract
Placed On:18th December 2020
Closes:11th January 2021
Job Ref:BUK2217

This project will be based in the School of Computer Science and Electronic Engineering, in collaboration with Qioptiq Ltd.

Project ID:  BUK2217

Annual Stipend: £11,586

Application Deadline: 11/01/2021

Interviews: 14/01/2021

We are seeking a highly capable and motivated graduate for this exciting opportunity to undertake a Research Masters. This is a collaborative project between Bangor University and leading photonics company Qioptiq Ltd.

This project aims at developing next-generation filters based on metamaterial design and technology, i.e. artificially engineered material with unique properties not found in nature materials. The project will focus on the development of edge and bandpass filters which are building elements of many optical products and systems, whose main function is to limit stray light found in optical systems, especially in underwater environments.

This Research Masters project is expected to start in January 2021 immediately after the interview and will take one year to complete. The KESS 2 scholar will be based in the School of Computer Science and Electronic Engineering (CSEE) and will be supervised by academic supervisors Dr James Wang and Dr Liyang Yue, and the industrial supervisor Dr. James Monks of Qioptiq. The applicant should hold a good degree (at least 2:2) in engineering, physics, optics, materials or a related scientific discipline, have demonstrably excellent research skills, and relevant experience.

For more information, or informal enquiries, please email Dr James Wang : [email protected]

To apply, please send your CV and a cover letter to [email protected] and cc to Penny Dowdney [email protected]

Knowledge Economy Skills Scholarships (KESS 2) is a pan-Wales higher level skills initiative led by Bangor University on behalf of the HE sector in Wales. It is part funded by the Welsh Government’s European Social Fund (ESF) convergence programme for West Wales and the Valleys.

Due to ESF funding, eligibility restrictions apply to this scholarship. To be eligible, the successful candidate will need to be resident in the Convergence Area of Wales on University registration, and must have the right to work in the region on qualification. 

www.kess2.ac.uk

AI: Circle fill

jameszbw

https://www.youtube.com/watch?v=QieXB5JHXLk

This Tutorial will show you How to Fill Your Design with Random Objects and Color using Script in Adobe Illustrator. Welcome Again, In this video, I will show you how to use three useful illustrator scripts to fill your design with objects or colors randomly. here is the script.

Place scripts in folder: C:\Program Files\Adobe\Adobe Illustrator CC 2019\Presets\en_GB\Scripts

1. Circle Fill The script fills outline shapes with packed circles. Script Link: http://www.jongware.com/binaries/Circ…

2. Replace with Symbol This script can exchange several objects with a symbol selected from the symbol panel in Illustrator. Script Link: http://bcrockett.com/blog/2016/12/cha…

3. Random Swatches Fill fill your artwork with random color Script Link : https://github.com/robotwood/Random-S… Mock Up in The Video https://www.behance.net/gallery/10249…

Hope you enjoy this tutorial, Don’t forget to leave a like and comment. Share this tutorial if you think this tutorial useful. About me – Website http://www.nobudesign.space – Instagram https://www.instagram.com/iwainobuyuki/ – Facebook http://www.facebook.com/nobuyuki006 – Background Music by LAKEY INSPIRED – That Girl https://www.youtube.com/watch?v=RlVfB… LAKEY INSPIRED channel https://www.youtube.com/channel/UCOmy…

Accuracy of Bessel Functions in MATLAB

jameszbw

by PAVEL HOLOBORODKO on MAY 12, 2016

Accuracy of Bessel Functions in MATLAB

Contents [hide]

Introduction

In previous posts we studied accuracy of computation of modified Bessel functions: K1(x), K0(x), I0(x) and I1(x). In spite of the fact that modified Bessel functions are easy to compute (they are monotonous and do not cross x-axis) we saw that MATLAB provides accuracy much lower than expected for double precision. Please refer to the pages for more details.

Today we will investigate how accurately MATLAB computes Bessel functions of the first and second kind Yn(x) and Jn(x) in double precision. Along the way, we will also check accuracy of the commonly used open source libraries.

BesselJ Functions (1st Kind, n=0,1,2)

Having extended precision routines, accuracy check of any function needs only few commands:

% Check accuracy of sine using 1M random points on (0, 16]
>> mp.Digits(34);
>> x = 16*(rand(1000000,1));
 
>> f = sin(x);       % call built-in function for double precision sine
>> y = sin(mp(x));   % compute sine values in quadruple precision
>> z = abs(1-f./y);  % element-wise relative error
 
>> max(z./eps)       % max. relative error in terms of 'eps'
ans = 
    0.5682267303295349594044472141263213
 
>> -log10(max(z))    % number of correct decimal digits in result
ans = 
    15.89903811472552788729580391380739

Computed sine values have ~15.9 correct decimal places and differ from ‘true’ function values by ~0.5eps. This is the highest accuracy we can expect from double precision floating-point arithmetic.

The recent revision of IEEE 754 standard[1] recommends all elementary functions to have such highest accuracy and be correctly rounded: the result should be as if the calculation had been performed in infinite precision, and then rounded (see Table 9.1 of the standard). Computing elementary functions with correct rounding is a fascinating topic on its own, please refer to pioneering works of Gal[2, 3], Ziv[4], recent developments by Lefèvre, Muller, Zimmermann, et al.[5–8] and excellent handbooks on the subject[9–11].

Most modern programming languages follow the IEEE 754-2008 recommendation and their standard libraries[12,13,14] compute elementary functions with correct rounding and guaranteed accuracy as we see in test above.

However special functions are not covered by the standard and software libraries for its computations are considered to be faithfully-accurate. Here we study what libraries in fact can be trusted in computing Bessel functions in double precision.

MATLAB 2016a

Basic accuracy check applied for Bessel function of the second kind – Y0(x):

% Check accuracy of bessely(0,x) using 1M random points on (0, 16]
>> f = bessely(0,x);     % built-in double precision routine
>> y = bessely(0,mp(x)); % 'true' values computed using quadruple precision
>> z = abs(1-f./y);
 
>> max(z./eps)           % max. relative error in terms of 'eps'
ans = 
    1366834.014340578752744356956505545
 
>> -log10(max(z))        % number of correct decimal digits in result
ans = 
    9.517843996632803435430170368588769

The problematic areas can be located using error scatter plot. Red lines show the expected limits of relative error (for convenience, we added another interval to cover whole x axis):

The spikes of error clearly coincide with zeros of Bessel function (0.8935, 3.9576, 7.0860, ... ). To test accuracy near zeros in more detail we use the following function:

function func_accuracy_check(func, x0, near_zero, n)
%func_accuracy_check - Checks accuracy of 'func' near its zero or extremum.
%       
%  1. Routine searches for zero/extremum of 'func' starting from 'x0'
%     Extended precision is used to be sure zero/extremum is accurate. 
%    
%  2. Relative error of 'func' is evaluated using 'n' points in close 
%     vicinity of found zero/extremum. 
%
%  Parameters:
%   func - function handle 
%   x0 - starting point for zero/extremum search
%   near_zero - true/false, check accuracy near zero/extremum
%   n - number of test points.
%
%  Usage example. 
%    Find zero of Y0(x) near x = 2 and check accuracy of the MATLAB 
%    function bessely(0,x) near found zero:
%
%    mp.Digits(34);
%    u = @(x) bessely(0,x);
%    func_accuracy_check(u,mp(2),true,3)
%
%    Output:
%    Accuracy check near zero, x = 8.9357696627916752e-01:
%    x = 8.9357696627916761e-01	correct digits =   1.5	rel. error = 1.40e+14eps
%    x = 8.9357696627916772e-01	correct digits =   0.5	rel. error = 1.50e+15eps
%    x = 8.9357696627916783e-01	correct digits =   1.0	rel. error = 4.13e+14eps
 
% Find root/extremum of 'func' using extended precision:
options = optimset('TolX',mp('eps')); 
if near_zero
    r = fzero(func,mp(x0),options);
    fprintf('Accuracy check near zero, x = %.16e:\n',r);    
else
    %g = @(x) -func(x);    
    r = fminsearch(func,mp(x0),options);
    fprintf('Accuracy check near extremum, x = %.16e:\n',r);    
end;
 
% Correctly rounded zero/extremum in double precision:
x = double(r); 
for k=1:n
    % Get next floating-point value after x:
    x = nextafter(x);
 
    f = func(x);      % function value computed by MATLAB's built-in bessely    
    y = func(mp(x));  % 'true' function value computed in extended precision
 
    z = abs(1-f./y);  % relative accuracy
    fprintf('x = %.16e\tcorrect digits = %4.1f\trel. error = %5.2geps\n',x,-log10(z),z./eps);  
end

The nextafter(x) function computes next floating-point value just after x[15] towards infinity. We use it to generate test arguments in close neighborhood of given point.

Accuracy of bessely(0,x) near first zero of Y0(x):

>> mp.Digits(34);
>> u = @(x) bessely(0,x);
>> func_accuracy_check(u,mp(2),true,5)
Accuracy check near zero, x = 8.9357696627916752e-01:
x = 8.9357696627916761e-01  correct digits =  1.5  rel. error = 1.4e+14eps
x = 8.9357696627916772e-01  correct digits =  0.5  rel. error = 1.5e+15eps
x = 8.9357696627916783e-01  correct digits =  1.0  rel. error = 4.1e+14eps
x = 8.9357696627916794e-01  correct digits =  2.1  rel. error = 3.5e+13eps
x = 8.9357696627916805e-01  correct digits =  1.4  rel. error = 1.6e+14eps

Accuracy of bessely(0,x) near second extremum of Y0(x):

>> func_accuracy_check(u,mp(3),false,5)
Accuracy check near extremum, x = 5.4296810407941351e+00:
x = 5.4296810407941356e+00  correct digits = 15.6  rel. error =  1.10eps
x = 5.4296810407941365e+00  correct digits = 16.2  rel. error =  0.32eps
x = 5.4296810407941374e+00  correct digits = 16.0  rel. error =  0.42eps
x = 5.4296810407941383e+00  correct digits = 16.0  rel. error =  0.42eps
x = 5.4296810407941392e+00  correct digits = 15.6  rel. error =  1.10eps

MATLAB delivers minimal error near extremum but completely losses accuracy near function zero.

Same analysis can be easily repeated for J0(x):

Accuracy of besselj(0,x) near first zero of J0(x):

>> u = @(x) besselj(0,x);
>> func_accuracy_check(u,mp(2),true,5)
Accuracy check near zero, x = 2.4048255576957728e+00:
x = 2.4048255576957733e+00  correct digits =  0.6  rel. error = 1.2e+15eps
x = 2.4048255576957738e+00  correct digits =  1.0  rel. error = 4.7e+14eps
x = 2.4048255576957742e+00  correct digits =  0.8  rel. error = 7.4e+14eps
x = 2.4048255576957747e+00  correct digits =  1.9  rel. error = 5.9e+13eps
x = 2.4048255576957751e+00  correct digits =  1.4  rel. error = 1.6e+14eps

Accuracy of besselj(0,x) near first extremum of J0(x):

>> func_accuracy_check(u,mp(2),false,5)
Accuracy check near extremum, x = 3.8317059702075123e+00:
x = 3.8317059702075129e+00  correct digits = 15.8  rel. error =  0.71eps
x = 3.8317059702075134e+00  correct digits = 15.8  rel. error =  0.71eps
x = 3.8317059702075138e+00  correct digits = 15.5  rel. error =  1.30eps
x = 3.8317059702075142e+00  correct digits = 15.8  rel. error =  0.71eps
x = 3.8317059702075147e+00  correct digits = 15.9  rel. error =  0.53eps

Situation is the same – besselj(0,x) computes J0(x) incorrectly near zeros.

In fact, all routines in MATLAB for computing Bessel functions of integer order suffer from accuracy loss in vicinity of function zeros. Error can be arbitrary high, even to the point when none of the digits are correct.

Below we just show relative error plots with obvious peaks near function zeros – indication of accuracy loss (detailed analysis for every function can be easily done using routine above):

Certainly this situation is absolutely unacceptable, and TMW should re-consider its library for computing Bessel functions.

More surprisingly, the issue is endemic for nearly all libraries for computing Bessel functions in double precision! Below we present error plots for some commonly used open source and commercial libraries. All of them (except one) suffer from the same accuracy loss near function zeros – error may become arbitrary high.

Open Source Libraries

GNU GSL 2.1

GNU Scientific Library[16]

Boost 1.60.0

Free peer-reviewed portable C++ source libraries[17]

Cephes 2.8

C and C++ language special functions math library[18]

Numeric Recipes, 2007

3rd Edition: The Art of Scientific Computing[19]

Commercial Libraries

NAG Toolbox for MATLAB

Mark 24 – newest version[25]

Microsoft C/C++ Language and Standard Libraries

Part of several generations of MS Visual Studio [21]

Intel Fortran

Conclusion

All tested commercial (MATLAB, NAG, Intel and Microsoft) and open source libraries (GNU GSL, Boost, CEPHES and Numeric Recipes) exhibit severe accuracy degradation near zeros of Bessel functions. In fact, computed function values have no correct digits in close vicinity of zeros.

Intel mathematical library delivers accurate results for Y0(x), Y1(x), J0(x) and J1(x) but suffers from the same accuracy loss for higher orders n=2,3,....

The difficulty of approximating Bessel functions near zeros is well known[22] and efficient algorithms were developed for both cases – double[23] and arbitrary precision[24]. We hope maintainers of the libraries will take this analysis into account and will provide fixes in future releases.

Until then, please rely on high-accuracy tools, e.g. Advanpix toolbox which is the only way to compute Bessels correctly in MATLAB at the moment.

References

  1. IEEE Computer Society (2008), IEEE Standard for Binary Floating-Point Arithmetic.
  2. S. Gal. Computing elementary functions: A new approach for achieving high accuracy and good performance. In Accurate Scientific Computations. Lecture Notes in Computer Science, volume 235, pages 1–16. Springer-Verlag, Berlin, 1986.
  3. S. Gal and B. Bachelis. An accurate elementary mathematical library for the IEEE floating point standard. ACM Transactions on Mathematical Software, 17(1):26–45, March 1991.
  4. A. Ziv, Fast Evaluation of Elementary Mathematical Functions with Correctly Rounded Last Bit, ACM Trans. Math. Software, vol. 17, no. 3, pp. 410-423, 1991.
  5. V. Lefevre and J.-M. Muller, Worst Cases for Correct Rounding of the Elementary Functions in Double Precision, Proc. 15th IEEE Symp. Computer Arithmetic (ARITH 15), N. Burgess and L. Ciminiera, eds., pp. 111-118, 2001
  6. D. Defour, G. Hanrot, V. Lefevre, J.-M. Muller, N. Revol, and P. Zimmermann, Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic, Numerical Algorithms,vol. 37, nos. 1-4, pp. 367-375, 2004
  7. D. Stehle and P. Zimmermann. Gal’s accurate tables method revisited. In Proceedings of the 17th IEEE Symposium on Computer Arithmetic (ARITH-17). IEEE Computer Society Press, Los Alamitos, CA, 2005.
  8. Catherine Daramy-Loirat, David Defour, Florent de Dinechin, Matthieu Gallet, Nicolas Gast, and Jean-Michel Muller. CR-LIBM: A library of correctly rounded elementary functions in double-precision, February 29, 2009.
  9. P. W. Markstein. IA-64 and Elementary Functions: Speed and Precision. Hewlett Packard Professional Books. Prentice Hall, 2000.
  10. J.-M. Muller. Elementary Functions, Algorithms and Implementation. Birkhäuser Boston, 2nd Edition, 2006
  11. Muller, J.-M., Brisebarre, N., de Dinechin, F., Jeannerod, C.-P., Lefèvre, V., Melquiond, G., Revol, N., Stehlé, D., Torres, S.. Handbook of Floating-Point Arithmetic. Birkhäuser Boston, 2009.
  12. ISO/IEC 14882: Programming Language C++ (Latest revisions).
  13. ISO/IEC 9899: Programming language C (Latest revisions).
  14. JTC1/SC22/WG5: FORTRAN.
  15. Next floating-point number in MATLAB. See also C/C++ standards.
  16. M. Galassi et al, GNU Scientific Library Reference Manual, 3rd Edition (January 2009), ISBN 0954612078. Version 2.1, November 11, 2015.
  17. Boost – free peer-reviewed portable C++ source libraries. Version 1.60.0, December 17th, 2015.
  18. CEPHES C and C++ language special functions math library. Version 2.8, November 4, 2014.
  19. Press, William H. and Teukolsky, Saul A. and Vetterling, William T. and Flannery, Brian P. Numerical Recipes 3rd Edition: The Art of Scientific Computing, 2007, Cambridge University Press, New York, USA.
  20. GNU C Library, February 19, 2016
  21. Microsoft C/C++ Language and Standard Libraries.
  22. J. F. Hart, E. W. Cheney, C. L. Lawson, H. J. Maehly, C. K. Mesztenyi, J. R. Rice, H. G. Thatcher, and C. Witzgall. Computer Approximations. Robert E. Krieger, 1978.
  23. J. Harrison. Fast and Accurate Bessel Function Computation, Proceedings of ARITH19, the 19th IEEE Conference on Computer Arithmetic, IEEE Computer Society Press, 2009, pp. 104-113.
  24. Fousse, G. Hanrot, V. Lefevre, P. Pelissier, and P. Zimmermann, 2007, MPFR: A multiple-precision binary floating-point library with correct rounding, ACM TOMS, 33(2), 13:1–15
  25. The NAG Toolbox for MATLAB, Mark 24 (the newest version).

{ 1 comment… read it below or add one }StefanSeptember 18, 2020 at 6:29 pm

Absolutely amazing work!
This is a serious issue, that I run into with my students, when expanding functions in a Bessel series. I can’t tell you how often I have said “yes, that looks sh*t because Matlab’s Bessel function are sh*t. We have to stop after a few terms in the series”.

Semster days BU

jameszbw

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