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99 lines
4.3 KiB
99 lines
4.3 KiB
2 years ago
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RFEJER: Semi-Automatic Rational Fejer Quadrature
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AUTHORS: Karl Deckers, Ahlem Mougaida, and Hedi Belhadjsalah
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Software revision date: August 25, 2016
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GENERAL DESCRIPTION
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-------------------
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RFEJER computes the nodes and weights in the n-point rational Fejer quadrature
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rule for a given sequence of n-1 real or complex conjugate poles outside the
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interval [-1,1]. In addition it also provides an approximation of the integral of
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an array-valued function FUN from -1 to 1, together with an estimation of the
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relative error, based on the computed nodes and weights. 'Semi' in 'semi-
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automatic' refers to the fact that RFEJER does not use adaptive refinement of the
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interval, while the latter may be necessary to achieve an accuracy on the
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approximation of order machine precision. The algorithm, however, is implemented
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in such a way that it can easily be plugged into a larger program also containing
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an adaptive method.
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===================================================================================
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ALGORITHM PACKAGE
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-----------------
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The algorithm package contains:
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* Main files: rfejer.m
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with subroutines errW.m and transf.m
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* Example files: exampleX.m (for X=1,...,7)
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with subroutine nextpoint.m, output files exampleX.out (for X=1,...,7),
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and figures example7_PQ_figK.pdf (for PQ=11,12,31,32, and K=1,2,3)
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* Test files: expaper.m, and expaperWG.m
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* Third party supporting code: gauss.m, gauss_rational.m, lanczos.m, mcdis.m,
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quadrat.m, rcheb.m, r_jacobi.m, r_mod.m, and stieltjes.m,
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All files need to be stored in the same directory. The third party supporting code
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rcheb.m, which is used in the main file rfejer.m, has been updated to ensure
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compatibility with Octave (see further). The other third party supporting codes
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have not been updated and are only necessary for comparison with Gautschi's method
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in the example file example7.m and the test file expaperWG.m.
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===================================================================================
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PORTABILITY
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-------------
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All files were initially developed in MATLAB R2011b, and have been tested
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afterwards to work properly in MATLAB R2013b, R2015a, R2015b, and R2016a as well.
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However, the file example7.m with second input argument 'intquad==1' does not work
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in MATLAB R2011b, since the MATLAB function integral.m was introduced later, in
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release 2012a. The output files exampleX.out, for X=1,...,7, were obtained by
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executing the corresponding MATLAB file exampleX.m in MATLAB R2011b on a notebook
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with a 2.26 GHz Intel Core 2 Duo P8400 processor. The figures
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example7_PQ_figK.pdf, for PQ=11,12,31,32, and K=1,2,3, were obtained by executing
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example7.m in MATLAB R2015b with an Intel(R) Core(TM) i7-3740QM CPU @2.70 GHz
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processor.
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Compatibility with GNU Octave 4.0.3 requires:
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1) the creation of a file named fcnchk.m with the following contents:
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_______________________
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function f=fcnchk(x, n)
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f = x;
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end
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_______________________
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2) removing '%' in
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* example7.m, lines 134 and 136;
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* expaperWG.m, lines 116, 118, 140, and 142.
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The third party supporting code rcheb.m has been updated to ensure compatibility
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with Octave. Whenever the initial version of rcheb.m is used instead, one should
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also:
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3) remove '%' in rfejer.m, lines 198, 200, 231, and 233; and
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4) make the following two small adaptations in rcheb.m:
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* line 353: 'bnd(2,i) == xo(i)' should be replaced by 'bnd(2,i) == xo(1,i)'
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* line 355: 'bnd(1,i) == xo(i)' should be replaced by 'bnd(1,i) == xo(1,i)'
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Furthermore, when executing the test file expaper.m with input argument 'ex == 1,
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2, 3, or 8', messages of the form "warning: division by zero" can be avoided by
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using the alternatives from lines 325-353 instead on lines 38, 52, 76, and 110-113.
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Finally, the file example7.m with second input argument 'intquad==1' does not
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work in Octave, since the latter has no built-in automatic integrator called
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'integral'.
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===================================================================================
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DOCUMENTATION AND CONTACT INFORMATION
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-------------------------------------
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Deckers, K., Mougaida, A., and Belhadjsalah, H.
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Algorithm XXX: Extended Rational Fejer Quadrature Rules based on Chebyshev
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Orthogonal Rational Functions.
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ACM Trans. Math. Software (2016), 29 pages.
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Corresponding author: Karl Deckers (karl.deckers@math.univ-lille1.fr)
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