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cut at Conjugate gradient. Also copied in Remez, Zolotarev, Chebyshev from Mike Clark, Tony Kennedy and my BFM package respectively since we know we will need these. I wanted the structure of algorithms/approx algorithms/iterative etc.. to start taking shape.
81 lines
3.8 KiB
Plaintext
81 lines
3.8 KiB
Plaintext
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PAB. Took Mike Clark's AlgRemez from GitHub and (modified a little) include.
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This is open source and license and readme and comments are preserved consistent
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with the license. Mike, thankyou!
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AlgRemez
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The archive downloadable here contains an implementation of the Remez
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algorithm which calculates optimal rational (and polynomial)
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approximations to the nth root over a given spectral range. The Remez
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algorithm, although in principle is extremely straightforward to
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program, is quite difficult to get completely correct, e.g., the Maple
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implementation of the algorithm does not always converge to the
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correct answer.
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To use this algorithm you need to install GMP, the GNU Multiple
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Precision Library, and when configuring the install, you must include
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the --enable-mpfr option (see the GMP manual for more details). You
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also have to edit the Makefile for AlgRemez appropriately for your
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system, namely to point corrrectly to the location of the GMP library.
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The simple main program included with this archive invokes the
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AlgRemez class to calculate an approximation given by command line
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arguments. It is invoked by the following
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./test y z n d lambda_low lambda_high precision,
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where the function to be approximated is f(x) = x^(y/z), with degree
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(n,d) over the spectral range [lambda_low, lambda_high], using
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precision digits of precision in the arithmetic. So an example would
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be
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./test 1 2 5 5 0.0004 64 40
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which corresponds to constructing a rational approximation to the
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square root function, with degree (5,5) over the range [0.0004,64]
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with 40 digits of precision used for the arithmetic. The parameters y
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and z must be positive, the approximation to f(x) = x^(-y/z) is simply
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the inverse of the approximation to f(x) = x^(y/z). After the
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approximation has been constructed, the roots and poles of the
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rational function are found, and then the partial fraction expansion
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of both the rational function and it's inverse are found, the results
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of which are output to a file called "approx.dat". In addition, the
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error function of the approximation is output to "error.dat", where it
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can be checked that the resultant approximation satisfies Chebychev's
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criterion, namely all error maxima are equal in magnitude, and
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adjacent maxima are oppostie in sign. There are some caveats here
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however, the optimal polynomial approximation has complex roots, and
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the root finding implemented here cannot (yet) handle complex roots.
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In addition, the partial fraction expansion of rational approximations
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is only found for the case n = d, i.e., the degree of numerator
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polynomial equals that of the denominator polynomial. The convention
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for the partial fraction expansion is that polar shifts are always
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written added to x, not subtracted.
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To do list
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1. Include an exponential dampening factor in the function to be
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approximated. This may sound trivial to implement, but for some
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parameters, the algorithm seems to breakdown. Also, the roots in the
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rational approximation sometimes become complex, which currently
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breaks the stupidly simple root finding code.
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2. Make the algorithm faster - it's too slow when running on qcdoc.
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3. Add complex root finding.
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4. Add more options for error minimisation - currently the code
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minimises the relative error, should add options for absolute error,
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and other norms.
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There will be a forthcoming publication concerning the results
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generated by this software, but in the meantime, if you use this
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software, please cite it as
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"M.A. Clark and A.D. Kennedy, https://github.com/mikeaclark/AlgRemez, 2005".
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If you have any problems using the software, then please email scientist.mike@gmail.com.
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