├── .gitignore ├── LICENSE ├── Makefile ├── README.md ├── doc ├── Makefile ├── fuchsia.bib ├── fuchsia.tex ├── include │ ├── commands.tex │ ├── layout.tex │ ├── summary.tex │ └── title.tex └── refs │ ├── barkatou_2008.pdf │ ├── barkatou_issac10_1.pdf │ ├── barkatou_issac10_2.pdf │ ├── barkatou_issac10_3.pdf │ ├── henn_2014.pdf │ ├── lee_2014.pdf │ └── moser_1960.pdf ├── examples ├── Makefile ├── README ├── eec.m ├── eec_eps.m ├── eec_eps_t.m ├── git_409.m ├── git_409_eps.m ├── git_409_eps_t.m ├── git_410.m ├── git_410_eps.m ├── git_410_eps_t.m ├── henn_324.m ├── henn_324_eps.m ├── henn_324_eps_t.m ├── henn_411.m ├── henn_413.m ├── lee_1.m ├── lee_1_y.m ├── lee_1_y_eps.m ├── lee_1_y_eps_t.m ├── lee_2.m ├── lee_2_y.m ├── lee_2_y_eps.m ├── lee_2_y_eps_t.m ├── lee_3.m ├── lee_3_eps.m ├── lee_3_eps_t.m ├── lee_81.m ├── lee_81_eps.m ├── lee_81_eps_t.m ├── lue_1.m ├── lue_1_eps.m ├── lue_1_eps_t.m ├── pap_1.m ├── pap_1_eps.m └── pap_1_eps_t.m ├── fuchsia.py ├── normalize.py └── test ├── __init__.py ├── data ├── bolibrukh.mtx ├── git_409.m ├── git_410.m ├── henn_324.m ├── lee_3.m ├── pap_1.m └── pap_3_52.mtx ├── test_block_triangular.py ├── test_cli.py ├── test_epsilon_form.py ├── test_fuchsia.py ├── test_fuchsify_by_blocks.py ├── test_normalize.py ├── test_normalize_by_blocks.py └── test_reduce.py /.gitignore: -------------------------------------------------------------------------------- 1 | build/ 2 | dist/ 3 | temp/ 4 | *.dvi 5 | *.nb 6 | *.pyc 7 | *.swn 8 | *.swo 9 | *.swp 10 | *.egg-info 11 | .cache 12 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Copyright (c) 2016, O.Gituliar and V.Magerya 2 | 3 | Permission to use, copy, modify, and/or distribute this software for any 4 | purpose with or without fee is hereby granted, provided that the above 5 | copyright notice and this permission notice appear in all copies. 6 | 7 | THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 8 | WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 9 | MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 10 | ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 11 | WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION 12 | OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN 13 | CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 14 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | .PHONY: dist test 2 | 3 | # Test actions 4 | 5 | test-maxima: 6 | env SAGE_PATH="$(CURDIR)" \ 7 | sage -python -munittest -fv test.test_suite_maxima \ 8 | 2>&1 | tee test/test_suite_maxima.log 9 | 10 | test-maple: 11 | env SAGE_PATH="$(CURDIR)" \ 12 | sage -python -munittest -fv test.test_suite_maple \ 13 | 2>&1 | tee test/test_suite_maple.log 14 | 15 | fuchsia.py: test/__init__.py 16 | 17 | test/*.py:: 18 | env SAGE_PATH="$(CURDIR)" \ 19 | sage -python -munittest -fv test.$(basename $(notdir $@)) 20 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | ## Description 2 | 3 | *Fuchsia* reduces differential equations for Feynman master integrals to the epsilon form. 4 | 5 | To illustrate, let us say we have a system of differential equations of this form: 6 | 7 | ∂f(x,ϵ)/∂x = 𝕄(x,ϵ) f(x,ϵ), 8 | 9 | where `𝕄(x,ϵ)` is a given matrix of rational functions in `x` and `ϵ`, i.e, a free variable and an infinitesimal parameter. 10 | Our ultimate goal is to find a column vector of unknown functions `f(x,ϵ)` as a Laurent series in `ϵ`, which satisfies the equations above. 11 | 12 | With the help of *Fuchsia* we can find a transformation matrix `𝕋(x,ϵ)` which turns our system to the equivalent Fuchsian system of this form: 13 | 14 | ∂g(x,ϵ)/∂x = ϵ 𝕊(x) g(x,ϵ) 15 | 16 | where `𝕊(x) = ∑ᵢ 𝕊ᵢ/(x-xᵢ)` and `f(x,ϵ) = 𝕋(x,ϵ) g(x,ϵ)`. 17 | 18 | Such a transformation is useful because we can easily solve the equivalent system for `g(x,ϵ)` (see [1]) and then, multiplying it by `𝕋(x,ϵ)`, find `f(x,ϵ)`. 19 | 20 | You can learn about the algorithm used in *Fuchsia* to find such transformations from Roman Lee's paper [2]. 21 | 22 | *Fuchsia* is available both as a command line utility and as a (Python) library for SageMath [3]. 23 | It will run on most Unix-like operating systems. 24 | 25 | Documentation with more information, installation and usage details is here [4]. 26 | 27 | ## Publications 28 | 29 | *Fuchsia* has been announced in proceedings of The Loops and Legs Conference 2016, PoS LL2016 (2016) 030, [arXiv:1607.00759](http://arxiv.org/abs/arXiv:1607.00759). 30 | Since then it is available on-line. 31 | A complete description of the program has been published in [Comput. Phys. Commun.](http://www.sciencedirect.com/science/article/pii/S0010465517301340), [arXiv:1701.04269](http://arxiv.org/abs/arXiv:1701.04269). 32 | 33 | If you find *Fuchsia* useful for your research, please, consider to cite both articles. 34 | 35 | ## References 36 | 37 | * [1] https://arxiv.org/abs/1304.1806 38 | * [2] https://arxiv.org/abs/1411.0911 39 | * [3] http://www.sagemath.org/ 40 | * [4] https://arxiv.org/abs/1701.04269 41 | -------------------------------------------------------------------------------- /doc/Makefile: -------------------------------------------------------------------------------- 1 | .PHONY: all 2 | 3 | all: 4 | latexmk -dvi- -pdf fuchsia.tex 5 | latexmk -bibtex -c fuchsia.tex 6 | -------------------------------------------------------------------------------- /doc/fuchsia.bib: -------------------------------------------------------------------------------- 1 | @article{Pra17, 2 | author = "Prausa, M.", 3 | title = "{epsilon: A tool to find a canonical basis of master 4 | integrals}", 5 | year = "2017", 6 | eprint = "1701.00725", 7 | archivePrefix = "arXiv", 8 | primaryClass = "hep-ph", 9 | reportNumber = "TTK-17-01", 10 | SLACcitation = "%%CITATION = ARXIV:1701.00725;%%" 11 | } 12 | 13 | @manual{sagemath, 14 | Key = {SageMath}, 15 | Author = {The Sage Developers}, 16 | Title = {{S}ageMath, the {S}age {M}athematics {S}oftware {S}ystem ({V}ersion 7.0)}, 17 | note = {\url{http://www.sagemath.org}}, 18 | Year = {2016}, 19 | } 20 | 21 | @article{GLZ16, 22 | author = "Georgoudis, A. and Larsen, K. J. and Zhang, Y.", 23 | title = "{Azurite: An algebraic geometry based package for finding 24 | bases of loop integrals}", 25 | year = "2016", 26 | eprint = "1612.04252", 27 | archivePrefix = "arXiv", 28 | primaryClass = "hep-th", 29 | SLACcitation = "%%CITATION = ARXIV:1612.04252;%%" 30 | } 31 | 32 | @article{GM16, 33 | author = "Gituliar, O. and Magerya, V.", 34 | title = "{Fuchsia and master integrals for splitting functions 35 | from differential equations in QCD}", 36 | booktitle = "{Proceedings, 13th DESY Workshop on Elementary Particle 37 | Physics: Loops and Legs in Quantum Field Theory (LL2016): 38 | Leipzig, Germany, April 24-29, 2016}", 39 | journal = "PoS", 40 | volume = "LL2016", 41 | year = "2016", 42 | pages = "030", 43 | eprint = "1607.00759", 44 | archivePrefix = "arXiv", 45 | primaryClass = "hep-ph", 46 | reportNumber = "IFJPAN-IV-2016-17", 47 | SLACcitation = "%%CITATION = ARXIV:1607.00759;%%" 48 | } 49 | 50 | @article{Mey16b, 51 | author = "Meyer, C.", 52 | title = "{Transforming differential equations of multi-loop 53 | Feynman integrals into canonical form}", 54 | year = "2016", 55 | eprint = "1611.01087", 56 | archivePrefix = "arXiv", 57 | primaryClass = "hep-ph", 58 | reportNumber = "HU-EP-16-24", 59 | SLACcitation = "%%CITATION = ARXIV:1611.01087;%%" 60 | } 61 | 62 | @article{Mey16a, 63 | author = "Meyer, C.", 64 | title = "{Evaluating multi-loop Feynman integrals using 65 | differential equations: automatizing the transformation to 66 | a canonical basis}", 67 | booktitle = "{Proceedings, 13th DESY Workshop on Elementary Particle 68 | Physics: Loops and Legs in Quantum Field Theory (LL2016): 69 | Leipzig, Germany, April 24-29, 2016}", 70 | journal = "PoS", 71 | volume = "LL2016", 72 | year = "2016", 73 | pages = "028", 74 | url = "http://pos.sissa.it/cgi-bin/reader/contribution.cgi?id=260/028", 75 | SLACcitation = "%%CITATION = POSCI,LL2016,028;%%" 76 | } 77 | 78 | @article{ABB15, 79 | author = "Ablinger, J. and Behring, A. and Bl{\"u}mlein, J. and De 80 | Freitas, A. and von Manteuffel, A. and Schneider, C.", 81 | title = "{Calculating Three Loop Ladder and V-Topologies for 82 | Massive Operator Matrix Elements by Computer Algebra}", 83 | journal = "Comput. 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Commun.", 84 | volume = "202", 85 | year = "2016", 86 | pages = "33-112", 87 | eprint = "1509.08324", 88 | archivePrefix = "arXiv", 89 | primaryClass = "hep-ph", 90 | reportNumber = "DESY-15-049, DO-TH-15-06, MITP-15-080, DESY-15--049, 91 | DO--TH-15-06", 92 | SLACcitation = "%%CITATION = ARXIV:1509.08324;%%" 93 | } 94 | 95 | @article{Git15, 96 | author = "Gituliar, O.", 97 | title = "{Master integrals for splitting functions from 98 | differential equations in QCD}", 99 | journal = "JHEP", 100 | volume = "02", 101 | year = "2016", 102 | pages = "017", 103 | doi = "10.1007/JHEP02(2016)017", 104 | eprint = "1512.02045", 105 | archivePrefix = "arXiv", 106 | primaryClass = "hep-ph", 107 | reportNumber = "IFJPAN-IV-2015-21", 108 | SLACcitation = "%%CITATION = ARXIV:1512.02045;%%" 109 | } 110 | @article{Lee15, 111 | author = "Lee, R.", 112 | title = "Reducing differential equations for multiloop master integrals", 113 | journal = "JHEP", 114 | volume = "04", 115 | year = "2015", 116 | pages = "108", 117 | eprint = "1411.0911", 118 | archivePrefix = "arXiv", 119 | primaryClass = "hep-ph" 120 | } 121 | @article{Tan15, 122 | author = "Tancredi, L.", 123 | title = "{Integration by parts identities in integer numbers of 124 | dimensions. 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Phys.", 391 | volume = "B192", 392 | year = "1981", 393 | pages = "159-204", 394 | doi = "10.1016/0550-3213(81)90199-1", 395 | SLACcitation = "%%CITATION = NUPHA,B192,159;%%" 396 | } 397 | 398 | @article{Mos59, 399 | author={Moser, J.}, 400 | title={The order of a singularity in Fuchs' theory}, 401 | year={1959}, 402 | issn={0025-5874}, 403 | journal={Mathematische Zeitschrift}, 404 | volume={72}, 405 | number={1}, 406 | doi={10.1007/BF01162962}, 407 | publisher={Springer-Verlag}, 408 | pages={379-398}, 409 | language={English} 410 | } 411 | 412 | 413 | @ELECTRONIC{maxima, 414 | author = {Maxima}, 415 | title = {Maxima, a Computer Algebra System. Version 5.35.1}, 416 | year = {2014}, 417 | month = {December}, 418 | address = {http://maxima.sourceforge.net/}, 419 | url = {http://maxima.sourceforge.net/}, 420 | owner = {maxima}, 421 | timestamp = {2014.12.13} 422 | } 423 | -------------------------------------------------------------------------------- /doc/include/commands.tex: -------------------------------------------------------------------------------- 1 | \newcommand{\abs}[1]{\left|#1\right|} 2 | \newcommand{\D}{\mathrm{d}} 3 | \newcommand{\eps}{\epsilon} 4 | \newcommand{\fuchsia}{\textcolor{fuchsia}{\texttt{Fuchsia}}\xspace} 5 | \newcommand{\linux}{\texttt{Linux}\xspace} 6 | \newcommand{\maple}{\texttt{Maple}\xspace} 7 | \newcommand{\maxima}{\texttt{Maxima}\xspace} 8 | \newcommand{\maximasage}{\texttt{Maxima/Sage}\xspace} 9 | \newcommand{\python}{\texttt{Python}\xspace} 10 | \newcommand{\rank}{\mathrm{rank}} 11 | \newcommand{\sage}{\texttt{SageMath}\xspace} 12 | \newcommand{\code}[1]{\texttt{#1}} 13 | \newcommand{\F}[1]{\texttt{#1}} % use this to style function names 14 | \newcommand{\M}[1]{\mathbb{#1}} % use this to style matrix names 15 | \newcommand{\V}[1]{\mathbf{#1}} % use this to style vector names 16 | \newcommand{\prompt}[2]{\textcolor{prompt}{#1} \textcolor{command}{#2}} 17 | \newcommand{\functionitem}[2]{\item[$\F{#1}(#2)$\hfill\textit{(function)}]} 18 | \newcommand{\classitem}[1]{\item[$#1$\hfill\textit{(class)}]} 19 | -------------------------------------------------------------------------------- /doc/include/layout.tex: -------------------------------------------------------------------------------- 1 | 2 | % Page layout definitions 3 | % ---------------------- 4 | \newlength{\dinwidth} \newlength{\dinmargin} 5 | \setlength{\dinwidth}{21.0cm} %width of A4 paper 6 | \setlength{\textheight}{23.9cm} %height of text 7 | \setlength{\textwidth}{15.7cm} %width of text 8 | % Now calculate the margin as 0.5*(paperwidth-textwidth} 9 | \setlength{\dinmargin}{\dinwidth} 10 | \addtolength{\dinmargin}{-\textwidth} 11 | \setlength{\dinmargin}{0.5\dinmargin} 12 | % For twoside prints you often want the oddsidemargin to be larger 13 | % than the evensidemargin because staples or punchholes endup at the 14 | % lefthand side of odd numbered pages 15 | \setlength{\oddsidemargin}{1.2\dinmargin} 16 | \setlength{\evensidemargin}{0.8\dinmargin} 17 | % Latex defines the margin one inch from the paper border 18 | \addtolength{\oddsidemargin}{-1in} 19 | \addtolength{\evensidemargin}{-1in} 20 | % 21 | \setlength{\marginparwidth}{0.9\dinmargin} \setlength{\itemsep}{0mm} 22 | \setlength{\parsep}{0mm} \marginparsep 8pt \marginparpush 5pt 23 | \topmargin -42pt \headheight 12pt \headsep 30pt \footskip 24pt 24 | \parskip 2mm \parindent 0mm \font\bollox=cmmib10 scaled \magstep4 25 | -------------------------------------------------------------------------------- /doc/include/summary.tex: -------------------------------------------------------------------------------- 1 | {\bf PROGRAM SUMMARY} 2 | 3 | \begin{small} 4 | \noindent 5 | 6 | {\em Program Title:} 7 | \fuchsia 8 | 9 | {\em Authors:} 10 | O.~Gituliar and V.~Magerya 11 | 12 | {\em Program obtainable from:} 13 | \url{https://github.com/gituliar/fuchsia/} 14 | 15 | {\em Journal Reference:} 16 | %Leave blank, supplied by Elsevier. 17 | 18 | {\em Catalog identifier:} 19 | %Leave blank, supplied by Elsevier. 20 | 21 | {\em Licensing provisions:} 22 | ISC license 23 | 24 | {\em Programming language:} 25 | \python 2.7 26 | 27 | {\em Operating system:} 28 | \linux, \texttt{Unix}-like 29 | 30 | {\em RAM:} 31 | Dependent upon the input data. Expect hundreds of megabytes. 32 | 33 | {\em Keywords:} 34 | Computer algebra, Feynman integrals, differential equations, epsilon form, Fuchsian form, Moser reduction 35 | 36 | {\em Classification:} 37 | 5 Computer Algebra, 11.1 High Energy Physics and Computing 38 | 39 | {\em External routines/libraries:} 40 | \href{http://www.sagemath.org/}{\sage} (7.0 or higher), \maple (optional) 41 | 42 | {\em Nature of problem:} 43 | Feynman master integrals may be calculated from solutions of a linear system of differential equations with rational coefficients. 44 | Such a system can be easily solved as an $\eps$-series when its epsilon form is known. 45 | Hence, a tool which is able to find the epsilon form transformations can be used to evaluate Feynman master integrals. 46 | 47 | {\em Solution method:} 48 | The solution method is based on the Lee algorithm \cite{Lee15} which consists of three main steps: fuchsification, normalization, and factorization. 49 | During the fuchsification step a given system of differential equations is transformed into the Fuchsian form with the help of the Moser method \cite{Mos59}. 50 | Next, during the normalization step the system is transformed to the form where eigenvalues of all residues are proportional to the dimensional regulator $\eps$. 51 | Finally, the system is factorized to the epsilon form by finding an unknown transformation which satisfies a system of linear equations. 52 | 53 | {\em Restrictions:} 54 | Systems of single-variable differential equations are considered. 55 | A system needs to be reducible to Fuchsian form and eigenvalues of its residues must be of the form $n + m\,\eps$, where $n$ is integer. 56 | 57 | {\em Running time:} 58 | Around an hour in total for an example $74\times74$ matrix with 20 singular points on a PC with a 1.7GHz Intel Core i5 CPU, but dependent upon input matrix, its size, number of singular points and their degrees. 59 | An additional slowdown is to be expected for matrices with complex and/or irrational singular point locations, as these are particularly difficult for symbolic algebra software to handle. 60 | 61 | \end{small} 62 | -------------------------------------------------------------------------------- /doc/include/title.tex: -------------------------------------------------------------------------------- 1 | \title{ 2 | \begin{flushright} 3 | \tt\normalsize{DESY-16-219}\\ 4 | % \tt\normalsize{arXiv:16xx.xxxxx} 5 | \end{flushright} 6 | \vspace{1cm} 7 | \Large \bf \fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form\\ 8 | \vspace{0.5cm} 9 | \normalsize \bf Version 16.11.14 10 | } 11 | 12 | \author[a]{Oleksandr Gituliar% 13 | \thanks{Corresponding author; email address: 14 | \href{mailto:oleksandr.gituliar@desy.de} 15 | {oleksandr.gituliar@desy.de}}} 16 | 17 | \author[b]{Vitaly Magerya} 18 | 19 | \affil[a]{II. Institut f\"ur Theoretische Physik, Universit\"at Hamburg, 20 | Luruper Chaussee 149, D-22761 Hamburg, Germany} 21 | \affil[b]{Novomoskovsk, Ukarine} 22 | 23 | %\date{\today\\\vspace{0.5cm}{\bf \LARGE DRAFT}} 24 | -------------------------------------------------------------------------------- /doc/refs/barkatou_2008.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gituliar/fuchsia/19e16fdd736ad4731b087f372770a21bfc7093bf/doc/refs/barkatou_2008.pdf -------------------------------------------------------------------------------- /doc/refs/barkatou_issac10_1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gituliar/fuchsia/19e16fdd736ad4731b087f372770a21bfc7093bf/doc/refs/barkatou_issac10_1.pdf -------------------------------------------------------------------------------- /doc/refs/barkatou_issac10_2.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gituliar/fuchsia/19e16fdd736ad4731b087f372770a21bfc7093bf/doc/refs/barkatou_issac10_2.pdf -------------------------------------------------------------------------------- /doc/refs/barkatou_issac10_3.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gituliar/fuchsia/19e16fdd736ad4731b087f372770a21bfc7093bf/doc/refs/barkatou_issac10_3.pdf -------------------------------------------------------------------------------- /doc/refs/henn_2014.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gituliar/fuchsia/19e16fdd736ad4731b087f372770a21bfc7093bf/doc/refs/henn_2014.pdf -------------------------------------------------------------------------------- /doc/refs/lee_2014.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gituliar/fuchsia/19e16fdd736ad4731b087f372770a21bfc7093bf/doc/refs/lee_2014.pdf -------------------------------------------------------------------------------- /doc/refs/moser_1960.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/gituliar/fuchsia/19e16fdd736ad4731b087f372770a21bfc7093bf/doc/refs/moser_1960.pdf -------------------------------------------------------------------------------- /examples/Makefile: -------------------------------------------------------------------------------- 1 | .PRECIOUS: %_norm.m %_fuchs.m 2 | 3 | FUCHSIA = sage -python ../fuchsia.py 4 | 5 | # Examples from R.Lee 6 | # 7 | lee_1_y.m: lee_1.m 8 | $(FUCHSIA) changevar -m $@ $< "(1+y^2)/(1-y^2)" 9 | lee_1_y_eps.m: lee_1_y.m 10 | $(FUCHSIA) reduce --use-maple -v -x y -m $@ -t $(basename $@)_t.m $< 11 | 12 | lee_2_y.m: lee_2.m 13 | $(FUCHSIA) changevar -m $@ $< "(1-y)^2/(y)" 14 | lee_2_y_eps.m: lee_2_y.m 15 | $(FUCHSIA) reduce --use-maple -v -x y -m $@ -t $(basename $@)_t.m $< 16 | 17 | lee_3_eps.m: lee_3.m 18 | $(FUCHSIA) reduce --use-maple -m $@ -t $(basename $@)_t.m $< 19 | 20 | # Example from C.Papadopolous 21 | # 22 | pap_1_eps.m: pap_1.m 23 | $(FUCHSIA) reduce --use-maple -e ep -m $@ -t $(basename $@)_t.m $< 24 | 25 | # Other examples 26 | # 27 | %_eps.m: %.m 28 | $(FUCHSIA) reduce -m $@ -t $(basename $@)_t.m $< 29 | %_fuchs.m: %.m 30 | $(FUCHSIA) fuchsify -m $@ -t $(basename $@)_t.m $< 31 | %_norm.m: %_fuchs.m 32 | $(FUCHSIA) normalize -m $@ -t $(basename $@)_t.m $< 33 | %_fact.m: %_norm.m 34 | $(FUCHSIA) factorize -m $@ -t $(basename $@)_t.m $< 35 | -------------------------------------------------------------------------------- /examples/README: -------------------------------------------------------------------------------- 1 | This directory contains example of systems of differential equations and their 2 | equivalent versions reduced to an epsilon form by Fuchsia. The exact commands 3 | used to obtain these examples are collected in Makefile. Using this information 4 | it is easy to deduce the content of every file in this directory. 5 | 6 | -------------------------------------------------------------------------------- /examples/eec.m: -------------------------------------------------------------------------------- 1 | {{-((eps*(1+x*(-2+z))+(-1+x)*x*z)/((-1+x)*x*(-1+x*z))),0,0},{-((1-5*eps+6*eps^2)/(eps*x*z-eps*x^2*z)),-((-1+2*eps+x)/((-1+x)*x)),0},{((-1+2*eps)*(-1+3*eps)*(-3+x+2*x*z))/(eps*(-1+x)^2*x*(-1+z)^2*z),(-2*eps*x)/((-1+x)^2*(-1+x*z)),(1-2*eps*(-1+z)-x*z)/((-1+x)*(-1+x*z))}} 2 | -------------------------------------------------------------------------------- /examples/eec_eps.m: -------------------------------------------------------------------------------- 1 | {{2*eps*z/(x*z-1)-eps/(x-1)-eps/x,0,0},{-12*eps*z/((x*z-1)*(z-1))+12*eps/((x-1)*(z-1)),-2*eps/(x-1)+2*eps/x,0},{12*eps*z/((x*z-1)*(z^2-2*z+1))+24*eps/((z^2-2*z+1)*(x-1))-36*eps/((z^2-2*z+1)*x),2*eps*z/((x*z-1)*(z-1))-2*eps/((x-1)*(z-1)),2*eps*z/(x*z-1)-2*eps/(x-1)}} -------------------------------------------------------------------------------- /examples/eec_eps_t.m: -------------------------------------------------------------------------------- 1 | {{-24*eps^2*z/((6*eps^2-5*eps+1)*(x*z-1)),0,0},{0,-2/x,0},{0,0,-2/(x-1)}} -------------------------------------------------------------------------------- /examples/git_409.m: -------------------------------------------------------------------------------- 1 | {{-4*eps/(x-1)+1/(x-1)+2*eps/((x-1)*x)-1/((x-1)*x),0,0,0,0,0},{0,-4*eps/(x-1)+1/(x-1)+3*eps/((x-1)*x)-1/((x-1)*x),0,0,0,0},{0,0,-3*eps/(x-1)+1/(x-1)+2*eps/((x-1)*x)-1/((x-1)*x),0,0,0},{0,6*eps^3/((x-1)*x)-5*eps^2/((x-1)*x)+eps/((x-1)*x),-4*eps^3/((x-1)*x)+4*eps^2/((x-1)*x)-eps/((x-1)*x),-2*eps/(x-1),0,0},{0,-6*eps^3/((x-1)^2*x^2)+18*eps^3/((x-1)^2*x^3)+5*eps^2/((x-1)^2*x^2)-15*eps^2/((x-1)^2*x^3)-eps/((x-1)^2*x^2)+3*eps/((x-1)^2*x^3),-8*eps^3/((x-1)^2*x^2)+8*eps^2/((x-1)^2*x^2)-2*eps/((x-1)^2*x^2),-2*eps/((x-1)^2*x),-4*eps/(x-1)-2/(x-1)+2*eps/((x-1)*x)+1/((x-1)*x),0},{0,12*eps^3/((x-1)*x^2)-10*eps^2/((x-1)*x^2)+2*eps/((x-1)*x^2),0,4*eps/((x-1)*x),0,-4*eps/x-1/x}} -------------------------------------------------------------------------------- /examples/git_409_eps.m: -------------------------------------------------------------------------------- 1 | {{-2*eps/(x-1)-2*eps/x,0,0,0,0,0},{0,-eps/(x-1)-3*eps/x,0,0,0,0},{0,0,-eps/(x-1)-2*eps/x,0,0,0},{0,-35*eps/(x-1),25*eps/(x-1),-2*eps/(x-1),0,0},{0,-70*eps/(x-1)+105*eps/x,50*eps/(x-1),-2*eps/(x-1),-2*eps/(x-1)-2*eps/x,0},{0,70*eps/(x-1),0,-4*eps/(x-1),0,-4*eps/x}} -------------------------------------------------------------------------------- /examples/git_409_eps_t.m: -------------------------------------------------------------------------------- 1 | {{-x,0,0,0,0,0},{0,70*x/(6*eps^2-5*eps+1),0,0,0,0},{0,0,50*x/(4*eps^2-4*eps+1),0,0,0},{0,0,0,-2,0,0},{2/(x-1)-2/x,0,0,0,-2/(x-1)+2/x,0},{0,0,0,0,0,2/x}} -------------------------------------------------------------------------------- /examples/git_410.m: -------------------------------------------------------------------------------- 1 | {{-4*eps/(x-1)+2/(x-1)+2*eps/((x-1)*x)-1/((x-1)*x),0,0,0,0,0,0,0},{-3*eps/((x-1)*x)+2/((x-1)*x),-3*eps/x+1/x,0,0,0,0,0,0},{0,6*eps^2/((x-1)*x)-5*eps/((x-1)*x)+1/((x-1)*x),-2*eps/(x-1),0,0,0,0,0},{0,0,0,-4*eps/(x-1)-2/(x-1)+2*eps/((x-1)*x)+1/((x-1)*x),0,0,0,0},{18*eps^4/((x+1)*x^2)-27*eps^3/((x+1)*x^2)+13*eps^2/((x+1)*x^2)-2*eps/((x+1)*x^2),-6*eps^4/x^2+5*eps^3/x^2-eps^2/x^2,0,0,-2*eps*x/((x+1)*(x-1))-6*eps/((x+1)*(x-1))+4*eps/((x+1)*(x-1)*x),12*eps^2/((x+1)*x)-2*eps/((x+1)*x),0,0},{18*eps^3/((x+1)*(x-1)*x)-27*eps^2/((x+1)*(x-1)*x)+13*eps/((x+1)*(x-1)*x)-2/((x+1)*(x-1)*x),0,0,0,2/(x+1),-6*eps/(x+1)+1/(x+1),0,0},{72*eps^4/((x+1)*(x-1)^3*x^3)-108*eps^3/((x+1)*(x-1)^3*x^3)+52*eps^2/((x+1)*(x-1)^3*x^3)-8*eps/((x+1)*(x-1)^3*x^3),-12*eps^4/((x-1)^2*x^2)+24*eps^4/((x-1)^2*x^3)+10*eps^3/((x-1)^2*x^2)-20*eps^3/((x-1)^2*x^3)-2*eps^2/((x-1)^2*x^2)+4*eps^2/((x-1)^2*x^3),-4*eps^3/((x-1)^2*x),0,4*eps/((x+1)*(x-1)^2)+4*eps/((x+1)*(x-1)^2*x^2),12*eps^2/((x+1)*(x-1)*x^2)-2*eps/((x+1)*(x-1)*x^2),-4*eps/(x-1)-2/(x-1)+2*eps/((x-1)*x)+1/((x-1)*x),0},{-36*eps^3/((x+1)^3*(x-1)*x)-144*eps^3/((x+1)^3*(x-1)*x^2)+54*eps^2/((x+1)^3*(x-1)*x)-36*eps^3/((x+1)^3*(x-1)*x^3)+216*eps^2/((x+1)^3*(x-1)*x^2)-26*eps/((x+1)^3*(x-1)*x)+54*eps^2/((x+1)^3*(x-1)*x^3)-104*eps/((x+1)^3*(x-1)*x^2)+4/((x+1)^3*(x-1)*x)-26*eps/((x+1)^3*(x-1)*x^3)+16/((x+1)^3*(x-1)*x^2)+4/((x+1)^3*(x-1)*x^3),12*eps^3/((x+1)^2*x^2)-12*eps^3/((x+1)^2*x^3)-10*eps^2/((x+1)^2*x^2)+10*eps^2/((x+1)^2*x^3)+2*eps/((x+1)^2*x^2)-2*eps/((x+1)^2*x^3),0,0,4/(x+1)^3+4/((x+1)^3*x^2),12*eps/((x+1)^3*x)-12*eps/((x+1)^3*x^2)-2/((x+1)^3*x)+2/((x+1)^3*x^2),0,-4*eps/(x+1)-2/(x+1)-2*eps/((x+1)*x)-1/((x+1)*x)}} -------------------------------------------------------------------------------- /examples/git_410_eps.m: -------------------------------------------------------------------------------- 1 | {{-2*eps/(x-1)-2*eps/x,0,0,0,0,0,0,0},{2*eps/x,-3*eps/x,0,0,0,0,0,0},{0,-15/2*eps/(x-1),-2*eps/(x-1),0,0,0,0,0},{0,0,0,-2*eps/(x-1)-2*eps/x,0,0,0,0},{-210/11*eps/x,-105/11*eps/x,0,0,-2*eps/(x-1)+4*eps/x,14*eps/x,0,0},{30/11*eps/(x+1)-30/11*eps/(x-1)+60/11*eps/x,-30/11*eps/(x+1)+60/11*eps/x,0,0,-6/7*eps/(x+1)+6/7*eps/(x-1)-16/7*eps/x,-2*eps/(x+1)-8*eps/x,0,0},{120*eps/(x-1)-120*eps/x,-60*eps/(x-1)+120*eps/x,-16*eps/(x-1),0,-88/7*eps/(x-1)-88/7*eps/x,-44*eps/x,-2*eps/(x-1)-2*eps/x,0},{60*eps/(x+1)-120*eps/x,-60*eps/(x+1)+30*eps/x,0,0,-132/7*eps/(x+1)+44/7*eps/x,-44*eps/(x+1)+22*eps/x,0,-2*eps/(x+1)-2*eps/x}} -------------------------------------------------------------------------------- /examples/git_410_eps_t.m: -------------------------------------------------------------------------------- 1 | {{-30*(eps*x^2-eps*x)/(18*eps^3-27*eps^2+13*eps-2),0,0,0,0,0,0,0},{0,15*x/(6*eps^2-5*eps+1),0,0,0,0,0,0},{0,0,-2/eps,0,0,0,0,0},{1/(x-1)-1/x,0,0,-1/(x-1)+1/x,0,0,0,0},{0,0,0,0,11/7*eps,0,0,0},{-30*eps/(6*eps-1),0,0,0,22/7*(eps*x+2*eps)/(6*eps-1),11*(eps*x+eps)/(6*eps-1),0,0},{1/(x-1)-1/x,0,0,2/(x-1)-2/x,0,0,-1/2*eps/(x-1)+1/2*eps/x,0},{0,0,0,0,0,0,0,1/(x+1)-1/x}} -------------------------------------------------------------------------------- /examples/henn_324.m: -------------------------------------------------------------------------------- 1 | {{eps/x,0},{-1/x^2,eps/(x+1)}} -------------------------------------------------------------------------------- /examples/henn_324_eps.m: -------------------------------------------------------------------------------- 1 | {{eps/x,0},{-eps/(x+1)+eps/x,eps/(x+1)}} -------------------------------------------------------------------------------- /examples/henn_324_eps_t.m: -------------------------------------------------------------------------------- 1 | {{eps-1,0},{-1/x,-1}} -------------------------------------------------------------------------------- /examples/henn_411.m: -------------------------------------------------------------------------------- 1 | {{0,0},{-eps/(x-4),-(eps*x+2)/((x-4)*x)}} -------------------------------------------------------------------------------- /examples/henn_413.m: -------------------------------------------------------------------------------- 1 | {{-2*eps/x,0,0},{0,1/x,-1/4*(2*eps+1)/x},{-27/(9*x+1)+3/(x+1),9*(3*eps+1)/(9*x+1)+3*(3*eps+1)/(x+1),-9*(2*eps+1)/(9*x+1)-(2*eps+1)/(x+1)-(eps-1)/x}} -------------------------------------------------------------------------------- /examples/lee_1.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_1_y.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_1_y_eps.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_2.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_2_y.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_2_y_eps.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_2_y_eps_t.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_3.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_3_eps.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_3_eps_t.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /examples/lee_81.m: -------------------------------------------------------------------------------- 1 | {{3*eps/(x+1)-(eps+1)/x,-eps/((5*eps+1)*(x+1)),(eps+1)*(1/(x+1)-1/x)/(5*eps+1)},{2*(5*eps+1)*(4*eps+1)/x,-(eps+1)/(x+1)-(3*eps+1)/x,-2*(eps+1)*(1/(x+1)-1/x)},{(5*eps+1)*(4*eps+1)*(2*eps+1)/((eps+1)*x),-(2*eps+1)*eps/((eps+1)*(x+1)),-eps/(x+1)+(5*eps+1)/x}} -------------------------------------------------------------------------------- /examples/lee_81_eps.m: -------------------------------------------------------------------------------- 1 | {{3*eps/(x+1)+eps/x,-1/11*eps/(x+1)+1/11*eps/x,3/22*eps/(x+1)-3/22*eps/x},{90*eps/(x+1)-20*eps/x,-54/11*eps/(x+1)+4*eps/x,-69/22*eps/(x+1)+3/2*eps/x},{-6*eps/(x+1)+70/3*eps/x,8/11*eps/(x+1)-4/3*eps/x,32/11*eps/(x+1)-4*eps/x}} -------------------------------------------------------------------------------- /examples/lee_81_eps_t.m: -------------------------------------------------------------------------------- 1 | {{-22*(eps+1)/((eps-2)*eps),0,0},{-44*(eps+1)/(eps-2)-99*(eps+1)/((eps-2)*(x+1))+55*(eps+1)/((eps-2)*x),7*(eps+1)/((eps-2)*(x+1))-3*(eps+1)/((eps-2)*x),6*(eps+1)/((eps-2)*(x+1))-12*(eps+1)/((eps-2)*x)},{22*(12*eps^2+(12*eps^2+7*eps+1)*x+7*eps+1)/(eps^2-2*eps)-99*eps/((eps-2)*(x+1)),2*(3*eps+1)/(eps-2)+7*eps/((eps-2)*(x+1)),-3*(3*eps+1)/(eps-2)+6*eps/((eps-2)*(x+1))}} -------------------------------------------------------------------------------- /examples/lue_1.m: -------------------------------------------------------------------------------- 1 | {{0, (-2*(-1 + x))/(1 + x)^3, 0, 0}, {-(((-1 + 2*eps)*(-2 + 3*eps)*(1 + x))/((-1 + x)*x)), (eps + 6*x - 14*eps*x + eps*x^2)/((-1 + x)*x*(1 + x)), 0, 0}, {0, (-1 + x)/(x*(1 + x)), ((-1 + 2*eps)*(-1 + x))/(x*(1 + x)), 0}, {-((-2 + 3*eps)*(-1 + x)*(1 + x))/(2*x^2), ((-1 + x)*(1 + x^2))/(2*x^2*(1 + x)), ((-1 + eps)*(-1 + x)*(1 + x))/(2*x^2), (eps*(-1 + x))/(x*(1 + x))}} 2 | -------------------------------------------------------------------------------- /examples/lue_1_eps.m: -------------------------------------------------------------------------------- 1 | {{2*eps/(x+1)-6*eps/(x-1)-14*eps/x,-16/5*eps/(x+1)-24/5*eps/(x-1)-91/10*eps/x,0,0},{20*eps/x,6*eps/(x+1)+13*eps/x,0,0},{-245*eps/x,-287/6*eps/(x+1)-2065/12*eps/x,4*eps/(x+1)-2*eps/x,0},{0,-49/6*eps/(x+1)+49/12*eps/x,-2/3*eps/(x+1)+1/3*eps/x,2*eps/(x+1)-eps/x}} -------------------------------------------------------------------------------- /examples/lue_1_eps_t.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /normalize.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/env sage-python 2 | import logging 3 | import os.path 4 | import sys 5 | 6 | import fuchsia 7 | fuchsia.USE_MAPLE = True 8 | 9 | import sage.all 10 | print "Welcome to the Normalize Assistant!" 11 | 12 | def print_usage(): 13 | print "Usage:" 14 | print " normalize.py " 15 | 16 | if len(sys.argv) != 2: 17 | print "invalid arguments: normalize.py needs exactly 1 argument" 18 | print_usage() 19 | exit(1) 20 | fname = sys.argv[1] 21 | 22 | if not os.path.isfile(fname): 23 | print "'%s' does not exist or is not a file" % fname 24 | exit(1) 25 | 26 | fuchsia.logger.setLevel(logging.INFO) 27 | m0 = fuchsia.import_matrix_from_file(fname) 28 | x, ep = sage.all.var("x ep") 29 | 30 | print "You are normalizing matrix '%s' in (%s, %s)" % (fname, x, ep) 31 | 32 | m = fuchsia.FuchsianSystem.from_M(m0, x, ep) 33 | na = fuchsia.NormalizeAssistant(m) 34 | na.start() 35 | -------------------------------------------------------------------------------- /test/__init__.py: -------------------------------------------------------------------------------- 1 | import doctest 2 | import time 3 | import unittest 4 | 5 | import fuchsia 6 | 7 | class TimedTextTestResult(unittest.TextTestResult): 8 | def startTest(self, test): 9 | super(unittest.TextTestResult, self).startTest(test) 10 | test._startTime = time.time() 11 | self.stream.write(self.getDescription(test)) 12 | self.stream.write(" ... ") 13 | self.stream.flush() 14 | 15 | def addSuccess(self, test): 16 | super(unittest.TextTestResult, self).addSuccess(test) 17 | duration = time.time() - test._startTime 18 | self.stream.writeln("ok, %.3f sec" % duration) 19 | 20 | def addError(self, test, err): 21 | super(unittest.TextTestResult, self).addError(test, err) 22 | duration = time.time() - test._startTime 23 | self.stream.writeln("ERROR, %.3f sec" % duration) 24 | 25 | def addFailure(self, test, err): 26 | super(unittest.TextTestResult, self).addFailure(test, err) 27 | duration = time.time() - test._startTime 28 | self.stream.writeln("FAIL, %.3f sec" % duration) 29 | 30 | def addSkip(self, test, reason): 31 | super(unittest.TextTestResult, self).addSkip(test, reason) 32 | self.stream.writeln("skipped {0!r}".format(reason)) 33 | 34 | def addExpectedFailure(self, test, err): 35 | super(unittest.TextTestResult, self).addExpectedFailure(test, err) 36 | duration = time.time() - test._startTime 37 | self.stream.writeln("expected failure, %.3f sec" % duration) 38 | 39 | def addUnexpectedSuccess(self, test): 40 | super(unittest.TextTestResult, self).addUnexpectedSuccess(test) 41 | duration = time.time() - test._startTime 42 | self.stream.writeln("unexpected success, %.3f sec" % duration) 43 | 44 | unittest.TextTestRunner.resultclass = TimedTextTestResult 45 | 46 | class FastTestLoader(unittest.TestLoader): 47 | def getTestCaseNames(self, testCaseClass): 48 | fnNames = super(FastTestLoader, self).getTestCaseNames(testCaseClass) 49 | return [name for name in fnNames if "slow" not in name] 50 | 51 | def load_tests(loader, tests, ignore): 52 | tests.addTests(doctest.DocTestSuite(fuchsia)) 53 | return tests 54 | 55 | def test_suite_maxima(): 56 | loader = FastTestLoader() 57 | tests = doctest.DocTestSuite(fuchsia) 58 | tests.addTests(loader.discover("test")) 59 | return tests 60 | 61 | def full_test_suite(): 62 | loader = unittest.TestLoader() 63 | tests = doctest.DocTestSuite(fuchsia) 64 | tests.addTests(loader.discover("test")) 65 | return tests 66 | 67 | def test_suite_maple(): 68 | fuchsia.setup_fuchsia(use_maple=True) 69 | return full_test_suite() 70 | -------------------------------------------------------------------------------- /test/data/bolibrukh.mtx: -------------------------------------------------------------------------------- 1 | %%MatrixMarket matrix array Maple[symbolic] general 2 | 3 3 3 | 0 4 | 0 5 | 0 6 | -2/(2*x-1)+1/(x+1)+1/x^2 7 | -2/3/(2*x-1)-1/6/(x+1)-1/2/(x-1)+1/x 8 | -2/3/(2*x-1)-1/6/(x+1)+1/2/(x-1) 9 | -2/(2*x-1)+1/(x-1) 10 | 2/3/(2*x-1)+1/6/(x+1)-1/2/(x-1) 11 | 2/3/(2*x-1)+1/6/(x+1)+1/2/(x-1)-1/x 12 | -------------------------------------------------------------------------------- /test/data/git_409.m: -------------------------------------------------------------------------------- 1 | {{-4*eps/(x-1)+1/(x-1)+2*eps/((x-1)*x)-1/((x-1)*x),0,0,0,0,0},{0,-4*eps/(x-1)+1/(x-1)+3*eps/((x-1)*x)-1/((x-1)*x),0,0,0,0},{0,0,-3*eps/(x-1)+1/(x-1)+2*eps/((x-1)*x)-1/((x-1)*x),0,0,0},{0,6*eps^3/((x-1)*x)-5*eps^2/((x-1)*x)+eps/((x-1)*x),-4*eps^3/((x-1)*x)+4*eps^2/((x-1)*x)-eps/((x-1)*x),-2*eps/(x-1),0,0},{0,-6*eps^3/((x-1)^2*x^2)+18*eps^3/((x-1)^2*x^3)+5*eps^2/((x-1)^2*x^2)-15*eps^2/((x-1)^2*x^3)-eps/((x-1)^2*x^2)+3*eps/((x-1)^2*x^3),-8*eps^3/((x-1)^2*x^2)+8*eps^2/((x-1)^2*x^2)-2*eps/((x-1)^2*x^2),-2*eps/((x-1)^2*x),-4*eps/(x-1)-2/(x-1)+2*eps/((x-1)*x)+1/((x-1)*x),0},{0,12*eps^3/((x-1)*x^2)-10*eps^2/((x-1)*x^2)+2*eps/((x-1)*x^2),0,4*eps/((x-1)*x),0,-4*eps/x-1/x}} -------------------------------------------------------------------------------- /test/data/git_410.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /test/data/henn_324.m: -------------------------------------------------------------------------------- 1 | {{eps/x,0},{-1/x^2,eps/(x+1)}} -------------------------------------------------------------------------------- /test/data/lee_3.m: -------------------------------------------------------------------------------- 1 | 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-------------------------------------------------------------------------------- /test/data/pap_3_52.mtx: -------------------------------------------------------------------------------- 1 | %%MatrixMarket matrix array Fuchsia[symbolic] general 2 | 2 2 3 | -(540*(eps-2)*x^4-10950*(eps-2)*x^3+1440*x^4+27486*(eps-2)*x^2-25134*x^3-21175*(eps-2)*x+62247*x^2+5929*eps-47971*x)/((30*x^2-87*x+77)*(6*x-7)*(x-11)*x) 4 | -45*(24*x^2-28*x-21)*eps/((30*x^2-87*x+77)*(10*x-7)*(3*x-8)*x) 5 | 154*(24*x^2-28*x-21)*(eps-1)*x/((30*x^2-87*x+77)*(6*x-7)*(x-11)) 6 | -(2700*(eps-2)*x^4-13470*(eps-2)*x^3+8100*x^4+27444*(eps-2)*x^2-41250*x^3-23331*(eps-2)*x+79812*x^2+4312*eps-64183*x+4312)/((30*x^2-87*x+77)*(10*x-7)*(3*x-8)*x) 7 | -------------------------------------------------------------------------------- /test/test_block_triangular.py: -------------------------------------------------------------------------------- 1 | import unittest 2 | import random 3 | 4 | from sage.all import SR, matrix 5 | import fuchsia 6 | from fuchsia import (block_triangular_form, transform) 7 | 8 | class Test(unittest.TestCase): 9 | def assertIsTriangular(t, M1, M2, x, T, B): 10 | t.assertEqual(M2, transform(M1, x, T)) 11 | 12 | def test_block_triangular_form_1(t): 13 | # Test with no transformation needed 14 | M = matrix([ 15 | [1, 0, 0], 16 | [2, 3, 0], 17 | [4, 5, 6] 18 | ]) 19 | MM, T, B = block_triangular_form(M) 20 | t.assertEqual(MM, M) 21 | t.assertEqual(T, matrix.identity(3)) 22 | t.assertEqual(sorted(B), [(0, 1), (1, 1), (2, 1)]) 23 | 24 | def test_block_triangular_form_2(t): 25 | # Test with no transformation possible 26 | M = matrix([ 27 | [1, 2, 3, 4], 28 | [5, 6, 7, 8], 29 | [9, 1, 2, 3], 30 | [4, 5, 6, 7] 31 | ]) 32 | MM, T, B = block_triangular_form(M) 33 | t.assertEqual(MM, M) 34 | t.assertEqual(T, matrix.identity(4)) 35 | t.assertEqual(B, [(0, 4)]) 36 | 37 | def test_block_triangular_form_3(t): 38 | m = matrix([ 39 | [1, 0, 1, 0], 40 | [0, 1, 0, 1], 41 | [1, 0, 1, 0], 42 | [0, 0, 0, 1] 43 | ]) 44 | mt, tt, b = block_triangular_form(m) 45 | x = SR.var("dummy") 46 | t.assertEqual(mt, transform(m, x, tt)) 47 | t.assertEqual(matrix([ 48 | [1, 1, 0, 0], 49 | [1, 1, 0, 0], 50 | [0, 0, 1, 0], 51 | [0, 0, 1, 1] 52 | ]), mt) 53 | 54 | def test_block_triangular_form_4(t): 55 | M = matrix([ 56 | [1, 2, 3, 0, 0, 0], 57 | [4, 5, 6, 0, 0, 0], 58 | [7, 8, 9, 0, 0, 0], 59 | [2, 0, 0, 1, 2, 0], 60 | [0, 2, 0, 3, 4, 0], 61 | [0, 0, 2, 0, 0, 1] 62 | ]) 63 | x = SR.var("dummy") 64 | T = matrix.identity(6)[random.sample(xrange(6), 6),:] 65 | M = transform(M, x, T) 66 | MM, T, B = block_triangular_form(M) 67 | t.assertEqual(MM, transform(M, x, T)) 68 | t.assertEqual(sorted(s for o, s in B), [1, 2, 3]) 69 | for o, s in B: 70 | for i in xrange(s): 71 | for j in xrange(s): 72 | MM[o + i, o + j] = 0 73 | for i in xrange(6): 74 | for j in xrange(i): 75 | MM[i, j] = 0 76 | t.assertEqual(MM, matrix(6)) 77 | -------------------------------------------------------------------------------- /test/test_cli.py: -------------------------------------------------------------------------------- 1 | import os 2 | import subprocess 3 | import tempfile 4 | import unittest 5 | 6 | import fuchsia 7 | from sage.all import SR 8 | 9 | class Temp: 10 | def __enter__(self): 11 | fd, self.name = tempfile.mkstemp(prefix="fuchsia-") 12 | os.close(fd) 13 | return self.name 14 | 15 | def __exit__(self, exc_type, exc_value, traceback): 16 | os.remove(self.name) 17 | 18 | def sh(*cmd): 19 | p = subprocess.Popen(cmd, stdout=subprocess.PIPE, stderr=subprocess.PIPE) 20 | stdout, stderr = p.communicate() 21 | if p.returncode != 0: 22 | raise Exception("Command %s exited with code %s" % (cmd, p.returncode)) 23 | return stdout, stderr 24 | 25 | class Test(unittest.TestCase): 26 | def assertTransformation(t, m1_path, x_name, t_path, m2_path): 27 | M1 = fuchsia.import_matrix_from_file(m1_path) 28 | T = fuchsia.import_matrix_from_file(t_path) 29 | M2 = fuchsia.import_matrix_from_file(m2_path) 30 | t.assertEqual(M2.simplify_rational(), 31 | fuchsia.transform(M1, SR.var(x_name), T).simplify_rational()) 32 | 33 | def assertIsFuchsian(t, m_path, x_name): 34 | M = fuchsia.import_matrix_from_file(m_path) 35 | x = SR.var(x_name) 36 | pranks = fuchsia.singularities(M, x).values() 37 | t.assertEqual(pranks, [0]*len(pranks)) 38 | 39 | def assertIsReduced(t, m_path, x_name, eps_name): 40 | M = fuchsia.import_matrix_from_file(m_path) 41 | x = SR.var(x_name) 42 | eps = SR.var(eps_name) 43 | pranks = fuchsia.singularities(M, x).values() 44 | t.assertEqual(pranks, [0]*len(pranks)) 45 | t.assertTrue(eps not in (M/eps).simplify_rational().variables()) 46 | 47 | def test_help(t): 48 | sh("sage", "-python", "fuchsia.py", "-h") 49 | 50 | def test_fuchsify_henn_324(t): 51 | with Temp() as mfile, Temp() as tfile: 52 | sh("sage", "-python", "fuchsia.py", "fuchsify", "-m", mfile, "-t", tfile, 53 | "test/data/henn_324.m") 54 | t.assertTransformation("test/data/henn_324.m", "x", tfile, mfile) 55 | t.assertIsFuchsian(mfile, "x") 56 | 57 | def test_fuchsify_git_409(t): 58 | with Temp() as mfile, Temp() as tfile: 59 | sh("sage", "-python", "fuchsia.py", "fuchsify", "-m", mfile, "-t", tfile, 60 | "test/data/git_409.m") 61 | t.assertTransformation("test/data/git_409.m", "x", tfile, mfile) 62 | t.assertIsFuchsian(mfile, "x") 63 | 64 | def test_reduce_git_410_slow(t): 65 | with Temp() as mfile, Temp() as tfile: 66 | sh("sage", "-python", "fuchsia.py", "reduce", "--use-maple", "-m", mfile, "-t", tfile, 67 | "test/data/git_410.m") 68 | t.assertTransformation("test/data/git_410.m", "x", tfile, mfile) 69 | t.assertIsReduced(mfile, "x", "eps") 70 | 71 | def test_reduce_lee_3_slow(t): 72 | with Temp() as mfile, Temp() as tfile: 73 | sh("sage", "-python", "fuchsia.py", "reduce", "--use-maple", "-m", mfile, "-t", tfile, 74 | "test/data/lee_3.m") 75 | t.assertTransformation("test/data/lee_3.m", "x", tfile, mfile) 76 | t.assertIsReduced(mfile, "x", "eps") 77 | 78 | def test_reduce_pap_1_slow(t): 79 | with Temp() as mfile, Temp() as tfile: 80 | sh("sage", "-python", "fuchsia.py", "reduce", "--use-maple", "-m", mfile, "-t", tfile, 81 | "-f", "m", "-e", "ep", "test/data/pap_1.m") 82 | t.assertTransformation("test/data/pap_1.m", "x", tfile, mfile) 83 | t.assertIsReduced(mfile, "x", "ep") 84 | -------------------------------------------------------------------------------- /test/test_epsilon_form.py: -------------------------------------------------------------------------------- 1 | import os.path 2 | import unittest 3 | 4 | from sage.all import SR 5 | from fuchsia import (epsilon_form, import_matrix_from_file, is_fuchsian, is_normalized, 6 | transform, singularities) 7 | 8 | class Test(unittest.TestCase): 9 | def assertReductionWorks(test, filename, fuchsian=False): 10 | m = import_matrix_from_file(filename) 11 | x, eps = SR.var("x eps") 12 | test.assertIn(x, m.variables()) 13 | 14 | if not fuchsian: 15 | m_pranks = singularities(m, x).values() 16 | test.assertNotEqual(m_pranks, [0]*len(m_pranks)) 17 | 18 | mt, t = epsilon_form(m, x, eps) 19 | test.assertTrue((mt-transform(m, x, t)).simplify_rational().is_zero()) 20 | test.assertTrue(is_fuchsian(mt, x)) 21 | test.assertTrue(is_normalized(mt, x, eps)) 22 | test.assertNotIn(eps, (mt/eps).simplify_rational().variables()) 23 | 24 | def test_git_409(t): 25 | t.assertReductionWorks(os.path.join(os.path.dirname(__file__), 26 | "data", "git_409.m")) 27 | 28 | def test_git_410(t): 29 | t.assertReductionWorks(os.path.join(os.path.dirname(__file__), 30 | "data", "git_410.m")) 31 | 32 | def test_henn_324(t): 33 | t.assertReductionWorks(os.path.join(os.path.dirname(__file__), 34 | "data", "henn_324.m")) 35 | 36 | def test_pap_3_50_slow(t): 37 | t.assertReductionWorks(os.path.join(os.path.dirname(__file__), 38 | "data", "pap_3_50.m"), fuchsian=True) 39 | -------------------------------------------------------------------------------- /test/test_fuchsia.py: -------------------------------------------------------------------------------- 1 | import unittest 2 | from random import randint 3 | from StringIO import StringIO 4 | 5 | from sage.all import SR 6 | import fuchsia 7 | from fuchsia import \ 8 | balance, balance_transform, identity_matrix, import_matrix_mathematica, \ 9 | import_matrix_matrixmarket, import_matrix_from_file, \ 10 | export_matrix_mathematica, export_matrix_matrixmarket, factorize, \ 11 | fuchsify, limit_fixed, matrix, matrix_complexity, oo, Rational, \ 12 | reduce_at_one_point, simplify_by_jordanification, singularities, \ 13 | transform, FuchsiaError 14 | 15 | def randpoly(x, maxrank=3): 16 | return sum(randint(-3, 3)*x**i for i in range(maxrank + 1)) 17 | 18 | def randrat(x, maxrank=3): 19 | return randpoly(x, maxrank)/randpoly(x, maxrank) 20 | 21 | def randpolym(x, size, maxrank=3): 22 | return matrix([ 23 | [randpoly(x, maxrank) for j in range(size)] 24 | for i in range(size) 25 | ]) 26 | 27 | def randratm(x, size, maxrank=3): 28 | return matrix([ 29 | [randrat(x, maxrank) for j in range(size)] 30 | for i in range(size) 31 | ]) 32 | 33 | class Test(unittest.TestCase): 34 | def test_import_matrix_from_file_1(t): 35 | x, eps = SR.var("x eps") 36 | m = import_matrix_from_file("test/data/henn_324.m") 37 | t.assertEqual(set(m.variables()), set([x, eps])) 38 | t.assertEqual(m, matrix([[eps/x, 0], [-1/x**2, eps/(x + 1)]])) 39 | 40 | def test_import_export_matrixmarket(t): 41 | a, b = SR.var("v1 v2") 42 | M = matrix([[1, a, b], [a + b, Rational((2, 3)), a/b]]) 43 | fout = StringIO() 44 | export_matrix_matrixmarket(fout, M) 45 | MM = import_matrix_matrixmarket(StringIO(fout.getvalue())) 46 | t.assertEqual(M, MM) 47 | 48 | def test_import_export_mathematica(t): 49 | a, b = SR.var("v1 v2") 50 | M = matrix([[1, a, b], [a + b, Rational((2, 3)), a/b]]) 51 | fout = StringIO() 52 | export_matrix_mathematica(fout, M) 53 | MM = import_matrix_mathematica(StringIO(fout.getvalue())) 54 | t.assertEqual(M, MM) 55 | 56 | def test_transform_1(t): 57 | # transform(M, x, I) == M 58 | x = SR.var("x") 59 | M = randpolym(x, 3) 60 | MM = transform(M, x, identity_matrix(M.nrows())) 61 | t.assertEqual(MM, M) 62 | 63 | def test_transform_2(t): 64 | # transform(transform(M, x, I), x, I^-1) == M 65 | x = SR.var("x") 66 | M = randpolym(x, 2) 67 | T = randpolym(x, 2) 68 | invT = T.inverse() 69 | M1 = transform(M, x, T) 70 | M2 = transform(M1, x, invT) 71 | t.assertEqual(M2.simplify_rational(), M) 72 | 73 | def test_balance_1(t): 74 | # balance(P, x1, x2, x)*balance(P, x2, x1, x) == I 75 | x = SR.var("x") 76 | P = matrix([[1, 1], [0, 0]]) 77 | x1 = randint(-10, 10) 78 | x2 = randint(20, 30) 79 | b1 = balance(P, x1, x2, x) 80 | b2 = balance(P, x2, x1, x) 81 | t.assertEqual((b1*b2).simplify_rational(), identity_matrix(P.nrows())) 82 | t.assertEqual((b2*b1).simplify_rational(), identity_matrix(P.nrows())) 83 | 84 | def test_balance_2(t): 85 | # balance(P, x1, oo, x)*balance(P, oo, x1, x) == I 86 | x = SR.var("x") 87 | P = matrix([[1, 1], [0, 0]]) 88 | x1 = randint(-10, 10) 89 | b1 = balance(P, x1, oo, x) 90 | b2 = balance(P, oo, x1, x) 91 | t.assertEqual((b1*b2).simplify_rational(), identity_matrix(P.nrows())) 92 | t.assertEqual((b2*b1).simplify_rational(), identity_matrix(P.nrows())) 93 | 94 | def test_balance_transform_1(t): 95 | x = SR.var("x") 96 | M = randpolym(x, 2) 97 | P = matrix([[1, 1], [0, 0]]) 98 | x1 = randint(-10, 10) 99 | x2 = randint(20, 30) 100 | b1 = balance(P, x1, x2, x) 101 | 102 | M1 = balance_transform(M, P, x1, x2, x) 103 | M2 = transform(M, x, balance(P, x1, x2, x)) 104 | t.assertEqual(M1.simplify_rational(), M2.simplify_rational()) 105 | 106 | M1 = balance_transform(M, P, x1, oo, x) 107 | M2 = transform(M, x, balance(P, x1, oo, x)) 108 | t.assertEqual(M1.simplify_rational(), M2.simplify_rational()) 109 | 110 | M1 = balance_transform(M, P, oo, x2, x) 111 | M2 = transform(M, x, balance(P, oo, x2, x)) 112 | t.assertEqual(M1.simplify_rational(), M2.simplify_rational()) 113 | 114 | def test_reduce_at_one_point_1(t): 115 | x = SR.var("x") 116 | M0 = matrix([ 117 | [1/x, 4, 0, 5], 118 | [0, 2/x, 0, 0], 119 | [0, 0, 3/x, 6], 120 | [0, 0, 0, 4/x] 121 | ]) 122 | 123 | u = matrix([ 124 | [0, Rational((3, 5)), Rational((4, 5)), 0], 125 | [Rational((5, 13)), 0, 0, Rational((12, 13))] 126 | ]) 127 | M1 = transform(M0, x, balance(u.transpose()*u, 0, 1, x)) 128 | M1 = M1.simplify_rational() 129 | 130 | u = matrix([[8, 0, 15, 0]])/17 131 | M2 = transform(M1, x, balance(u.transpose()*u, 0, 2, x)) 132 | M2 = M2.simplify_rational() 133 | 134 | M2_sing = singularities(M2, x) 135 | t.assertIn(0, M2_sing) 136 | t.assertEqual(M2_sing[0], 2) 137 | 138 | M3, T23 = reduce_at_one_point(M2, x, 0, 2) 139 | M3 = M3.simplify_rational() 140 | t.assertEqual(M3, transform(M2, x, T23).simplify_rational()) 141 | 142 | M3_sing = singularities(M3, x) 143 | t.assertIn(0, M3_sing) 144 | t.assertEqual(M3_sing[0], 1) 145 | 146 | M4, T34 = reduce_at_one_point(M3, x, 0, 1) 147 | M4 = M4.simplify_rational() 148 | t.assertEqual(M4, transform(M3, x, T34).simplify_rational()) 149 | 150 | M4_sing = singularities(M4, x) 151 | t.assertIn(0, M4_sing) 152 | t.assertEqual(M4_sing[0], 0) 153 | 154 | def test_fuchsify_1(t): 155 | x = SR.var("x") 156 | M = matrix([ 157 | [1/x, 5, 0, 6], 158 | [0, 2/x, 0, 0], 159 | [0, 0, 3/x, 7], 160 | [0, 0, 0, 4/x] 161 | ]) 162 | 163 | u = matrix([ 164 | [0, Rational((3, 5)), Rational((4, 5)), 0], 165 | [Rational((5, 13)), 0, 0, Rational((12, 13))] 166 | ]) 167 | M = transform(M, x, balance(u.transpose()*u, 0, 1, x)) 168 | M = M.simplify_rational() 169 | 170 | u = matrix([[8, 0, 15, 0]])/17 171 | M = transform(M, x, balance(u.transpose()*u, 0, 2, x)) 172 | M = M.simplify_rational() 173 | 174 | Mx, T = fuchsify(M, x) 175 | Mx = Mx.simplify_rational() 176 | t.assertEqual(Mx, transform(M, x, T).simplify_rational()) 177 | 178 | pranks = singularities(Mx, x).values() 179 | t.assertEqual(pranks, [0]*len(pranks)) 180 | 181 | def test_fuchsify_2(t): 182 | x = SR.var("x") 183 | M = matrix([ 184 | [0, 1/x/(x-1), 0, 0], 185 | [0, 0, 0, 0], 186 | [0, 0, 0, 0], 187 | [0, 0, 0, 0] 188 | ]) 189 | u = matrix([[6, 3, 2, 0]])/7 190 | P = u.transpose()*u 191 | M = balance_transform(M, P, 1, 0, x).simplify_rational() 192 | M = balance_transform(M, P, 1, 0, x).simplify_rational() 193 | M = balance_transform(M, P, 1, 0, x).simplify_rational() 194 | M = balance_transform(M, P, 1, 0, x).simplify_rational() 195 | M = balance_transform(M, P, 1, 0, x).simplify_rational() 196 | 197 | MM, T = fuchsify(M, x) 198 | MM = MM.simplify_rational() 199 | t.assertEqual(MM, transform(M, x, T).simplify_rational()) 200 | 201 | pranks = singularities(MM, x).values() 202 | t.assertEqual(pranks, [0]*len(pranks)) 203 | 204 | def test_factorize_1(t): 205 | x = SR.var("x") 206 | e = SR.var("epsilon") 207 | M = matrix([[1/x, 0, 0], [0, 2/x, 0], [0, 0, 3/x]])*e 208 | M = transform(M, x, matrix([[1, 1, 0], [0, 1, 0],[1+2*e, 0, e]])) 209 | F,T = factorize(M, x, e) 210 | F = F.simplify_rational() 211 | for f in F.list(): 212 | t.assertEqual(limit_fixed(f, e, 0), 0) 213 | 214 | def test_factorize_2(t): 215 | x = SR.var("x") 216 | e = SR.var("epsilon") 217 | M = matrix([[e*e/x]]) 218 | with t.assertRaises(FuchsiaError): 219 | F,T = factorize(M, x, e) 220 | 221 | def test_simplify_by_jordanification(t): 222 | x = SR.var("x") 223 | M = matrix([ 224 | [4/(x+1), -1/(6*x*(x+1)), -1/(3*x*(x+1))], 225 | [6*(13*x+6)/(x*(x+1)),-5*(x+3)/(3*x*(x+1)),2*(x-6)/(3*x*(x+1))], 226 | [-63*(x-1)/(x*(x+1)), (5*x-9)/(6*x*(x+1)), -(x-18)/(3*x*(x+1))] 227 | ]).simplify_rational() 228 | MM, T = simplify_by_jordanification(M, x) 229 | MM = MM.simplify_rational() 230 | t.assertEqual(MM, transform(M, x, T).simplify_rational()) 231 | t.assertLess(matrix_complexity(MM), matrix_complexity(M)) 232 | -------------------------------------------------------------------------------- /test/test_fuchsify_by_blocks.py: -------------------------------------------------------------------------------- 1 | import os.path 2 | import unittest 3 | 4 | from sage.all import SR, matrix 5 | from fuchsia import (is_fuchsian, fuchsify_off_diagonal_blocks, transform, _parser) 6 | 7 | class Test(unittest.TestCase): 8 | def assert_fuchsify_by_blocks_works(test, m,b,x,eps): 9 | test.assertFalse(is_fuchsian(m, x)) 10 | 11 | mt, t = fuchsify_off_diagonal_blocks(m, x, eps, b=b) 12 | test.assertTrue((mt-transform(m, x, t)).simplify_rational().is_zero()) 13 | test.assertTrue(is_fuchsian(mt, x)) 14 | 15 | def test_fuchsify_by_blocks_01(test): 16 | x, eps = SR.var("x eps") 17 | m = matrix([ 18 | [ eps/x, 0, 0], 19 | [ 0, 2*eps/x, -eps/x], 20 | [1/x**2, 0, 3*eps/x], 21 | ]) 22 | b = [(0,1),(1,2)] 23 | test.assert_fuchsify_by_blocks_works(m, b, x, eps) 24 | 25 | def test_fuchsify_by_blocks_02(test): 26 | x, eps = SR.var("x eps") 27 | m = matrix([ 28 | [ eps/x/(x-1), 0, 0], 29 | [ 1/(x-1)**2, 2*eps/x, -eps/x], 30 | [ 1/x**2, 0, 3*eps/x], 31 | ]) 32 | b = [(0,1),(1,2)] 33 | test.assert_fuchsify_by_blocks_works(m, b, x, eps) 34 | 35 | def test_fuchsify_by_blocks_03(test): 36 | x, eps = SR.var("x eps") 37 | m = matrix([ 38 | [ eps/x, 0, 0], 39 | [ 1/x**2, 2*eps/x, 0], 40 | [ 1/x**2, 2/x**2, 3*eps/x], 41 | ]) 42 | b = [(0,1),(1,1),(2,1)] 43 | test.assert_fuchsify_by_blocks_works(m, b, x, eps) 44 | 45 | def test_fuchsify_by_blocks_04(test): 46 | x, eps = SR.var("x eps") 47 | m = matrix([ 48 | [ eps/x/(x-1), 0, 0], 49 | [ 1/x**2/(x-1)**2, 2*eps/x/(x-1), 0], 50 | [ 1/x**2/(x-1)**2, 2/x**2/(x-1)**2, 3*eps/x/(x-1)], 51 | ]) 52 | b = [(0,1),(1,1),(2,1)] 53 | test.assert_fuchsify_by_blocks_works(m, b, x, eps) 54 | 55 | def test_fuchsify_by_blocks_05(test): 56 | x, eps = SR.var("x eps") 57 | m = matrix([ 58 | [ eps/x, 0, 0], 59 | [ 0, 2*eps/x, -eps/x], 60 | [ x**2, 0, 3*eps/x], 61 | ]) 62 | b = [(0,1),(1,2)] 63 | test.assert_fuchsify_by_blocks_works(m, b, x, eps) 64 | 65 | 66 | def test_fuchsify_by_blocks_06(test): 67 | x, eps = SR.var("x eps") 68 | m = matrix([ 69 | [eps/(x - 1), 0, 0], 70 | [(x**3 + x**2 - 3*x - 3)/(2*x**3 - 2*x**2 - x + 1), 2*eps/(x - 3), 0], 71 | [-(x**3 + 3*x**2 - 3*x + 2)/(2*x**3 + x**2 - 2*x) + 2*(x**3 - x**2 - x + 1)/(2*x + 1), 72 | -3*(x**3 + x - 1)/(3*x**3 + x - 2), 73 | 4*eps/(x - 5)] 74 | ]) 75 | b = [(0,1),(1,1),(2,1)] 76 | test.assert_fuchsify_by_blocks_works(m, b, x, eps) 77 | 78 | def test_fuchsify_by_blocks_07(test): 79 | x, eps = SR.var("x ep") 80 | m = matrix([ 81 | [0,0], 82 | [1/(x**2-x+1)**2, 0] 83 | ]) 84 | b = [(0,1),(1,1)] 85 | test.assert_fuchsify_by_blocks_works(m, b, x, eps) 86 | -------------------------------------------------------------------------------- /test/test_normalize.py: -------------------------------------------------------------------------------- 1 | import unittest 2 | 3 | from sage.all import SR 4 | from fuchsia import (is_normalized, matrix, matrix_c0, 5 | normalize, singularities, transform, import_matrix_from_file, 6 | fuchsify, FuchsiaError) 7 | 8 | class Test(unittest.TestCase): 9 | 10 | def test_is_normalized_1(t): 11 | x = SR.var("x") 12 | e = SR.var("epsilon") 13 | t.assertFalse(is_normalized(matrix([[1/x/2]]), x, e)) 14 | t.assertFalse(is_normalized(matrix([[-1/x/2]]), x, e)) 15 | t.assertTrue (is_normalized(matrix([[1/x/3]]), x, e)) 16 | t.assertFalse(is_normalized(matrix([[x]]), x, e)) 17 | t.assertFalse(is_normalized(matrix([[1/x**2]]), x, e)) 18 | t.assertTrue (is_normalized( \ 19 | matrix([[(e+SR(1)/3)/x-SR(1)/2/(x-1)]]), x, e)) 20 | 21 | def test_normalize_1(t): 22 | # Test with apparent singularities at 0 and oo, but not at 1. 23 | x = SR.var("x") 24 | M = matrix([ 25 | [1/x, 5/(x-1), 0, 6/(x-1)], 26 | [0, 2/x, 0, 0], 27 | [0, 0, 3/x, 7/(x-1)], 28 | [6/(x-1), 0, 0, 1/x] 29 | ]) 30 | 31 | N, T = normalize(M, x, SR.var("epsilon")) 32 | N = N.simplify_rational() 33 | t.assertEqual(N, transform(M, x, T).simplify_rational()) 34 | for point, prank in singularities(N, x).iteritems(): 35 | R = matrix_c0(N, x, point, prank) 36 | evlist = R.eigenvalues() 37 | t.assertEqual(evlist, [0]*len(evlist)) 38 | 39 | def test_normalize_2(t): 40 | # Test with apparent singularities at 0, 1, and oo. 41 | x = SR.var("x") 42 | M = matrix([ 43 | [1/x, 5/(x-1), 0, 6/(x-1)], 44 | [0, 2/(x-1), 0, 0], 45 | [0, 0, 3/x, 7/(x-1)], 46 | [6/(x-1), 0, 0, 1/x] 47 | ]) 48 | 49 | N, T = normalize(M, x, SR.var("epsilon")) 50 | N = N.simplify_rational() 51 | t.assertEqual(N, transform(M, x, T).simplify_rational()) 52 | for point, prank in singularities(N, x).iteritems(): 53 | R = matrix_c0(N, x, point, prank) 54 | evlist = R.eigenvalues() 55 | t.assertEqual(evlist, [0]*len(evlist)) 56 | 57 | def test_normalize_3(t): 58 | # Test with non-zero normalized eigenvalues 59 | x = SR.var("x") 60 | e = SR.var("epsilon") 61 | M = matrix([ 62 | [(1-e)/x, 0], 63 | [0, (1+e)/3/x] 64 | ]) 65 | 66 | with t.assertRaises(FuchsiaError): 67 | N, T = normalize(M, x, e) 68 | 69 | def test_normalize_4(t): 70 | # Test with non-zero normalized eigenvalues 71 | x, e = SR.var("x eps") 72 | M = matrix([ 73 | [1/x/2, 0], 74 | [0, 0] 75 | ]) 76 | 77 | with t.assertRaises(FuchsiaError): 78 | N, T = normalize(M, x, e) 79 | 80 | def test_normalize_5(t): 81 | # An unnormalizable example by A. A. Bolibrukh 82 | x, e = SR.var("x eps") 83 | b = import_matrix_from_file("test/data/bolibrukh.mtx") 84 | f, ft = fuchsify(b, x) 85 | f_pranks = singularities(f, x).values() 86 | t.assertEqual(f_pranks, [0]*len(f_pranks)) 87 | 88 | with t.assertRaises(FuchsiaError): 89 | n, nt = normalize(f, x, e) 90 | 91 | def test_pap_3_52_slow(t): 92 | x, eps = SR.var("x eps") 93 | M = import_matrix_from_file("test/data/pap_3_52.mtx") 94 | N, T = normalize(M, x, eps) 95 | N = N.simplify_rational() 96 | t.assertEqual(N, transform(M, x, T).simplify_rational()) 97 | -------------------------------------------------------------------------------- /test/test_normalize_by_blocks.py: -------------------------------------------------------------------------------- 1 | import os.path 2 | import unittest 3 | 4 | from sage.all import SR 5 | from fuchsia import (block_triangular_form, import_matrix_from_file, is_normalized, 6 | are_diagonal_blocks_reduced, reduce_diagonal_blocks, transform, simplify_by_factorization, 7 | singularities) 8 | 9 | class Test(unittest.TestCase): 10 | def assertNormalizeBlocksWorks(test, filename): 11 | x, eps = SR.var("x eps") 12 | 13 | m = import_matrix_from_file(filename) 14 | test.assertIn(x, m.variables()) 15 | test.assertIn(eps, m.variables()) 16 | test.assertFalse(is_normalized(m, x, eps)) 17 | 18 | m_pranks = singularities(m, x).values() 19 | test.assertNotEqual(m_pranks, [0]*len(m_pranks)) 20 | 21 | m, t, b = block_triangular_form(m) 22 | mt, tt = reduce_diagonal_blocks(m, x, eps, b=b) 23 | t = t*tt 24 | test.assertTrue((mt-transform(m, x, t)).simplify_rational().is_zero()) 25 | test.assertTrue(are_diagonal_blocks_reduced(mt, b, x, eps)) 26 | 27 | def test_git_409(test): 28 | test.assertNormalizeBlocksWorks(os.path.join(os.path.dirname(__file__), 29 | "data", "git_409.m")) 30 | 31 | def test_git_410(test): 32 | test.assertNormalizeBlocksWorks(os.path.join(os.path.dirname(__file__), 33 | "data", "git_410.m")) 34 | -------------------------------------------------------------------------------- /test/test_reduce.py: -------------------------------------------------------------------------------- 1 | import os.path 2 | import unittest 3 | 4 | from sage.all import SR 5 | from fuchsia import (import_matrix_from_file, is_normalized, factorize, 6 | fuchsify, normalize, transform, simplify_by_factorization, singularities) 7 | 8 | class Test(unittest.TestCase): 9 | def assertReductionWorks(t, filename): 10 | M = import_matrix_from_file(filename) 11 | x, eps = SR.var("x eps") 12 | t.assertIn(x, M.variables()) 13 | M_pranks = singularities(M, x).values() 14 | t.assertNotEqual(M_pranks, [0]*len(M_pranks)) 15 | 16 | #1 Fuchsify 17 | m, t1 = simplify_by_factorization(M, x) 18 | Mf, t2 = fuchsify(m, x) 19 | Tf = t1*t2 20 | t.assertTrue((Mf-transform(M, x, Tf)).simplify_rational().is_zero()) 21 | Mf_pranks = singularities(Mf, x).values() 22 | t.assertEqual(Mf_pranks, [0]*len(Mf_pranks)) 23 | 24 | #2 Normalize 25 | t.assertFalse(is_normalized(Mf, x, eps)) 26 | m, t1 = simplify_by_factorization(Mf, x) 27 | Mn, t2 = normalize(m, x, eps) 28 | Tn = t1*t2 29 | t.assertTrue((Mn-transform(Mf, x, Tn)).simplify_rational().is_zero()) 30 | t.assertTrue(is_normalized(Mn, x, eps)) 31 | 32 | #3 Factorize 33 | t.assertIn(eps, Mn.variables()) 34 | m, t1 = simplify_by_factorization(Mn, x) 35 | Mc, t2 = factorize(m, x, eps, seed=3) 36 | Tc = t1*t2 37 | t.assertTrue((Mc-transform(Mn, x, Tc)).simplify_rational().is_zero()) 38 | t.assertNotIn(eps, (Mc/eps).simplify_rational().variables()) 39 | 40 | def test_git_409(t): 41 | t.assertReductionWorks(os.path.join(os.path.dirname(__file__), 42 | "data", "git_409.m")) 43 | 44 | def test_git_410_slow(t): 45 | t.assertReductionWorks(os.path.join(os.path.dirname(__file__), 46 | "data", "git_410.m")) 47 | 48 | def test_henn_324(t): 49 | t.assertReductionWorks(os.path.join(os.path.dirname(__file__), 50 | "data", "henn_324.m")) 51 | --------------------------------------------------------------------------------