diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,674 @@ +GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. By contrast, +the GNU General Public License is intended to guarantee your freedom to +share and change all versions of a program--to make sure it remains free +software for all its users. We, the Free Software Foundation, use the +GNU General Public License for most of our software; it applies also to +any other work released this way by its authors. You can apply it to +your programs, too. + + When we speak of free software, we are referring to freedom, not +price. Our General Public Licenses are designed to make sure that you +have the freedom to distribute copies of free software (and charge for +them if you wish), that you receive source code or can get it if you +want it, that you can change the software or use pieces of it in new +free programs, and that you know you can do these things. + + To protect your rights, we need to prevent others from denying you +these rights or asking you to surrender the rights. Therefore, you have +certain responsibilities if you distribute copies of the software, or if +you modify it: responsibilities to respect the freedom of others. + + For example, if you distribute copies of such a program, whether +gratis or for a fee, you must pass on to the recipients the same +freedoms that you received. You must make sure that they, too, receive +or can get the source code. And you must show them these terms so they +know their rights. + + Developers that use the GNU GPL protect your rights with two steps: +(1) assert copyright on the software, and (2) offer you this License +giving you legal permission to copy, distribute and/or modify it. + + For the developers' and authors' protection, the GPL clearly explains +that there is no warranty for this free software. For both users' and +authors' sake, the GPL requires that modified versions be marked as +changed, so that their problems will not be attributed erroneously to +authors of previous versions. + + Some devices are designed to deny users access to install or run +modified versions of the software inside them, although the manufacturer +can do so. This is fundamentally incompatible with the aim of +protecting users' freedom to change the software. The systematic +pattern of such abuse occurs in the area of products for individuals to +use, which is precisely where it is most unacceptable. Therefore, we +have designed this version of the GPL to prohibit the practice for those +products. If such problems arise substantially in other domains, we +stand ready to extend this provision to those domains in future versions +of the GPL, as needed to protect the freedom of users. + + Finally, every program is threatened constantly by software patents. +States should not allow patents to restrict development and use of +software on general-purpose computers, but in those that do, we wish to +avoid the special danger that patents applied to a free program could +make it effectively proprietary. To prevent this, the GPL assures that +patents cannot be used to render the program non-free. + + The precise terms and conditions for copying, distribution and +modification follow. + + TERMS AND CONDITIONS + + 0. Definitions. + + "This License" refers to version 3 of the GNU General Public License. + + "Copyright" also means copyright-like laws that apply to other kinds of +works, such as semiconductor masks. + + "The Program" refers to any copyrightable work licensed under this +License. Each licensee is addressed as "you". "Licensees" and +"recipients" may be individuals or organizations. + + To "modify" a work means to copy from or adapt all or part of the work +in a fashion requiring copyright permission, other than the making of an +exact copy. The resulting work is called a "modified version" of the +earlier work or a work "based on" the earlier work. + + A "covered work" means either the unmodified Program or a work based +on the Program. + + To "propagate" a work means to do anything with it that, without +permission, would make you directly or secondarily liable for +infringement under applicable copyright law, except executing it on a +computer or modifying a private copy. Propagation includes copying, +distribution (with or without modification), making available to the +public, and in some countries other activities as well. + + To "convey" a work means any kind of propagation that enables other +parties to make or receive copies. Mere interaction with a user through +a computer network, with no transfer of a copy, is not conveying. + + An interactive user interface displays "Appropriate Legal Notices" +to the extent that it includes a convenient and prominently visible +feature that (1) displays an appropriate copyright notice, and (2) +tells the user that there is no warranty for the work (except to the +extent that warranties are provided), that licensees may convey the +work under this License, and how to view a copy of this License. If +the interface presents a list of user commands or options, such as a +menu, a prominent item in the list meets this criterion. + + 1. Source Code. + + The "source code" for a work means the preferred form of the work +for making modifications to it. "Object code" means any non-source +form of a work. + + A "Standard Interface" means an interface that either is an official +standard defined by a recognized standards body, or, in the case of +interfaces specified for a particular programming language, one that +is widely used among developers working in that language. + + The "System Libraries" of an executable work include anything, other +than the work as a whole, that (a) is included in the normal form of +packaging a Major Component, but which is not part of that Major +Component, and (b) serves only to enable use of the work with that +Major Component, or to implement a Standard Interface for which an +implementation is available to the public in source code form. A +"Major Component", in this context, means a major essential component +(kernel, window system, and so on) of the specific operating system +(if any) on which the executable work runs, or a compiler used to +produce the work, or an object code interpreter used to run it. + + The "Corresponding Source" for a work in object code form means all +the source code needed to generate, install, and (for an executable +work) run the object code and to modify the work, including scripts to +control those activities. However, it does not include the work's +System Libraries, or general-purpose tools or generally available free +programs which are used unmodified in performing those activities but +which are not part of the work. For example, Corresponding Source +includes interface definition files associated with source files for +the work, and the source code for shared libraries and dynamically +linked subprograms that the work is specifically designed to require, +such as by intimate data communication or control flow between those +subprograms and other parts of the work. + + The Corresponding Source need not include anything that users +can regenerate automatically from other parts of the Corresponding +Source. + + The Corresponding Source for a work in source code form is that +same work. + + 2. Basic Permissions. + + All rights granted under this License are granted for the term of +copyright on the Program, and are irrevocable provided the stated +conditions are met. This License explicitly affirms your unlimited +permission to run the unmodified Program. The output from running a +covered work is covered by this License only if the output, given its +content, constitutes a covered work. This License acknowledges your +rights of fair use or other equivalent, as provided by copyright law. + + You may make, run and propagate covered works that you do not +convey, without conditions so long as your license otherwise remains +in force. You may convey covered works to others for the sole purpose +of having them make modifications exclusively for you, or provide you +with facilities for running those works, provided that you comply with +the terms of this License in conveying all material for which you do +not control copyright. Those thus making or running the covered works +for you must do so exclusively on your behalf, under your direction +and control, on terms that prohibit them from making any copies of +your copyrighted material outside their relationship with you. + + Conveying under any other circumstances is permitted solely under +the conditions stated below. Sublicensing is not allowed; section 10 +makes it unnecessary. + + 3. Protecting Users' Legal Rights From Anti-Circumvention Law. + + No covered work shall be deemed part of an effective technological +measure under any applicable law fulfilling obligations under article +11 of the WIPO copyright treaty adopted on 20 December 1996, or +similar laws prohibiting or restricting circumvention of such +measures. + + When you convey a covered work, you waive any legal power to forbid +circumvention of technological measures to the extent such circumvention +is effected by exercising rights under this License with respect to +the covered work, and you disclaim any intention to limit operation or +modification of the work as a means of enforcing, against the work's +users, your or third parties' legal rights to forbid circumvention of +technological measures. + + 4. Conveying Verbatim Copies. + + You may convey verbatim copies of the Program's source code as you +receive it, in any medium, provided that you conspicuously and +appropriately publish on each copy an appropriate copyright notice; +keep intact all notices stating that this License and any +non-permissive terms added in accord with section 7 apply to the code; +keep intact all notices of the absence of any warranty; and give all +recipients a copy of this License along with the Program. + + You may charge any price or no price for each copy that you convey, +and you may offer support or warranty protection for a fee. + + 5. Conveying Modified Source Versions. + + You may convey a work based on the Program, or the modifications to +produce it from the Program, in the form of source code under the +terms of section 4, provided that you also meet all of these conditions: + + a) The work must carry prominent notices stating that you modified + it, and giving a relevant date. + + b) The work must carry prominent notices stating that it is + released under this License and any conditions added under section + 7. This requirement modifies the requirement in section 4 to + "keep intact all notices". + + c) You must license the entire work, as a whole, under this + License to anyone who comes into possession of a copy. This + License will therefore apply, along with any applicable section 7 + additional terms, to the whole of the work, and all its parts, + regardless of how they are packaged. This License gives no + permission to license the work in any other way, but it does not + invalidate such permission if you have separately received it. + + d) If the work has interactive user interfaces, each must display + Appropriate Legal Notices; however, if the Program has interactive + interfaces that do not display Appropriate Legal Notices, your + work need not make them do so. + + A compilation of a covered work with other separate and independent +works, which are not by their nature extensions of the covered work, +and which are not combined with it such as to form a larger program, +in or on a volume of a storage or distribution medium, is called an +"aggregate" if the compilation and its resulting copyright are not +used to limit the access or legal rights of the compilation's users +beyond what the individual works permit. Inclusion of a covered work +in an aggregate does not cause this License to apply to the other +parts of the aggregate. + + 6. Conveying Non-Source Forms. + + You may convey a covered work in object code form under the terms +of sections 4 and 5, provided that you also convey the +machine-readable Corresponding Source under the terms of this License, +in one of these ways: + + a) Convey the object code in, or embodied in, a physical product + (including a physical distribution medium), accompanied by the + Corresponding Source fixed on a durable physical medium + customarily used for software interchange. + + b) Convey the object code in, or embodied in, a physical product + (including a physical distribution medium), accompanied by a + written offer, valid for at least three years and valid for as + long as you offer spare parts or customer support for that product + model, to give anyone who possesses the object code either (1) a + copy of the Corresponding Source for all the software in the + product that is covered by this License, on a durable physical + medium customarily used for software interchange, for a price no + more than your reasonable cost of physically performing this + conveying of source, or (2) access to copy the + Corresponding Source from a network server at no charge. + + c) Convey individual copies of the object code with a copy of the + written offer to provide the Corresponding Source. This + alternative is allowed only occasionally and noncommercially, and + only if you received the object code with such an offer, in accord + with subsection 6b. + + d) Convey the object code by offering access from a designated + place (gratis or for a charge), and offer equivalent access to the + Corresponding Source in the same way through the same place at no + further charge. You need not require recipients to copy the + Corresponding Source along with the object code. If the place to + copy the object code is a network server, the Corresponding Source + may be on a different server (operated by you or a third party) + that supports equivalent copying facilities, provided you maintain + clear directions next to the object code saying where to find the + Corresponding Source. Regardless of what server hosts the + Corresponding Source, you remain obligated to ensure that it is + available for as long as needed to satisfy these requirements. + + e) Convey the object code using peer-to-peer transmission, provided + you inform other peers where the object code and Corresponding + Source of the work are being offered to the general public at no + charge under subsection 6d. + + A separable portion of the object code, whose source code is excluded +from the Corresponding Source as a System Library, need not be +included in conveying the object code work. + + A "User Product" is either (1) a "consumer product", which means any +tangible personal property which is normally used for personal, family, +or household purposes, or (2) anything designed or sold for incorporation +into a dwelling. In determining whether a product is a consumer product, +doubtful cases shall be resolved in favor of coverage. For a particular +product received by a particular user, "normally used" refers to a +typical or common use of that class of product, regardless of the status +of the particular user or of the way in which the particular user +actually uses, or expects or is expected to use, the product. A product +is a consumer product regardless of whether the product has substantial +commercial, industrial or non-consumer uses, unless such uses represent +the only significant mode of use of the product. + + "Installation Information" for a User Product means any methods, +procedures, authorization keys, or other information required to install +and execute modified versions of a covered work in that User Product from +a modified version of its Corresponding Source. The information must +suffice to ensure that the continued functioning of the modified object +code is in no case prevented or interfered with solely because +modification has been made. + + If you convey an object code work under this section in, or with, or +specifically for use in, a User Product, and the conveying occurs as +part of a transaction in which the right of possession and use of the +User Product is transferred to the recipient in perpetuity or for a +fixed term (regardless of how the transaction is characterized), the +Corresponding Source conveyed under this section must be accompanied +by the Installation Information. But this requirement does not apply +if neither you nor any third party retains the ability to install +modified object code on the User Product (for example, the work has +been installed in ROM). + + The requirement to provide Installation Information does not include a +requirement to continue to provide support service, warranty, or updates +for a work that has been modified or installed by the recipient, or for +the User Product in which it has been modified or installed. Access to a +network may be denied when the modification itself materially and +adversely affects the operation of the network or violates the rules and +protocols for communication across the network. + + Corresponding Source conveyed, and Installation Information provided, +in accord with this section must be in a format that is publicly +documented (and with an implementation available to the public in +source code form), and must require no special password or key for +unpacking, reading or copying. + + 7. Additional Terms. + + "Additional permissions" are terms that supplement the terms of this +License by making exceptions from one or more of its conditions. +Additional permissions that are applicable to the entire Program shall +be treated as though they were included in this License, to the extent +that they are valid under applicable law. If additional permissions +apply only to part of the Program, that part may be used separately +under those permissions, but the entire Program remains governed by +this License without regard to the additional permissions. + + When you convey a copy of a covered work, you may at your option +remove any additional permissions from that copy, or from any part of +it. (Additional permissions may be written to require their own +removal in certain cases when you modify the work.) You may place +additional permissions on material, added by you to a covered work, +for which you have or can give appropriate copyright permission. + + Notwithstanding any other provision of this License, for material you +add to a covered work, you may (if authorized by the copyright holders of +that material) supplement the terms of this License with terms: + + a) Disclaiming warranty or limiting liability differently from the + terms of sections 15 and 16 of this License; or + + b) Requiring preservation of specified reasonable legal notices or + author attributions in that material or in the Appropriate Legal + Notices displayed by works containing it; or + + c) Prohibiting misrepresentation of the origin of that material, or + requiring that modified versions of such material be marked in + reasonable ways as different from the original version; or + + d) Limiting the use for publicity purposes of names of licensors or + authors of the material; or + + e) Declining to grant rights under trademark law for use of some + trade names, trademarks, or service marks; or + + f) Requiring indemnification of licensors and authors of that + material by anyone who conveys the material (or modified versions of + it) with contractual assumptions of liability to the recipient, for + any liability that these contractual assumptions directly impose on + those licensors and authors. + + All other non-permissive additional terms are considered "further +restrictions" within the meaning of section 10. If the Program as you +received it, or any part of it, contains a notice stating that it is +governed by this License along with a term that is a further +restriction, you may remove that term. If a license document contains +a further restriction but permits relicensing or conveying under this +License, you may add to a covered work material governed by the terms +of that license document, provided that the further restriction does +not survive such relicensing or conveying. + + If you add terms to a covered work in accord with this section, you +must place, in the relevant source files, a statement of the +additional terms that apply to those files, or a notice indicating +where to find the applicable terms. + + Additional terms, permissive or non-permissive, may be stated in the +form of a separately written license, or stated as exceptions; +the above requirements apply either way. + + 8. Termination. + + You may not propagate or modify a covered work except as expressly +provided under this License. Any attempt otherwise to propagate or +modify it is void, and will automatically terminate your rights under +this License (including any patent licenses granted under the third +paragraph of section 11). + + However, if you cease all violation of this License, then your +license from a particular copyright holder is reinstated (a) +provisionally, unless and until the copyright holder explicitly and +finally terminates your license, and (b) permanently, if the copyright +holder fails to notify you of the violation by some reasonable means +prior to 60 days after the cessation. + + Moreover, your license from a particular copyright holder is +reinstated permanently if the copyright holder notifies you of the +violation by some reasonable means, this is the first time you have +received notice of violation of this License (for any work) from that +copyright holder, and you cure the violation prior to 30 days after +your receipt of the notice. + + Termination of your rights under this section does not terminate the +licenses of parties who have received copies or rights from you under +this License. If your rights have been terminated and not permanently +reinstated, you do not qualify to receive new licenses for the same +material under section 10. + + 9. Acceptance Not Required for Having Copies. + + You are not required to accept this License in order to receive or +run a copy of the Program. Ancillary propagation of a covered work +occurring solely as a consequence of using peer-to-peer transmission +to receive a copy likewise does not require acceptance. However, +nothing other than this License grants you permission to propagate or +modify any covered work. These actions infringe copyright if you do +not accept this License. Therefore, by modifying or propagating a +covered work, you indicate your acceptance of this License to do so. + + 10. Automatic Licensing of Downstream Recipients. + + Each time you convey a covered work, the recipient automatically +receives a license from the original licensors, to run, modify and +propagate that work, subject to this License. You are not responsible +for enforcing compliance by third parties with this License. + + An "entity transaction" is a transaction transferring control of an +organization, or substantially all assets of one, or subdividing an +organization, or merging organizations. If propagation of a covered +work results from an entity transaction, each party to that +transaction who receives a copy of the work also receives whatever +licenses to the work the party's predecessor in interest had or could +give under the previous paragraph, plus a right to possession of the +Corresponding Source of the work from the predecessor in interest, if +the predecessor has it or can get it with reasonable efforts. + + You may not impose any further restrictions on the exercise of the +rights granted or affirmed under this License. For example, you may +not impose a license fee, royalty, or other charge for exercise of +rights granted under this License, and you may not initiate litigation +(including a cross-claim or counterclaim in a lawsuit) alleging that +any patent claim is infringed by making, using, selling, offering for +sale, or importing the Program or any portion of it. + + 11. Patents. + + A "contributor" is a copyright holder who authorizes use under this +License of the Program or a work on which the Program is based. The +work thus licensed is called the contributor's "contributor version". + + A contributor's "essential patent claims" are all patent claims +owned or controlled by the contributor, whether already acquired or +hereafter acquired, that would be infringed by some manner, permitted +by this License, of making, using, or selling its contributor version, +but do not include claims that would be infringed only as a +consequence of further modification of the contributor version. For +purposes of this definition, "control" includes the right to grant +patent sublicenses in a manner consistent with the requirements of +this License. + + Each contributor grants you a non-exclusive, worldwide, royalty-free +patent license under the contributor's essential patent claims, to +make, use, sell, offer for sale, import and otherwise run, modify and +propagate the contents of its contributor version. + + In the following three paragraphs, a "patent license" is any express +agreement or commitment, however denominated, not to enforce a patent +(such as an express permission to practice a patent or covenant not to +sue for patent infringement). To "grant" such a patent license to a +party means to make such an agreement or commitment not to enforce a +patent against the party. + + If you convey a covered work, knowingly relying on a patent license, +and the Corresponding Source of the work is not available for anyone +to copy, free of charge and under the terms of this License, through a +publicly available network server or other readily accessible means, +then you must either (1) cause the Corresponding Source to be so +available, or (2) arrange to deprive yourself of the benefit of the +patent license for this particular work, or (3) arrange, in a manner +consistent with the requirements of this License, to extend the patent +license to downstream recipients. "Knowingly relying" means you have +actual knowledge that, but for the patent license, your conveying the +covered work in a country, or your recipient's use of the covered work +in a country, would infringe one or more identifiable patents in that +country that you have reason to believe are valid. + + If, pursuant to or in connection with a single transaction or +arrangement, you convey, or propagate by procuring conveyance of, a +covered work, and grant a patent license to some of the parties +receiving the covered work authorizing them to use, propagate, modify +or convey a specific copy of the covered work, then the patent license +you grant is automatically extended to all recipients of the covered +work and works based on it. + + A patent license is "discriminatory" if it does not include within +the scope of its coverage, prohibits the exercise of, or is +conditioned on the non-exercise of one or more of the rights that are +specifically granted under this License. You may not convey a covered +work if you are a party to an arrangement with a third party that is +in the business of distributing software, under which you make payment +to the third party based on the extent of your activity of conveying +the work, and under which the third party grants, to any of the +parties who would receive the covered work from you, a discriminatory +patent license (a) in connection with copies of the covered work +conveyed by you (or copies made from those copies), or (b) primarily +for and in connection with specific products or compilations that +contain the covered work, unless you entered into that arrangement, +or that patent license was granted, prior to 28 March 2007. + + Nothing in this License shall be construed as excluding or limiting +any implied license or other defenses to infringement that may +otherwise be available to you under applicable patent law. + + 12. No Surrender of Others' Freedom. + + If conditions are imposed on you (whether by court order, agreement or +otherwise) that contradict the conditions of this License, they do not +excuse you from the conditions of this License. If you cannot convey a +covered work so as to satisfy simultaneously your obligations under this +License and any other pertinent obligations, then as a consequence you may +not convey it at all. For example, if you agree to terms that obligate you +to collect a royalty for further conveying from those to whom you convey +the Program, the only way you could satisfy both those terms and this +License would be to refrain entirely from conveying the Program. + + 13. Use with the GNU Affero General Public License. + + Notwithstanding any other provision of this License, you have +permission to link or combine any covered work with a work licensed +under version 3 of the GNU Affero General Public License into a single +combined work, and to convey the resulting work. The terms of this +License will continue to apply to the part which is the covered work, +but the special requirements of the GNU Affero General Public License, +section 13, concerning interaction through a network will apply to the +combination as such. + + 14. Revised Versions of this License. + + The Free Software Foundation may publish revised and/or new versions of +the GNU General Public License from time to time. Such new versions will +be similar in spirit to the present version, but may differ in detail to +address new problems or concerns. + + Each version is given a distinguishing version number. If the +Program specifies that a certain numbered version of the GNU General +Public License "or any later version" applies to it, you have the +option of following the terms and conditions either of that numbered +version or of any later version published by the Free Software +Foundation. If the Program does not specify a version number of the +GNU General Public License, you may choose any version ever published +by the Free Software Foundation. + + If the Program specifies that a proxy can decide which future +versions of the GNU General Public License can be used, that proxy's +public statement of acceptance of a version permanently authorizes you +to choose that version for the Program. + + Later license versions may give you additional or different +permissions. However, no additional obligations are imposed on any +author or copyright holder as a result of your choosing to follow a +later version. + + 15. Disclaimer of Warranty. + + THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY +APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT +HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY +OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, +THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR +PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM +IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF +ALL NECESSARY SERVICING, REPAIR OR CORRECTION. + + 16. Limitation of Liability. + + IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING +WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS +THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY +GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE +USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF +DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD +PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), +EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF +SUCH DAMAGES. + + 17. Interpretation of Sections 15 and 16. + + If the disclaimer of warranty and limitation of liability provided +above cannot be given local legal effect according to their terms, +reviewing courts shall apply local law that most closely approximates +an absolute waiver of all civil liability in connection with the +Program, unless a warranty or assumption of liability accompanies a +copy of the Program in return for a fee. + + END OF TERMS AND CONDITIONS + + How to Apply These Terms to Your New Programs + + If you develop a new program, and you want it to be of the greatest +possible use to the public, the best way to achieve this is to make it +free software which everyone can redistribute and change under these terms. + + To do so, attach the following notices to the program. It is safest +to attach them to the start of each source file to most effectively +state the exclusion of warranty; and each file should have at least +the "copyright" line and a pointer to where the full notice is found. + + {one line to give the program's name and a brief idea of what it does.} + Copyright (C) {year} {name of author} + + This program is free software: you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation, either version 3 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. + + You should have received a copy of the GNU General Public License + along with this program. If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If the program does terminal interaction, make it output a short +notice like this when it starts in an interactive mode: + + {project} Copyright (C) {year} {fullname} + This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. Of course, your program's commands +might be different; for a GUI interface, you would use an "about box". + + You should also get your employer (if you work as a programmer) or school, +if any, to sign a "copyright disclaimer" for the program, if necessary. +For more information on this, and how to apply and follow the GNU GPL, see +. + + The GNU General Public License does not permit incorporating your program +into proprietary programs. If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. diff --git a/Math/ExpPairs.hs b/Math/ExpPairs.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs.hs @@ -0,0 +1,89 @@ +module Math.ExpPairs (optimize, LinearForm (..), RationalForm (..), IneqType (..), Constraint (..), InitPair, Path, simulateOptimize,simulateOptimize', RatioInf (..), RationalInf, OptimizeResult, optimalValue, optimalPair, optimalPath) where + +import Data.Ratio +import Data.Ord +import Data.List +import Data.Monoid + +import Math.ExpPairs.LinearForm +import Math.ExpPairs.Process +import Math.ExpPairs.Pair +import Math.ExpPairs.RatioInf + +fracs2proj :: (Rational, Rational) -> (Integer, Integer, Integer) +fracs2proj (q, r) = (k, l, m) where + dq = denominator q + dr = denominator r + m = lcm dq dr + k = numerator q * (m `div` dq) + l = numerator r * (m `div` dr) + +proj2fracs :: (Integer, Integer, Integer) -> (Rational, Rational) +proj2fracs (k, l, m) = (k%m, l%m) + + +evalFunctional :: [InitPair] -> [InitPair] -> [RationalForm Rational] -> [Constraint Rational] -> Path -> (RationalInf, InitPair) +evalFunctional corners interiors rfs cons path = if null rs then (InfPlus, undefined) else minimumBy (comparing fst) rs where + applyPath ips = map (evalPath path . fracs2proj . initPairToValue) ips `zip` ips + corners' = applyPath corners + interiors' = applyPath interiors + + predicate (p, _) = all (checkConstraint p) cons + qs = if all predicate corners' then corners' else filter predicate interiors' + + rs = map (\(p, ip) -> (maximum $ map (evalRF p) rfs, ip)) qs + +checkMConstraints :: Path -> [Constraint Rational] -> Bool +checkMConstraints path = all (\con -> any (\p -> checkConstraint (evalPath path p) con ) triangleT) where + triangleT = map fracs2proj [ (0%1,1%1), (0%1,1%2), (1%2,1%2)] + +data OptimizeResult = OptimizeResult { + optimalValue :: RationalInf, + optimalPair :: InitPair, + optimalPath :: Path + } + +instance Show OptimizeResult where + show (OptimizeResult r' ip p) = show' r' ++ "\n" ++ show ip ++ "\t" ++ show p where + show' (Finite r) = show (fromRational r :: Double) ++ " = " ++ show r + show' r = show r + +instance Eq OptimizeResult where + a==b = optimalValue a == optimalValue b + +instance Ord OptimizeResult where + compare a b = compare (optimalValue a) (optimalValue b) + +simulateOptimize :: Rational -> OptimizeResult +simulateOptimize r = OptimizeResult (Finite r) Corput01 mempty + +simulateOptimize' :: RationalInf -> OptimizeResult +simulateOptimize' r = OptimizeResult r Corput01 mempty + +optimize :: [RationalForm Rational] -> [Constraint Rational] -> OptimizeResult +optimize rfs cons = optimize' rfs cons (OptimizeResult r0 ip0 mempty) where + (r0, ip0) = evalFunctional [Corput01, Corput12] [Corput01, Corput12] rfs cons mempty + +optimize' :: [RationalForm Rational] -> [Constraint Rational] -> OptimizeResult -> OptimizeResult +optimize' rfs cons ret@(OptimizeResult r _ path) + | lengthPath path > 100 = ret + | otherwise = retBA where + ret0@(OptimizeResult r0 ip0 _) = if r0' < r then OptimizeResult r0' ip0' path else ret where + (r0', ip0') = evalFunctional corners interiors rfs cons path + corners = [Mix 1 0, Mix 0 1, Mix 0 0] + interiors = initPairs + + cons0 = if r0==InfPlus then cons else cons ++ map (consBuilder r0) rfs + + retA@(OptimizeResult r1 ip1 _) = if checkMConstraints patha cons0 && r1' < r0 then branchA else ret0 where + patha = path `mappend` aPath + branchA@(OptimizeResult r1' _ _) = optimize' rfs cons (OptimizeResult r0 ip0 patha) + + cons1 = if r1==r0 then cons0 else cons ++ map (consBuilder r1) rfs + + retBA = if checkMConstraints pathba cons1 && r2' < r1 then branchB else retA where + pathba = path `mappend` baPath + branchB@(OptimizeResult r2' _ _) = optimize' rfs cons (OptimizeResult r1 ip1 pathba) + + consBuilder rr (RationalForm num den) = Constraint (substituteLF (num, den, 1) (LinearForm (-1) (toRational rr) 0)) Strict + diff --git a/Math/ExpPairs/Ivic.hs b/Math/ExpPairs/Ivic.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/Ivic.hs @@ -0,0 +1,185 @@ +module Math.ExpPairs.Ivic where + +import Data.Ratio +import Data.List +import Data.Ord + +import Math.ExpPairs + +zetaOnS :: Rational -> OptimizeResult +zetaOnS s + | s >= 1 = simulateOptimize 0 + | s >= 1%2 = optimize + [RationalForm (LinearForm 1 1 (-s)) 2] + [Constraint (LinearForm (-1) 1 (-s)) NonStrict] + | otherwise = optRes {optimalValue = r} where + optRes = zetaOnS (1-s) + r = Finite (1%2 - s) + optimalValue optRes + +zetaOnHalf :: Rational +zetaOnHalf = 32%205 + +reverseZetaOnS :: Rational -> OptimizeResult +reverseZetaOnS mu + | mu >= 1%2 = simulateOptimize 0 + | mu > zetaOnHalf = optimize [RationalForm (LinearForm 1 (-1) 1) 1] [Constraint (LinearForm 0 (-2) (1+2*mu)) NonStrict] + | otherwise = optRes {optimalValue = negate $ optimalValue optRes} where + optRes = optimize [RationalForm (LinearForm 1 (-1) 0) 1] [Constraint (LinearForm 1 0 (-mu)) NonStrict, Constraint (LinearForm (-1) 1 (-1%2)) NonStrict] + +lemma82_f :: Rational -> Rational +lemma82_f s + | s < 1%2 = undefined + | s<= 2%3 = 2/(3-4*s) + | s<=11%14 = 10/(7-8*s) + | s<=13%15 = 34/(15-16*s) + | s<=57%62 = 98/(31-32*s) + | otherwise = 5/(1-s) + +-- Ivic, (8.97) +-- R << T V^{-2f(sigma)} + T^alpha1 V^beta1 + T^alpha2 V^beta2 +-- +-- If a<1 then T^a V^b << T V^{b+(a-1)/muS} +-- +-- (8.97) implies that alpha1 <= 1 for S >= 1/2 +-- and that alpha2 <= 1 for S >= 2/3 or S >= 5/8 and +-- (4S-2)k + (8S-6)l + 2S-1 >=0 + +mOnS :: Rational -> OptimizeResult +mOnS s + | s < 1%2 = simulateOptimize 0 + | s < 5%8 = simulateOptimize $ 4/(3-4*s) + | s>= 1 = simulateOptimize' InfPlus + | otherwise = minimumBy (comparing optimalValue) [x1, x2, simulateOptimize (lemma82_f s * 2)] where + + optRes = zetaOnS s + muS = toRational $ optimalValue optRes + alpha1 = (4-4*s)/(1+2*s) + beta1 = -12/(1+2*s) + x1 = optRes {optimalValue = Finite $ (1-alpha1)/muS - beta1} + + --alpha2 = 4*(1-s)*(k+l)/((2*m+4*l)*s-m+2*k-2*l) + --beta2 = -4*(m+2*k+2*l)/((2*m+4*l)*s-m+2*k-2*l) + --ratio = (1-alpha2)/muS - beta2 + --numer = numerator ratio + --denom = denominator ratio + numer = LinearForm + (-4*s + (-8*muS + 2)) + (-8*s + (-8*muS + 6)) + (-2*s + (-4*muS + 1)) + denom = LinearForm + (2*muS) + (4*muS*s - 2*muS) + (2*muS*s - muS) + + cons = if s >= 2%3 then [] else [Constraint + (LinearForm (4*s-2) (8*s-6) (2*s-1)) NonStrict + ] + + x2' = optimize [RationalForm numer denom] cons + x2 = x2' {optimalValue = negate $ optimalValue x2'} + +reverseMOnS m = reverseMOnS' from to where + from = 1 % 2 + to = 1 % 1 + reverseMOnS' a b + | b-a < 1%1000000 = a + | optimalValue (mOnS ((a+b)/2)) > m = reverseMOnS' a ((a+b)/2) + | otherwise = reverseMOnS' ((a+b)/2) b + +checkAbscissa :: [(Rational, Rational)] -> Rational -> Bool +checkAbscissa xs s = sum rs < Finite 1 where + qs = map (\(n,m) -> optimalValue (mOnS (n*s)) / Finite m) xs + rs = map (\q -> 1/q) qs + +searchMinAbscissa :: [(Rational, Rational)] -> Rational +searchMinAbscissa xs = searchMinAbscissa' from to where + from = 1 % 2 / minimum (map fst xs) + to = 1 % 1 + searchMinAbscissa' a b + | b-a < 1%1000000 = a + | checkAbscissa xs ((a+b)/2) = searchMinAbscissa' a ((a+b)/2) + | otherwise = searchMinAbscissa' ((a+b)/2) b + +-- % \begin{lemma}\label{l:pointwise-moments-on-1/2} +-- % For $A\ge12$ let +-- % $$ +-- % f(A) = 1+\inf \{ l/k \mid (4-A)k+4l+2 \ge 0 \}. +-- % $$ +-- % Then +-- % $$ +-- % R \ll T V^{-6} \log^8 T + T^{f(A)+\eps} V^{-A} +-- % \ll T^{\max \{ 1+32(A-6)/205, f(A) \}} V^{-A} +-- % $$ +-- % and thus +-- % $$ +-- % M(A) \le \max \{ 1+32(A-6)/205, f(A) \}. +-- % $$ +-- % \end{lemma} + +-- Constant +-- is produced by +-- optimize [RationalForm (LinearForm 4 4 2) (LinearForm 1 0 0)] [Constraint (LinearForm (-64) (-77) 64) Strict] + +mBigOnHalf a + | a < 4 = simulateOptimize 1 + | a < 12 = simulateOptimize $ 1+(a-4)/8 + | a > 41614060315296730740083860226662 % 2636743270445733804969041895717 = simulateOptimize $ 1 + 32*(a-6)/205 + | otherwise = if Finite x >= optimalValue optRes + then simulateOptimize x + else optRes where + optRes = optimize [RationalForm (LinearForm 1 1 0) (LinearForm 1 0 0)] + [Constraint (LinearForm (4-a) 4 2) NonStrict] + x = 1 + 32*(a-6)/205 + +reverseMBigOnHalf m + | m <= 2 = simulateOptimize $ (m-1)*8 + 4 + | otherwise = if Finite a <= optimalValue optRes + then simulateOptimize a + else optRes where + a = (m-1)*205/32 + 6 + optRes = optimize [RationalForm (LinearForm 4 4 2) (LinearForm 1 0 0)] [Constraint (LinearForm (1-m) 1 0) NonStrict] + + +f l lambda = (l-lambda)*32/205 + + ((toRational . optimalValue . mBigOnHalf) (4*l/(4-lambda)) - 1) * (4-lambda) /4 + +--bestLambda l = minimum $ map (f l) lambdas `zip` lambdas where +-- lambdas = [0,1%100..4-1%100] + +heckeZetaByHalf a = 1 - xt where + d = toRational 12.571624917200547 + ia 2 = 0 + ia 3 = 1%4 + ia a + | a<=12 = 32%205 * ((1 + 4 / d) * a - 4) + (toRational (optimalValue (mBigOnHalf d)) - 1) * a / d + | a<=15 = 32%205 * a + toRational (optimalValue (mBigOnHalf a)) - 1 + | otherwise = 32%205 * (2 * a - 6) + xt = 1%2 / (1 + ia a) + + +bestLambda l = (\x -> (x, fromRational $ f l x)) (bestLambda' 0 (3999%1000)) where + bestLambda' a b + | b-a < 1%1000000000 = a + | otherwise = if mx1 > mx2 then bestLambda' x1 b else bestLambda' a x2 where + x1 = (2*a+b)/3 + x2 = (a+2*b)/3 + ma = f l a + mb = f l b + mx1 = f l x1 + mx2 = f l x2 + +difur a = a * m' - m + (77%205) where + h = 1%(10^100) + m = toRational $ optimalValue $ mBigOnHalf a + mh = toRational $ optimalValue $ mBigOnHalf (a+h) + m' = (mh-m) / h + +solveD a b + | b-a < 1%(10^6) = a + | otherwise = if difur c > 0 then solveD a c else solveD c b where + c = (a+b)/2 + +{- +D = 12.571624917200547 +-} + diff --git a/Math/ExpPairs/Kratzel.hs b/Math/ExpPairs/Kratzel.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/Kratzel.hs @@ -0,0 +1,64 @@ +module Math.ExpPairs.Kratzel where + +import Data.Ratio +import Data.Ord +import Data.List + +import Math.ExpPairs + +data TauabTheorem = Kr511a | Kr511b | Kr512a | Kr512b + deriving (Show) + +tauab :: Integer -> Integer -> (TauabTheorem, OptimizeResult) +tauab a' b' = minimumBy (comparing (optimalValue . snd)) [kr511a, kr511b, kr512a, kr512b] where + a = a'%1 + b = b'%1 + kr511a = (Kr511a, optimize + [RationalForm (LinearForm 2 2 (-1)) (LinearForm 0 0 (a+b))] + [Constraint (LinearForm (-2*b) (2*a) (-a)) NonStrict]) + kr511b = (Kr511b, optimize + [RationalForm (LinearForm 1 0 0) (LinearForm b (-a) a)] + [Constraint (LinearForm (2*b) (-2*a) a) Strict]) + kr512a = (Kr512a, simulateOptimize r) where + r = if 11*a >= 8*b then 19/29/(a+b) else 1%1 + kr512b = if 11*a >= 8*b then kr512a else (Kr512b, optimize + [ + RationalForm (LinearForm (-11) 8 (-4)) (LinearForm (-29*b) (29*a) (4*b-20*a)) + ] + [ + Constraint (LinearForm (-2*b) (2*a) (-a)) NonStrict, + Constraint (LinearForm (-29) 0 4) Strict, + Constraint (LinearForm 29 29 (-24)) Strict + ]) + +data TauabcTheorem = Kolesnik | Kr61 | Kr62 | Kr63 | Kr64 | Kr65 | Kr66 | Tauab TauabTheorem + deriving (Show) + +tauabc :: Integer -> Integer -> Integer -> (TauabcTheorem, OptimizeResult) +tauabc 1 1 1 = (Kolesnik, simulateOptimize $ 43%96) +tauabc a' b' c' = minimumBy (comparing (optimalValue . snd)) [kr61, kr62, kr63, kr64, kr65, kr66] where + a = a'%1 + b = b'%1 + c = c'%1 + kr61 + | c= 2 * (b+c), (3*2^k-8)%1 * (a+b) >= (3*2^k-4*k+4)%1 * c]) + maxk = 4 `max` floor (logBase 2 (fromRational $ b+c)) + kr65 = (Kr65, simulateOptimize r) where + r = if 7*a>=2*(b+c) && 4*(a+b)>=5*c then 3%2/(a+b+c) else 1%1 + kr66 = (Kr66, simulateOptimize r) where + r = if 18*a>=7*(b+c) && 2*(a+b)>=3*c then 25%17/(a+b+c) else 1%1 diff --git a/Math/ExpPairs/LinearForm.hs b/Math/ExpPairs/LinearForm.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/LinearForm.hs @@ -0,0 +1,85 @@ +module Math.ExpPairs.LinearForm (LinearForm (..), RationalForm (..), IneqType (..), Constraint (..), checkConstraint, evalLF, evalRF, substituteLF) where + +import Data.List +import Data.Ratio +import Data.Monoid +import Math.ExpPairs.RatioInf + +data LinearForm t = LinearForm t t t + deriving (Eq) + +instance (Num t, Eq t, Show t) => Show (LinearForm t) where + show (LinearForm a b c) = if (a==0) && (b==0) && (c==0) + then "0" + else "(" ++ intercalate " + " (filter (/=[]) $ + [if a/= 0 then show a ++ "k" else []] ++ + [if b/= 0 then show b ++ "l" else []] ++ + [if c/= 0 then show c else []] ) ++ ")" -- where + -- show' :: Rational -> String + -- show' z = if denominator z==1 then show (numerator z) else show z + +instance Num t => Num (LinearForm t) where + (LinearForm a b c) + (LinearForm d e f) = LinearForm (a+d) (b+e) (c+f) + (*) = undefined + negate (LinearForm a b c) = LinearForm (negate a) (negate b) (negate c) + abs = undefined + signum = undefined + fromInteger n = LinearForm 0 0 (fromInteger n) + +instance Num t => Monoid (LinearForm t) where + mempty = 0 + mappend = (+) + +scaleLF :: (Num t, Eq t) => t -> LinearForm t -> LinearForm t +scaleLF 0 (LinearForm {}) = LinearForm 0 0 0 +scaleLF s (LinearForm a b c) = LinearForm (a*s) (b*s) (c*s) + +evalLF :: Num t => (t, t, t) -> LinearForm t -> t +evalLF (k, l, m) (LinearForm a b c) = a*k+l*b+m*c + +substituteLF :: (Eq t, Num t) => (LinearForm t, LinearForm t, LinearForm t) -> LinearForm t -> LinearForm t +substituteLF (k, l, m) (LinearForm a b c) = scaleLF a k + scaleLF b l + scaleLF c m + + +data RationalForm t = RationalForm (LinearForm t) (LinearForm t) + deriving (Show) + +instance Num t => Num (RationalForm t) where + (+) = undefined + (*) = undefined + negate (RationalForm a b) = RationalForm (negate a) b + abs = undefined + signum = undefined + fromInteger n = RationalForm (fromInteger n) 1 + +instance Num t => Fractional (RationalForm t) where + fromRational r = RationalForm (fromInteger $ numerator r) (fromInteger $ denominator r) + recip (RationalForm a b) = RationalForm b a + +evalRF :: (Real t, Num t) => (Integer, Integer, Integer) -> RationalForm t -> RationalInf +evalRF (k', l', m') (RationalForm num den) = if denom==0 then InfPlus else Finite (numer / denom) where + k = fromInteger k' + l = fromInteger l' + m = fromInteger m' + numer = toRational $ evalLF (k, l, m) num + denom = toRational $ evalLF (k, l, m) den + +substituteRF :: (Eq t, Num t) => (LinearForm t, LinearForm t, LinearForm t) -> RationalForm t -> RationalForm t +substituteRF (k, l, m) (RationalForm num den) = RationalForm (substituteLF (k, l, m) num) (substituteLF (k, l, m) den) + + +data IneqType = Strict | NonStrict + deriving (Eq, Show) + +data Constraint t = Constraint (LinearForm t) IneqType + deriving (Show) + +checkConstraint :: (Num t, Eq t) => (Integer, Integer, Integer) -> Constraint t -> Bool +checkConstraint (k', l', m') (Constraint lf ineq) + = if ineq==NonStrict + then signum numer /= -1 + else signum numer == 1 where + k = fromInteger k' + l = fromInteger l' + m = fromInteger m' + numer = evalLF (k, l, m) lf diff --git a/Math/ExpPairs/Matrix3.hs b/Math/ExpPairs/Matrix3.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/Matrix3.hs @@ -0,0 +1,166 @@ +module Math.ExpPairs.Matrix3 (Matrix3 (..), Vector3 (..), fromList, toList, normalize, prettyMatrix, multCol, det) where + +import qualified Data.List as List +import Data.Monoid + +data Vector3 t = Vector3 { + a1 :: t, + a2 :: t, + a3 :: t + } + deriving (Eq, Show) + +data Matrix3 t = Matrix3 { + a11 :: t, + a12 :: t, + a13 :: t, + a21 :: t, + a22 :: t, + a23 :: t, + a31 :: t, + a32 :: t, + a33 :: t + } + deriving (Eq, Show) + +instance Num t => Num (Matrix3 t) where + a + b = Matrix3 { + a11 = a11 a + a11 b, + a12 = a12 a + a12 b, + a13 = a13 a + a13 b, + a21 = a21 a + a21 b, + a22 = a22 a + a22 b, + a23 = a23 a + a23 b, + a31 = a31 a + a31 b, + a32 = a32 a + a32 b, + a33 = a33 a + a33 b + } + + -- intercalate ",\n" [ "a"++(show i)++(show j)++" = "++( intercalate " + " ["a"++(show i)++(show k)++" a * "++"a"++(show k)++(show j)++" b" | k<-[1..3]] ) | i<-[1..3], j<-[1..3]] + a * b = Matrix3 { + a11 = a11 a * a11 b + a12 a * a21 b + a13 a * a31 b, + a12 = a11 a * a12 b + a12 a * a22 b + a13 a * a32 b, + a13 = a11 a * a13 b + a12 a * a23 b + a13 a * a33 b, + a21 = a21 a * a11 b + a22 a * a21 b + a23 a * a31 b, + a22 = a21 a * a12 b + a22 a * a22 b + a23 a * a32 b, + a23 = a21 a * a13 b + a22 a * a23 b + a23 a * a33 b, + a31 = a31 a * a11 b + a32 a * a21 b + a33 a * a31 b, + a32 = a31 a * a12 b + a32 a * a22 b + a33 a * a32 b, + a33 = a31 a * a13 b + a32 a * a23 b + a33 a * a33 b + } + + negate a = Matrix3 { + a11 = - a11 a, + a12 = - a12 a, + a13 = - a13 a, + a21 = - a21 a, + a22 = - a22 a, + a23 = - a23 a, + a31 = - a31 a, + a32 = - a32 a, + a33 = - a33 a + } + + abs = undefined + + signum = undefined + + -- Multiplicative, not additive behaviour + fromInteger n = Matrix3 { + a11 = fromInteger n, + a12 = 0, + a13 = 0, + a21 = 0, + a22 = fromInteger n, + a23 = 0, + a31 = 0, + a32 = 0, + a33 = fromInteger n + } + +det :: (Num t) => Matrix3 t -> t +det a = + a11 a * (a22 a * a33 a - a32 a * a23 a) + - a12 a * (a21 a * a33 a - a23 a * a31 a) + + a13 a * (a21 a * a32 a - a22 a * a31 a) + +instance Fractional t => Fractional (Matrix3 t) where + -- Multiplicative, not additive behaviour + fromRational n = Matrix3 { + a11 = fromRational n, + a12 = 0, + a13 = 0, + a21 = 0, + a22 = fromRational n, + a23 = 0, + a31 = 0, + a32 = 0, + a33 = fromRational n + } + + recip a = Matrix3 { + a11 = (a22 a * a33 a - a32 a * a23 a) / d, + a12 = -(a21 a * a33 a - a23 a * a31 a) / d, + a13 = (a21 a * a32 a - a22 a * a31 a) / d, + a21 = -(a12 a * a33 a - a13 a * a32 a) / d, + a22 = (a11 a * a33 a - a13 a * a31 a) / d, + a23 = -(a11 a * a32 a - a12 a * a31 a) / d, + a31 = (a12 a * a23 a - a13 a * a22 a) / d, + a32 = -(a11 a * a23 a - a13 a * a21 a) / d, + a33 = (a11 a * a22 a - a12 a * a21 a) / d + } where d = det a + + +instance Num t => Monoid (Matrix3 t) where + mempty = 1 + mappend = (*) + +toList :: Matrix3 t -> [t] +toList a = [a11 a, a12 a, a13 a, a21 a, a22 a, a23 a, a31 a, a32 a, a33 a] + +fromList :: [t] -> Matrix3 t +fromList as = Matrix3 { + a11 = as!!0, + a12 = as!!1, + a13 = as!!2, + a21 = as!!3, + a22 = as!!4, + a23 = as!!5, + a31 = as!!6, + a32 = as!!7, + a33 = as!!8 + } + +instance Functor Matrix3 where + fmap f = fromList . List.map f . toList + +normalize :: Integral t => Matrix3 t -> Matrix3 t +normalize m = m' where + l = toList m + d = foldl1 gcd l + m' = if d==0 then m else fromList $ List.map (`div`d) l + +maximum :: Ord t => Matrix3 t -> t +maximum = List.maximum . toList + +prettyMatrix :: (Show t) => Matrix3 t -> String +prettyMatrix m = + show (a11 m) ++ " " ++ + show (a12 m) ++ " " ++ + show (a13 m) ++ "\n" ++ + show (a21 m) ++ " " ++ + show (a22 m) ++ " " ++ + show (a23 m) ++ "\n" ++ + show (a31 m) ++ " " ++ + show (a32 m) ++ " " ++ + show (a33 m) + +multCol :: (Num t) => Matrix3 t -> Vector3 t -> Vector3 t +multCol m v = Vector3 { + a1 = a11 m * a1 v + a12 m * a2 v + a13 m * a3 v, + a2 = a21 m * a1 v + a22 m * a2 v + a23 m * a3 v, + a3 = a31 m * a1 v + a32 m * a2 v + a33 m * a3 v + } + + + diff --git a/Math/ExpPairs/MenzerNowak.hs b/Math/ExpPairs/MenzerNowak.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/MenzerNowak.hs @@ -0,0 +1,15 @@ +module Math.ExpPairs.MenzerNowak where + +import Data.Ratio + +import Math.ExpPairs + +menzerNowak :: Integer -> Integer -> OptimizeResult +menzerNowak a' b' = optimize + [ + RationalForm (LinearForm 1 1 0) (LinearForm (a+b) 0 (a+b)), + RationalForm (LinearForm 1 0 0) (LinearForm (a+b) (-a) a) + ] + [] where + a = a'%1 + b = b'%1 diff --git a/Math/ExpPairs/Pair.hs b/Math/ExpPairs/Pair.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/Pair.hs @@ -0,0 +1,41 @@ +module Math.ExpPairs.Pair (InitPair' (..), InitPair, initPairs, initPairToValue) where + +import Data.Ratio + +data Triangle = Corput16 | HuxW87b1 | Hux05 + deriving (Show, Bounded, Enum, Eq, Ord) + +data InitPair' t = Corput01 | Corput12 | Mix t t + deriving (Eq) +type InitPair = InitPair' Rational + +instance (Show t, Num t, Eq t) => Show (InitPair' t) where + show Corput01 = "(0, 1)" + show Corput12 = "(1/2, 1/2)" + show (Mix r1 r2) = + s1 ++ (if s1/="" && (s2/=""||s3/="") then " + " else "") + ++ s2 ++ (if s2/="" && s3/="" then " + " else "") ++ s3 + where + r3 = 1 - r1 - r2 + f r t = if r==0 then "" else (if r==1 then "" else show r ++ " * ") ++ show t + s1 = f r1 Corput16 + s2 = f r2 HuxW87b1 + s3 = f r3 Hux05 + +sect :: Integer +sect = 30 + +initPairs :: [InitPair' (Ratio Integer)] +initPairs = Corput01 : Corput12 : [Mix (r1%sect) (r2%sect) | r1<-[0..sect], r2<-[0..sect-r1]] + +initPairToValue :: InitPair -> (Rational, Rational) +initPairToValue Corput01 = (0, 1) +initPairToValue Corput12 = (1%2, 1%2) +initPairToValue (Mix r1 r2) = (x, y) where + r3 = 1 - r1 - r2 + (x1, y1) = (1%6, 2%3) + (x2, y2) = ( 2 % 13, 35 % 52) + (x3, y3) = (32 % 205, 269 % 410) + x = x1*r1 + x2*r2 + x3*r3 + y = y1*r1 + y2*r2 + y3*r3 + diff --git a/Math/ExpPairs/Process.hs b/Math/ExpPairs/Process.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/Process.hs @@ -0,0 +1,117 @@ +{-# LANGUAGE TemplateHaskell #-} +module Math.ExpPairs.Process (Process, Path (), evalPath, lengthPath, aPath, baPath) where + +import Data.Monoid +import Data.List +import Data.Ord +import Data.Function.Memoize + +import qualified Math.ExpPairs.Matrix3 as Mx + +data Process = A | BA + deriving (Eq, Show, Read, Ord, Enum) + +deriveMemoizable ''Process + +type ProcessMatrix = Mx.Matrix3 Integer + +process2matrix :: Process -> ProcessMatrix +process2matrix A = Mx.Matrix3 1 0 0 1 1 1 2 0 2 +process2matrix BA = Mx.Matrix3 0 1 0 2 0 1 2 0 2 + +data Path = Path ProcessMatrix [Process] + +aPath :: Path +aPath = Path (process2matrix A) [ A] +baPath :: Path +baPath = Path (process2matrix BA) [BA] + +instance Monoid Path where + mempty = Path 1 [] + mappend (Path m1 l1) (Path m2 l2) = Path (Mx.normalize $ m1*m2) (l1++l2) + +instance Show Path where + show (Path m l) = prettyProcesses l -- ++ "\n" ++ Mx.prettyMatrix m + +instance Read Path where + readsPrec _ zs = [reads' zs] where + reads' ('A':xs) = (aPath `mappend` path, ys) where + (path, ys) = reads' xs + reads' ('B':'A':xs) = (baPath `mappend` path, ys) where + (path, ys) = reads' xs + reads' ('B':xs) = (baPath, xs) + reads' xs = (mempty, xs) + +instance Eq Path where + (Path m1 _) == (Path m2 _) = Mx.normalize m1 == Mx.normalize m2 + +instance Ord Path where + (Path _ q1) <= (Path _ q2) = cmp q1 q2 where + cmp (A:p1) (A:p2) = cmp p1 p2 + cmp (BA:p1) (BA:p2) = cmp p2 p1 + cmp (A:_) (BA:_) = True + cmp (BA:_) (A:_) = False + cmp [] _ = True + cmp _ [] = False + +evalPath :: (Num t) => Path -> (t, t, t) -> (t, t, t) +evalPath (Path m _) (a,b,c) = (a',b',c') where + m' = fmap fromInteger m + (Mx.Vector3 a' b' c') = Mx.multCol m' (Mx.Vector3 a b c) + +lengthPath :: Path -> Int +lengthPath (Path _ xs) = length xs + +symbolWidth :: Int +symbolWidth = 10 +bracketWidth :: Int +bracketWidth = 4 +subscriptWidth :: Int +subscriptWidth = 4 + +-- Пусть строка из n символов является повторением строки из l символов +-- Какова длина такой записи? +len0 :: [Process] -> Int -> (Int, String) +len0 xs 1 = (lxs, pxs) where + lxs = length pxs * symbolWidth + pxs = concatMap show xs +len0 [A] n = (symbolWidth + subscriptWidth, show A ++ "^" ++ show n) +len0 [BA] n = (symbolWidth + bracketWidth*2 + subscriptWidth, "(" ++ show BA ++ ")^" ++ show n) +len0 xs n = (lxs + bracketWidth*2 + subscriptWidth, "(" ++ pxs ++ ")^" ++ show n) where + (lxs, pxs) = len2M xs + +len0M :: [Process] -> Int -> (Int, String) +len0M = memoize len0 + +-- Простейшая оптимизация: строка as, целиком состоящая из n повторений подстроки bs, может быть записана как bs^n +len1 :: [Process] -> (Int, String) +len1 as = if null inner + then len0M as 1 + else len0M as 1 `min` minimumBy (comparing fst) inner where + l = length as + bs n = take n as + cs m xs = concat (replicate m xs) + inner = [len0M (bs n) (l`div`n) | n<-[1..l-1], l`mod`n==0, cs (l`div`n) (bs n) == as] + +len1M :: [Process] -> (Int, String) +len1M = memoize len1 + +-- Перебираем все способы разбить строку на две части и применить к каждой из них len1 +len2 :: [Process] -> (Int, String) +len2 as = if null inner + then len1M as + else len1M as `min` minimumBy (comparing fst) inner where + l = length as + bs n = take n as + cs n = drop n as + add (x, xs) (y, ys) = (x+y, xs++ys) + inner = [ len2M (bs n) `add` len2M (cs n) | n<-[1..l-1] ] + +len2M :: [Process] -> (Int, String) +len2M = memoize len2 + +prettyProcesses :: [Process] -> String +prettyProcesses = snd . len2M + + + diff --git a/Math/ExpPairs/RatioInf.hs b/Math/ExpPairs/RatioInf.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/RatioInf.hs @@ -0,0 +1,97 @@ +{-| +Module : ExpPairs.RatioInf +Description : Rational numbers with infinities +Copyright : (c) Andrew Lelechenko, 2014-2015 +License : GPL-3 +Maintainer : andrew.lelechenko@gmail.com +Stability : experimental +Portability : POSIX + +Provides types and necessary instances for rational numbers, extended with infinite values. Just use @RationalInf@ instead of @Rational@. +-} +module Math.ExpPairs.RatioInf (RatioInf (..), RationalInf) where + +import Data.Ratio + +-- |Extends a rational type with positive and negative +-- infinities. +data RatioInf t = InfMinus | Finite (Ratio t) | InfPlus + deriving (Ord, Eq) + +-- |Arbitrary-precision rational numbers with positive and negative +-- infinities +type RationalInf = RatioInf Integer + +instance (Integral t, Show t) => Show (RatioInf t) where + show InfMinus = "-Inf" + show (Finite x) = show x + show InfPlus = "+Inf" + +instance (Integral t) => Num (RatioInf t) where + InfMinus + InfPlus = error "Cannot add up negative and positive infinities" + InfPlus + InfMinus = error "Cannot add up negative and positive infinities" + InfMinus + _ = InfMinus + InfPlus + _ = InfPlus + _ + InfMinus = InfMinus + _ + InfPlus = InfPlus + (Finite a) + (Finite b) = Finite (a+b) + + fromInteger n = Finite (fromInteger n) + + signum InfMinus = Finite (-1) + signum InfPlus = Finite 1 + signum (Finite r) = Finite (signum r) + + abs InfMinus = InfPlus + abs InfPlus = InfPlus + abs (Finite r) = Finite (abs r) + + negate InfMinus = InfPlus + negate InfPlus = InfMinus + negate (Finite r) = Finite (negate r) + + InfMinus * a + | signum a == Finite 0 = error "Cannot multiply infinity by zero" + | signum a == Finite 1 = InfMinus + | signum a == Finite (-1) = InfPlus + InfPlus * a + | signum a == Finite 0 = error "Cannot multiply infinity by zero" + | signum a == Finite 1 = InfPlus + | signum a == Finite (-1) = InfMinus + a * InfMinus + | signum a == Finite 0 = error "Cannot multiply infinity by zero" + | signum a == Finite 1 = InfMinus + | signum a == Finite (-1) = InfPlus + a * InfPlus + | signum a == Finite 0 = error "Cannot multiply infinity by zero" + | signum a == Finite 1 = InfPlus + | signum a == Finite (-1) = InfMinus + (Finite a) * (Finite b) = Finite (a*b) + +instance (Integral t) => Fractional (RatioInf t) where + fromRational = Finite . fromRational + + InfMinus / InfMinus = error "Cannot divide infinity by infinity" + InfMinus / InfPlus = error "Cannot divide infinity by infinity" + InfMinus / (Finite a) + | signum a == 0 = error "Cannot divide infinity by zero" + | signum a == 1 = InfMinus + | signum a == -1 = InfPlus + + InfPlus / InfMinus = error "Cannot divide infinity by infinity" + InfPlus / InfPlus = error "Cannot divide infinity by infinity" + InfPlus / (Finite a) + | signum a == 0 = error "Cannot divide infinity by zero" + | signum a == 1 = InfPlus + | signum a == -1 = InfMinus + + (Finite _) / InfPlus = Finite 0 + (Finite _) / InfMinus = Finite 0 + + (Finite _) / (Finite 0) = error "Cannot divide finite value by zero" + (Finite a) / (Finite b) = Finite (a/b) + +instance (Integral t) => Real (RatioInf t) where + toRational (Finite r) = toRational r + toRational InfPlus = error "Cannot map infinity into Rational" + toRational InfMinus = error "Cannot map infinity into Rational" diff --git a/Math/ExpPairs/Tests.hs b/Math/ExpPairs/Tests.hs new file mode 100644 --- /dev/null +++ b/Math/ExpPairs/Tests.hs @@ -0,0 +1,19 @@ +--module Math.ExpPairs.Tests where + +import qualified Math.ExpPairs.Tests.RatioInf as RatioInf (testSuite) +import qualified Math.ExpPairs.Tests.Pair as Pair (testSuite) +import qualified Math.ExpPairs.Tests.Ivic as Ivic (testSuite) +import qualified Math.ExpPairs.Tests.MenzerNowak as MenzerNowak (testSuite) +import qualified Math.ExpPairs.Tests.Matrix3 as Matrix3 (testSuite) +import qualified Math.ExpPairs.Tests.Kratzel as Kratzel (testSuite) +import qualified Math.ExpPairs.Tests.LinearForm as LinearForm (testSuite) + +main :: IO () +main = do + RatioInf.testSuite + Pair.testSuite + Ivic.testSuite + MenzerNowak.testSuite + Matrix3.testSuite + Kratzel.testSuite + LinearForm.testSuite diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/exp-pairs.cabal b/exp-pairs.cabal new file mode 100644 --- /dev/null +++ b/exp-pairs.cabal @@ -0,0 +1,41 @@ +name: exp-pairs +version: 0.1.0.0 +synopsis: Linear programming over exponent pairs +description: Package implements an algorithm to minimize rational objective function over the set of exponent pairs +homepage: https://github.com/Bodigrim/exp-pairs +license: GPL-3 +license-file: LICENSE +author: Andrew Lelechenko +maintainer: andrew.lelechenko@gmail.com +category: Math +build-type: Simple +cabal-version: >=1.8 + +source-repository head + type: git + location: git://github.com/Bodigrim/exp-pairs.git + +library + exposed-modules: Math.ExpPairs, + Math.ExpPairs.Ivic, + Math.ExpPairs.Kratzel, + Math.ExpPairs.MenzerNowak + other-modules: Math.ExpPairs.LinearForm, + Math.ExpPairs.Matrix3, + Math.ExpPairs.Pair, + Math.ExpPairs.Process, + Math.ExpPairs.RatioInf + build-depends: base >=4 && <5, + memoize >=0.1, + matrix >=0.1 + +test-suite tests + ghc-options: -Wall + type: exitcode-stdio-1.0 + main-is: Math/ExpPairs/Tests.hs + build-depends: base >=4 && <5, + QuickCheck >=2.4.2, + smallcheck >=0.2.1, + exp-pairs, + memoize >=0.1, + matrix >=0.1