hspray 0.1.2.0 → 0.1.3.0
raw patch · 7 files changed
+1222/−1154 lines, 7 filessetup-changedPVP ok
version bump matches the API change (PVP)
API changes (from Hackage documentation)
+ Math.Algebra.Hspray: Powers :: Seq Int -> Int -> Powers
+ Math.Algebra.Hspray: [exponents] :: Powers -> Seq Int
+ Math.Algebra.Hspray: [nvariables] :: Powers -> Int
+ Math.Algebra.Hspray: data Powers
Files
- CHANGELOG.md +34/−27
- LICENSE +674/−674
- README.md +130/−74
- Setup.hs +2/−2
- hspray.cabal +53/−52
- src/Math/Algebra/Hspray.hs +256/−252
- tests/Main.hs +73/−73
CHANGELOG.md view
@@ -1,27 +1,34 @@-# Changelog for `hspray`--All notable changes to this project will be documented in this file.--The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),-and this project adheres to the-[Haskell Package Versioning Policy](https://pvp.haskell.org/).---## 0.1.0.0 - 2022-12-11--First release.---## 0.1.1.0 - 2022-12-12--* New functions `toList`, `sprayTerms` and `bombieriSpray`.--* New operation `.^`, to multiply a spray by an integer.--* Added some unit tests.---## 0.1.2.0 - 2023-02-24--New function `derivSpray`, to differentiate a spray.-+# Changelog for `hspray` + +All notable changes to this project will be documented in this file. + +The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/), +and this project adheres to the +[Haskell Package Versioning Policy](https://pvp.haskell.org/). + + +## 0.1.0.0 - 2022-12-11 + +First release. + + +## 0.1.1.0 - 2022-12-12 + +* New functions `toList`, `sprayTerms` and `bombieriSpray`. + +* New operation `.^`, to multiply a spray by an integer. + +* Added some unit tests. + + +## 0.1.2.0 - 2023-02-24 + +New function `derivSpray`, to differentiate a spray. + + +## 0.1.3.0 - 2023-08-29 + +* `Powers(..)` is now exported. + +* Completed the README to show how to deal with symbolic coefficients. +
LICENSE view
@@ -1,674 +1,674 @@- GNU GENERAL PUBLIC LICENSE- Version 3, 29 June 2007-- Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>- 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 <https://www.gnu.org/licenses/>.--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:-- <program> Copyright (C) <year> <name of author>- 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-<https://www.gnu.org/licenses/>.-- 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-<https://www.gnu.org/licenses/why-not-lgpl.html>.+ GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/> + 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 <https://www.gnu.org/licenses/>. + +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: + + <program> Copyright (C) <year> <name of author> + 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 +<https://www.gnu.org/licenses/>. + + 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 +<https://www.gnu.org/licenses/why-not-lgpl.html>.
README.md view
@@ -1,74 +1,130 @@-# hspray--<!-- badges: start -->-[](https://github.com/stla/hspray/actions/workflows/Stack-lts.yml)-[](https://github.com/stla/hspray/actions/workflows/Stack-nightly.yml)-<!-- badges: end -->--Simple multivariate polynomials in Haskell.--___---```haskell-import Math.Algebra.Hspray-x = lone 1 :: Spray Double-y = lone 2 :: Spray Double-z = lone 3 :: Spray Double-poly = (2 *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ (4 *^ (x ^*^ y ^*^ z))-prettySpray show "X" poly--- "(4.0) * X^(3, 1, 1) + (8.0) * X^(4, 2, 2)"-```--More generally, one can use the type `Spray a` as long as the type `a` has -the instances `Eq` and `Algebra.Ring` (defined in the **numeric-prelude** -library). For example `a = Rational`:--```haskell-import Math.Algebra.Hspray-import Data.Ratio-x = lone 1 :: Spray Rational-y = lone 2 :: Spray Rational-z = lone 3 :: Spray Rational-poly = ((2%3) *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ ((7%4) *^ (x ^*^ y ^*^ z))-prettySpray show "X" poly--- "(7 % 4) * X^(3, 1, 1) + (7 % 6) * X^(4, 2, 2)"-```--Or `a = Spray Double`:--```haskell-import Math.Algebra.Hspray-p = lone 1 :: Spray Double-x = lone 1 :: Spray (Spray Double)-y = lone 2 :: Spray (Spray Double)-poly = ((p *^ x) ^+^ (p *^ y))^**^2 -prettySpray (prettySpray show "a") "X" poly--- "((1.0) * a^(2)) * X^(0, 2) + ((2.0) * a^(2)) * X^(1, 1) + ((1.0) * a^(2)) * X^(2)"-```--Evaluation:--```haskell-import Math.Algebra.Hspray-x = lone 1 :: Spray Double-y = lone 2 :: Spray Double-z = lone 3 :: Spray Double-poly = 2 *^ (x ^*^ y ^*^ z) --- evaluate poly at x=2, y=1, z=2-evalSpray poly [2, 1, 2]--- 8.0-```--Differentiation:--```haskell-import Math.Algebra.Hspray-x = lone 1 :: Spray Double-y = lone 2 :: Spray Double-z = lone 3 :: Spray Double-poly = 2 *^ (x ^*^ y ^*^ z) ^+^ (3 *^ x^**^2)--- derivate with respect to x-prettySpray show "X" $ derivSpray 1 poly--- "(2.0) * X^(0, 1, 1) + (6.0) * X^(1)"-```+# hspray + +<!-- badges: start --> +[](https://github.com/stla/hspray/actions/workflows/Stack-lts.yml) +[](https://github.com/stla/hspray/actions/workflows/Stack-nightly.yml) +<!-- badges: end --> + +Simple multivariate polynomials in Haskell. + +___ + + +```haskell +import Math.Algebra.Hspray +x = lone 1 :: Spray Double +y = lone 2 :: Spray Double +z = lone 3 :: Spray Double +poly = (2 *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ (4 *^ (x ^*^ y ^*^ z)) +prettySpray show "X" poly +-- "(4.0) * X^(3, 1, 1) + (8.0) * X^(4, 2, 2)" +``` + +More generally, one can use the type `Spray a` as long as the type `a` has +the instances `Eq` and `Algebra.Ring` (defined in the **numeric-prelude** +library). For example `a = Rational`: + +```haskell +import Math.Algebra.Hspray +import Data.Ratio +x = lone 1 :: Spray Rational +y = lone 2 :: Spray Rational +z = lone 3 :: Spray Rational +poly = ((2%3) *^ (x^**^3 ^*^ y ^*^ z) ^+^ x^**^2) ^*^ ((7%4) *^ (x ^*^ y ^*^ z)) +prettySpray show "X" poly +-- "(7 % 4) * X^(3, 1, 1) + (7 % 6) * X^(4, 2, 2)" +``` + +Or `a = Spray Double`: + +```haskell +import Math.Algebra.Hspray +p = lone 1 :: Spray Double +x = lone 1 :: Spray (Spray Double) +y = lone 2 :: Spray (Spray Double) +poly = ((p *^ x) ^+^ (p *^ y))^**^2 +prettySpray (prettySpray show "a") "X" poly +-- "((1.0) * a^(2)) * X^(0, 2) + ((2.0) * a^(2)) * X^(1, 1) + ((1.0) * a^(2)) * X^(2)" +``` + +Evaluation: + +```haskell +import Math.Algebra.Hspray +x = lone 1 :: Spray Double +y = lone 2 :: Spray Double +z = lone 3 :: Spray Double +poly = 2 *^ (x ^*^ y ^*^ z) +-- evaluate poly at x=2, y=1, z=2 +evalSpray poly [2, 1, 2] +-- 8.0 +``` + +Differentiation: + +```haskell +import Math.Algebra.Hspray +x = lone 1 :: Spray Double +y = lone 2 :: Spray Double +z = lone 3 :: Spray Double +poly = 2 *^ (x ^*^ y ^*^ z) ^+^ (3 *^ x^**^2) +-- derivate with respect to x +prettySpray show "X" $ derivSpray 1 poly +-- "(2.0) * X^(0, 1, 1) + (6.0) * X^(1)" +``` + +## Easier usage + +To construct a polynomial using the ordinary symbols `+`, `*` and `-`, +one can hide these operators from **Prelude** and import them from +the **numeric-prelude** library: + +```haskell +import Prelude hiding ((*), (+), (-)) +import qualified Prelude as P +import Algebra.Additive +import Algebra.Module +import Algebra.Ring +import Math.Algebra.Hspray +``` + +Or, maybe better (I didn't try yet), follow the "Usage" section on the +[Hackage page](https://hackage.haskell.org/package/numeric-prelude-0.4.4#usage) +of **numeric-prelude**. + +## Symbolic coefficients + +Assume you have the polynomial `a * (x² + y²) + 2b/3 * z`, +where `a` and `b` are symbolic coefficients. +You can define this polynomial as a `Spray` as follows: + +```haskell +import Prelude hiding ((*), (+), (-)) +import qualified Prelude as P +import Algebra.Additive +import Algebra.Module +import Algebra.Ring +import Math.Algebra.Hspray +import Data.Ratio + +x = lone 1 :: Spray (Spray Rational) +y = lone 2 :: Spray (Spray Rational) +z = lone 3 :: Spray (Spray Rational) +a = lone 1 :: Spray Rational +b = lone 2 :: Spray Rational + +poly = a *^ (x*x + y*y) + ((2%3) *^ b) *^ z +prettySpray (prettySpray show "a") "X" poly +-- "((2 % 3) * a^(0, 1)) * X^(0, 0, 1) + ((1 % 1) * a^(1)) * X^(0, 2) + ((1 % 1) * a^(1)) * X^(2)" +``` + +The `prettySpray` function shows the expansion of the polynomial. +You can extract the powers and the coefficients as follows: + +```haskell +l = toList poly +map fst l +-- [[0,0,1],[2],[0,2]] +map toList $ map snd l +-- [[([0,1],2 % 3)],[([1],1 % 1)],[([1],1 % 1)]] +```
Setup.hs view
@@ -1,2 +1,2 @@-import Distribution.Simple-main = defaultMain+import Distribution.Simple +main = defaultMain
hspray.cabal view
@@ -1,52 +1,53 @@-name: hspray-version: 0.1.2.0-synopsis: Multivariate polynomials.-description: Manipulation of multivariate polynomials on a ring.-homepage: https://github.com/stla/hspray#readme-license: GPL-3-license-file: LICENSE-author: Stéphane Laurent-maintainer: laurent_step@outlook.fr-copyright: 2022 Stéphane Laurent-category: Math, Algebra-build-type: Simple-extra-source-files: README.md- CHANGELOG.md-cabal-version: >=1.10--library- hs-source-dirs: src- exposed-modules: Math.Algebra.Hspray- build-depends: base >= 4.7 && < 5- , containers >= 0.6.4.1- , hashable >= 1.3.4.0- , unordered-containers >= 0.2.17.0- , numeric-prelude >= 0.4.4- , text >= 1.2.5.0- other-extensions: FlexibleInstances- , FlexibleContexts- , MultiParamTypeClasses- default-language: Haskell2010- ghc-options: -Wall- -Wcompat- -Widentities- -Wincomplete-record-updates- -Wincomplete-uni-patterns- -Wmissing-export-lists- -Wmissing-home-modules- -Wpartial-fields- -Wredundant-constraints--test-suite unit-tests- type: exitcode-stdio-1.0- main-is: Main.hs- hs-source-dirs: tests/- Build-Depends: base >= 4.7 && < 5- , tasty- , tasty-hunit- , hspray- Default-Language: Haskell2010--source-repository head- type: git- location: https://github.com/stla/hspray+name: hspray +version: 0.1.3.0 +synopsis: Multivariate polynomials. +description: Manipulation of multivariate polynomials on a ring. +homepage: https://github.com/stla/hspray#readme +license: GPL-3 +license-file: LICENSE +author: Stéphane Laurent +maintainer: laurent_step@outlook.fr +copyright: 2022 Stéphane Laurent +category: Math, Algebra +build-type: Simple +extra-source-files: README.md + CHANGELOG.md +cabal-version: >=1.10 + +library + hs-source-dirs: src + exposed-modules: Math.Algebra.Hspray + build-depends: base >= 4.7 && < 5 + , containers >= 0.6.4.1 + , hashable >= 1.3.4.0 + , unordered-containers >= 0.2.17.0 + , numeric-prelude >= 0.4.4 + , text >= 1.2.5.0 + other-extensions: FlexibleInstances + , FlexibleContexts + , MultiParamTypeClasses + , InstanceSigs + default-language: Haskell2010 + ghc-options: -Wall + -Wcompat + -Widentities + -Wincomplete-record-updates + -Wincomplete-uni-patterns + -Wmissing-export-lists + -Wmissing-home-modules + -Wpartial-fields + -Wredundant-constraints + +test-suite unit-tests + type: exitcode-stdio-1.0 + main-is: Main.hs + hs-source-dirs: tests/ + Build-Depends: base >= 4.7 && < 5 + , tasty + , tasty-hunit + , hspray + Default-Language: Haskell2010 + +source-repository head + type: git + location: https://github.com/stla/hspray
src/Math/Algebra/Hspray.hs view
@@ -1,252 +1,256 @@-{-# LANGUAGE FlexibleInstances #-}-{-# LANGUAGE FlexibleContexts #-}-{-# LANGUAGE MultiParamTypeClasses #-}--module Math.Algebra.Hspray- ( Spray- , lone- , unitSpray- , constantSpray- , fromList- , toList- , sprayTerms- , (*^)- , (.^)- , (^+^)- , (^-^)- , (^*^)- , (^**^)- , evalSpray- , prettySpray- , composeSpray- , bombieriSpray- , derivSpray- ) where-import qualified Algebra.Additive as AlgAdd-import qualified Algebra.Module as AlgMod-import qualified Algebra.Ring as AlgRing-import qualified Data.Foldable as DF-import Data.Function ( on )-import Data.HashMap.Strict ( HashMap )-import qualified Data.HashMap.Strict as HM-import Data.Hashable ( Hashable(hashWithSalt) )-import Data.List ( sortBy )-import qualified Data.Sequence as S-import Data.Sequence ( (><)- , Seq- , dropWhileR- , (|>)- , index- , adjust- )-import Data.Text ( Text- , append- , cons- , intercalate- , pack- , snoc- , unpack- )---infixr 7 *^, .^--infixl 6 ^+^, ^-^--infixl 7 ^*^--infixr 8 ^**^---data Powers = Powers- { exponents :: Seq Int- , nvariables :: Int- }- deriving Show--growSequence :: Seq Int -> Int -> Int -> Seq Int-growSequence s m n = s >< t where t = S.replicate (n - m) 0--harmonize :: (Powers, Powers) -> (Powers, Powers)-harmonize (pows1, pows2) = (Powers e1' n, Powers e2' n)- where- e1 = exponents pows1- e2 = exponents pows2- n1 = nvariables pows1- n2 = nvariables pows2- (e1', e2', n) = if n1 < n2- then (growSequence e1 n1 n2, e2, n2)- else (e1, growSequence e2 n2 n1, n1)--instance Eq Powers where- pows1 == pows2 = exponents pows1' == exponents pows2'- where (pows1', pows2') = harmonize (pows1, pows2)--instance Hashable Powers where- hashWithSalt k pows = hashWithSalt k (exponents pows, nvariables pows)--type Spray a = HashMap Powers a--instance (AlgAdd.C a, Eq a) => AlgAdd.C (Spray a) where- p + q = addSprays p q- zero = HM.empty- negate = negateSpray--instance (AlgRing.C a, Eq a) => AlgMod.C a (Spray a) where- lambda *> p = scaleSpray lambda p--instance (AlgRing.C a, Eq a) => AlgRing.C (Spray a) where- p * q = multSprays p q- one = lone 0---- | Addition of two sprays-(^+^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a-(^+^) p q = p AlgAdd.+ q---- | Substraction of two sprays-(^-^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a-(^-^) p q = p AlgAdd.- q---- | Multiply two sprays-(^*^) :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a-(^*^) p q = p AlgRing.* q---- | Power of a spray-(^**^) :: (AlgRing.C a, Eq a) => Spray a -> Int -> Spray a-(^**^) p n = AlgRing.product (replicate n p)---- | Scale spray by a scalar-(*^) :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a-(*^) lambda pol = lambda AlgMod.*> pol---- | Scale spray by an integer-(.^) :: (AlgAdd.C a, Eq a) => Int -> Spray a -> Spray a-(.^) k pol = if k >= 0- then AlgAdd.sum (replicate k pol)- else AlgAdd.negate $ AlgAdd.sum (replicate (-k) pol)--simplifyPowers :: Powers -> Powers-simplifyPowers pows = Powers s (S.length s)- where s = dropWhileR (== 0) (exponents pows)--simplifySpray :: Spray a -> Spray a-simplifySpray = HM.mapKeys simplifyPowers--cleanSpray :: (AlgAdd.C a, Eq a) => Spray a -> Spray a-cleanSpray p = HM.filter (/= AlgAdd.zero) (simplifySpray p)--addSprays :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a-addSprays p q = cleanSpray $ HM.foldlWithKey' f p q- where f s powers coef = HM.insertWith (AlgAdd.+) powers coef s--negateSpray :: AlgAdd.C a => Spray a -> Spray a-negateSpray = HM.map AlgAdd.negate--scaleSpray :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a-scaleSpray lambda p = cleanSpray $ HM.map (lambda AlgRing.*) p--derivMonomial :: AlgRing.C a => Int -> (Powers, a) -> (Powers, a) -derivMonomial i (pows, coef) = if i' >= S.length expts - then (Powers S.empty 0, AlgAdd.zero)- else (pows', coef')- where- i' = i - 1- expts = exponents pows- expt_i = expts `index` i'- expts' = adjust (subtract 1) i' expts- coef' = AlgAdd.sum (replicate expt_i coef)- pows' = Powers expts' (nvariables pows) --derivSpray :: (AlgRing.C a, Eq a) => Int -> Spray a -> Spray a-derivSpray i p = cleanSpray $ HM.fromListWith (AlgAdd.+) monomials- where- p' = HM.toList p- monomials = [ derivMonomial i mp | mp <- p' ]---multMonomial :: AlgRing.C a => (Powers, a) -> (Powers, a) -> (Powers, a)-multMonomial (pows1, coef1) (pows2, coef2) = (pows, coef1 AlgRing.* coef2)- where- (pows1', pows2') = harmonize (pows1, pows2)- expts = S.zipWith (+) (exponents pows1') (exponents pows2')- pows = Powers expts (nvariables pows1')--multSprays :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a-multSprays p q = cleanSpray $ HM.fromListWith (AlgAdd.+) prods- where- p' = HM.toList p- q' = HM.toList q- prods = [ multMonomial mp mq | mp <- p', mq <- q' ]---- | Spray corresponding to polynomial x_n-lone :: AlgRing.C a => Int -> Spray a-lone n = HM.singleton pows AlgRing.one- where- pows = if n == 0- then Powers S.empty 0- else Powers (S.replicate (n - 1) AlgAdd.zero |> AlgRing.one) n---- | Unit spray-unitSpray :: AlgRing.C a => Spray a-unitSpray = lone 0---- | Constant spray-constantSpray :: (AlgRing.C a, Eq a) => a -> Spray a-constantSpray c = c *^ lone 0--evalMonomial :: AlgRing.C a => [a] -> (Powers, a) -> a-evalMonomial xyz (powers, coeff) = coeff- AlgRing.* AlgRing.product (zipWith (AlgRing.^) xyz pows)- where pows = DF.toList (fromIntegral <$> exponents powers)---- | Evaluate a spray-evalSpray :: AlgRing.C a => Spray a -> [a] -> a-evalSpray p xyz = AlgAdd.sum $ map (evalMonomial xyz) (HM.toList p)--identify :: (AlgRing.C a, Eq a) => Spray a -> Spray (Spray a)-identify = HM.map constantSpray---- | Compose a spray with a change of variables-composeSpray :: (AlgRing.C a, Eq a) => Spray a -> [Spray a] -> Spray a-composeSpray p = evalSpray (identify p)---- | Create a spray from list of terms-fromList :: (AlgRing.C a, Eq a) => [([Int], a)] -> Spray a-fromList x = cleanSpray $ HM.fromList $ map- (\(expts, coef) -> (Powers (S.fromList expts) (length expts), coef)) x--prettyPowers :: String -> [Int] -> Text-prettyPowers var pows = append (pack x) (cons '(' $ snoc string ')')- where- x = " " ++ var ++ "^"- string = intercalate (pack ", ") (map (pack . show) pows)---- | Pretty form of a spray-prettySpray :: (a -> String) -> String -> Spray a -> String-prettySpray prettyCoef var p = unpack $ intercalate (pack " + ") stringTerms- where- stringTerms = map stringTerm (sortBy (compare `on` fexpts) (HM.toList p))- fexpts term = exponents $ fst term- stringTerm term = append- (snoc (snoc (cons '(' $ snoc stringCoef ')') ' ') '*')- (prettyPowers var pows)- where- pows = DF.toList $ exponents (fst term)- stringCoef = pack $ prettyCoef (snd term)---- | Terms of a spray-sprayTerms :: Spray a -> HashMap (Seq Int) a-sprayTerms = HM.mapKeys exponents---- | Spray as list-toList :: Spray a -> [([Int], a)]-toList p = HM.toList $ HM.mapKeys (DF.toList . exponents) p---- | Bombieri spray-bombieriSpray :: AlgAdd.C a => Spray a -> Spray a-bombieriSpray = HM.mapWithKey f- where- f pows = times (pfactorial $ exponents pows)- pfactorial pows = product $ DF.toList $ factorial <$> S.filter (/= 0) pows- factorial n = product [1 .. n]- times k x = AlgAdd.sum (replicate k x)+{-# LANGUAGE FlexibleInstances #-} +{-# LANGUAGE FlexibleContexts #-} +{-# LANGUAGE MultiParamTypeClasses #-} +{-# LANGUAGE InstanceSigs #-} + +module Math.Algebra.Hspray + ( Powers (..) + , Spray + , lone + , unitSpray + , constantSpray + , fromList + , toList + , sprayTerms + , (*^) + , (.^) + , (^+^) + , (^-^) + , (^*^) + , (^**^) + , evalSpray + , prettySpray + , composeSpray + , bombieriSpray + , derivSpray + ) where +import qualified Algebra.Additive as AlgAdd +import qualified Algebra.Module as AlgMod +import qualified Algebra.Ring as AlgRing +import qualified Data.Foldable as DF +import Data.Function ( on ) +import Data.HashMap.Strict ( HashMap ) +import qualified Data.HashMap.Strict as HM +import Data.Hashable ( Hashable(hashWithSalt) ) +import Data.List ( sortBy ) +import qualified Data.Sequence as S +import Data.Sequence ( (><) + , Seq + , dropWhileR + , (|>) + , index + , adjust + ) +import Data.Text ( Text + , append + , cons + , intercalate + , pack + , snoc + , unpack + ) + + +infixr 7 *^, .^ + +infixl 6 ^+^, ^-^ + +infixl 7 ^*^ + +infixr 8 ^**^ + + +data Powers = Powers + { exponents :: Seq Int + , nvariables :: Int + } + deriving Show + +growSequence :: Seq Int -> Int -> Int -> Seq Int +growSequence s m n = s >< t where t = S.replicate (n - m) 0 + +harmonize :: (Powers, Powers) -> (Powers, Powers) +harmonize (pows1, pows2) = (Powers e1' n, Powers e2' n) + where + e1 = exponents pows1 + e2 = exponents pows2 + n1 = nvariables pows1 + n2 = nvariables pows2 + (e1', e2', n) = if n1 < n2 + then (growSequence e1 n1 n2, e2, n2) + else (e1, growSequence e2 n2 n1, n1) + +instance Eq Powers where + (==) :: Powers -> Powers -> Bool + pows1 == pows2 = exponents pows1' == exponents pows2' + where (pows1', pows2') = harmonize (pows1, pows2) + +instance Hashable Powers where + hashWithSalt :: Int -> Powers -> Int + hashWithSalt k pows = hashWithSalt k (exponents pows, nvariables pows) + +type Spray a = HashMap Powers a + +instance (AlgAdd.C a, Eq a) => AlgAdd.C (Spray a) where + p + q = addSprays p q + zero = HM.empty + negate = negateSpray + +instance (AlgRing.C a, Eq a) => AlgMod.C a (Spray a) where + lambda *> p = scaleSpray lambda p + +instance (AlgRing.C a, Eq a) => AlgRing.C (Spray a) where + p * q = multSprays p q + one = lone 0 + +-- | Addition of two sprays +(^+^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a +(^+^) p q = p AlgAdd.+ q + +-- | Substraction of two sprays +(^-^) :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a +(^-^) p q = p AlgAdd.- q + +-- | Multiply two sprays +(^*^) :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a +(^*^) p q = p AlgRing.* q + +-- | Power of a spray +(^**^) :: (AlgRing.C a, Eq a) => Spray a -> Int -> Spray a +(^**^) p n = AlgRing.product (replicate n p) + +-- | Scale spray by a scalar +(*^) :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a +(*^) lambda pol = lambda AlgMod.*> pol + +-- | Scale spray by an integer +(.^) :: (AlgAdd.C a, Eq a) => Int -> Spray a -> Spray a +(.^) k pol = if k >= 0 + then AlgAdd.sum (replicate k pol) + else AlgAdd.negate $ AlgAdd.sum (replicate (-k) pol) + +simplifyPowers :: Powers -> Powers +simplifyPowers pows = Powers s (S.length s) + where s = dropWhileR (== 0) (exponents pows) + +simplifySpray :: Spray a -> Spray a +simplifySpray = HM.mapKeys simplifyPowers + +cleanSpray :: (AlgAdd.C a, Eq a) => Spray a -> Spray a +cleanSpray p = HM.filter (/= AlgAdd.zero) (simplifySpray p) + +addSprays :: (AlgAdd.C a, Eq a) => Spray a -> Spray a -> Spray a +addSprays p q = cleanSpray $ HM.foldlWithKey' f p q + where f s powers coef = HM.insertWith (AlgAdd.+) powers coef s + +negateSpray :: AlgAdd.C a => Spray a -> Spray a +negateSpray = HM.map AlgAdd.negate + +scaleSpray :: (AlgRing.C a, Eq a) => a -> Spray a -> Spray a +scaleSpray lambda p = cleanSpray $ HM.map (lambda AlgRing.*) p + +derivMonomial :: AlgRing.C a => Int -> (Powers, a) -> (Powers, a) +derivMonomial i (pows, coef) = if i' >= S.length expts + then (Powers S.empty 0, AlgAdd.zero) + else (pows', coef') + where + i' = i - 1 + expts = exponents pows + expt_i = expts `index` i' + expts' = adjust (subtract 1) i' expts + coef' = AlgAdd.sum (replicate expt_i coef) + pows' = Powers expts' (nvariables pows) + +derivSpray :: (AlgRing.C a, Eq a) => Int -> Spray a -> Spray a +derivSpray i p = cleanSpray $ HM.fromListWith (AlgAdd.+) monomials + where + p' = HM.toList p + monomials = [ derivMonomial i mp | mp <- p' ] + + +multMonomial :: AlgRing.C a => (Powers, a) -> (Powers, a) -> (Powers, a) +multMonomial (pows1, coef1) (pows2, coef2) = (pows, coef1 AlgRing.* coef2) + where + (pows1', pows2') = harmonize (pows1, pows2) + expts = S.zipWith (+) (exponents pows1') (exponents pows2') + pows = Powers expts (nvariables pows1') + +multSprays :: (AlgRing.C a, Eq a) => Spray a -> Spray a -> Spray a +multSprays p q = cleanSpray $ HM.fromListWith (AlgAdd.+) prods + where + p' = HM.toList p + q' = HM.toList q + prods = [ multMonomial mp mq | mp <- p', mq <- q' ] + +-- | Spray corresponding to polynomial x_n +lone :: AlgRing.C a => Int -> Spray a +lone n = HM.singleton pows AlgRing.one + where + pows = if n == 0 + then Powers S.empty 0 + else Powers (S.replicate (n - 1) AlgAdd.zero |> AlgRing.one) n + +-- | Unit spray +unitSpray :: AlgRing.C a => Spray a +unitSpray = lone 0 + +-- | Constant spray +constantSpray :: (AlgRing.C a, Eq a) => a -> Spray a +constantSpray c = c *^ lone 0 + +evalMonomial :: AlgRing.C a => [a] -> (Powers, a) -> a +evalMonomial xyz (powers, coeff) = coeff + AlgRing.* AlgRing.product (zipWith (AlgRing.^) xyz pows) + where pows = DF.toList (fromIntegral <$> exponents powers) + +-- | Evaluate a spray +evalSpray :: AlgRing.C a => Spray a -> [a] -> a +evalSpray p xyz = AlgAdd.sum $ map (evalMonomial xyz) (HM.toList p) + +identify :: (AlgRing.C a, Eq a) => Spray a -> Spray (Spray a) +identify = HM.map constantSpray + +-- | Compose a spray with a change of variables +composeSpray :: (AlgRing.C a, Eq a) => Spray a -> [Spray a] -> Spray a +composeSpray p = evalSpray (identify p) + +-- | Create a spray from list of terms +fromList :: (AlgRing.C a, Eq a) => [([Int], a)] -> Spray a +fromList x = cleanSpray $ HM.fromList $ map + (\(expts, coef) -> (Powers (S.fromList expts) (length expts), coef)) x + +prettyPowers :: String -> [Int] -> Text +prettyPowers var pows = append (pack x) (cons '(' $ snoc string ')') + where + x = " " ++ var ++ "^" + string = intercalate (pack ", ") (map (pack . show) pows) + +-- | Pretty form of a spray +prettySpray :: (a -> String) -> String -> Spray a -> String +prettySpray prettyCoef var p = unpack $ intercalate (pack " + ") stringTerms + where + stringTerms = map stringTerm (sortBy (compare `on` fexpts) (HM.toList p)) + fexpts term = exponents $ fst term + stringTerm term = append + (snoc (snoc (cons '(' $ snoc stringCoef ')') ' ') '*') + (prettyPowers var pows) + where + pows = DF.toList $ exponents (fst term) + stringCoef = pack $ prettyCoef (snd term) + +-- | Terms of a spray +sprayTerms :: Spray a -> HashMap (Seq Int) a +sprayTerms = HM.mapKeys exponents + +-- | Spray as list +toList :: Spray a -> [([Int], a)] +toList p = HM.toList $ HM.mapKeys (DF.toList . exponents) p + +-- | Bombieri spray +bombieriSpray :: AlgAdd.C a => Spray a -> Spray a +bombieriSpray = HM.mapWithKey f + where + f pows = times (pfactorial $ exponents pows) + pfactorial pows = product $ DF.toList $ factorial <$> S.filter (/= 0) pows + factorial n = product [1 .. n] + times k x = AlgAdd.sum (replicate k x)
tests/Main.hs view
@@ -1,73 +1,73 @@-module Main where-import Data.Ratio ( (%) )-import Math.Algebra.Hspray ( Spray,- (^+^),- (^*^),- (^**^),- (*^),- lone,- unitSpray,- evalSpray,- composeSpray,- fromList,- toList,- bombieriSpray,- derivSpray )-import Test.Tasty ( defaultMain- , testGroup- )-import Test.Tasty.HUnit ( assertEqual- , testCase- )--main :: IO ()-main = defaultMain $ testGroup- "Testing hspray"-- [ testCase "bombieriSpray" $ do- let- x = lone 1 :: Spray Rational- y = lone 2 :: Spray Rational- z = lone 3 :: Spray Rational- poly =- (2 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray- bpoly =- (24 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray- assertEqual "" bpoly (bombieriSpray poly),-- testCase "composeSpray" $ do- let- x = lone 1 :: Spray Int- y = lone 2 :: Spray Int- z = lone 3 :: Spray Int- p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray- px = x ^+^ y ^+^ z- py = x ^*^ y ^*^ z- pz = y ^**^ 2- q = composeSpray p [px, py, pz]- xyz = [2, 3, 4]- pxyz = map (`evalSpray` xyz) [px, py, pz]- assertEqual "" (evalSpray p pxyz) (evalSpray q xyz),-- testCase "fromList . toList = identity" $ do- let- x = lone 1 :: Spray Int- y = lone 2 :: Spray Int- z = lone 3 :: Spray Int- p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray- assertEqual "" p (fromList . toList $ p),-- testCase "derivSpray" $ do- let- x = lone 1 :: Spray Int- y = lone 2 :: Spray Int- z = lone 3 :: Spray Int- p1 = x ^+^ y ^*^ z ^**^ 3- p2 = (x ^*^ y ^*^ z) ^+^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2))- q = p1 ^*^ p2- p1' = derivSpray 1 p1- p2' = derivSpray 1 p2- q' = derivSpray 1 q- assertEqual "" q' ((p1' ^*^ p2) ^+^ (p1 ^*^ p2'))-- ]+module Main where +import Data.Ratio ( (%) ) +import Math.Algebra.Hspray ( Spray, + (^+^), + (^*^), + (^**^), + (*^), + lone, + unitSpray, + evalSpray, + composeSpray, + fromList, + toList, + bombieriSpray, + derivSpray ) +import Test.Tasty ( defaultMain + , testGroup + ) +import Test.Tasty.HUnit ( assertEqual + , testCase + ) + +main :: IO () +main = defaultMain $ testGroup + "Testing hspray" + + [ testCase "bombieriSpray" $ do + let + x = lone 1 :: Spray Rational + y = lone 2 :: Spray Rational + z = lone 3 :: Spray Rational + poly = + (2 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray + bpoly = + (24 % 1) *^ ((2 % 1) *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ (4 % 1) *^ z ^+^ (5 % 1) *^ unitSpray + assertEqual "" bpoly (bombieriSpray poly), + + testCase "composeSpray" $ do + let + x = lone 1 :: Spray Int + y = lone 2 :: Spray Int + z = lone 3 :: Spray Int + p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray + px = x ^+^ y ^+^ z + py = x ^*^ y ^*^ z + pz = y ^**^ 2 + q = composeSpray p [px, py, pz] + xyz = [2, 3, 4] + pxyz = map (`evalSpray` xyz) [px, py, pz] + assertEqual "" (evalSpray p pxyz) (evalSpray q xyz), + + testCase "fromList . toList = identity" $ do + let + x = lone 1 :: Spray Int + y = lone 2 :: Spray Int + z = lone 3 :: Spray Int + p = 2 *^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) ^+^ 4 *^ z ^+^ 5 *^ unitSpray + assertEqual "" p (fromList . toList $ p), + + testCase "derivSpray" $ do + let + x = lone 1 :: Spray Int + y = lone 2 :: Spray Int + z = lone 3 :: Spray Int + p1 = x ^+^ y ^*^ z ^**^ 3 + p2 = (x ^*^ y ^*^ z) ^+^ (2 *^ (x ^**^ 3 ^*^ y ^**^ 2)) + q = p1 ^*^ p2 + p1' = derivSpray 1 p1 + p2' = derivSpray 1 p2 + q' = derivSpray 1 q + assertEqual "" q' ((p1' ^*^ p2) ^+^ (p1 ^*^ p2')) + + ]