- Commit
- 9c7392a31b59f45a69505c0415b9b7e6387205d5
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Initial commit
numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
Diffstat
5 files changed, 1060 insertions, 0 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Added | .gitignore | 5 | 5 | 0 |
| Added | LICENSE | 675 | 675 | 0 |
| Added | getrf.py | 106 | 106 | 0 |
| Added | linear-solvers.ipynb | 196 | 196 | 0 |
| Added | test_getrf.py | 78 | 78 | 0 |
diff --git a/.gitignore b/.gitignore @@ -0,0 +1,5 @@ +testlogs.json +lapack + +.ipynb_checkpoints +__pycache__
diff --git a/LICENSE b/LICENSE @@ -0,0 +1,675 @@ + 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>. +
diff --git a/getrf.py b/getrf.py @@ -0,0 +1,106 @@ +# "SciPy-transpiled" version of LAPACK's GETRF family of subroutines! +import numpy as np +import scipy.linalg as la + +def getrf(A: np.ndarray) -> (np.ndarray, np.ndarray, np.ndarray): + """Returns the P, L, U + + * A is m by n + * P is m by m + * L is m by n if m >= n and m by m if m <= n + * U is n by n if m >= n and m by n if m <= n + """ + m, n = A.shape + + # A is a row + if m == 1: + return np.eye(1), np.eye(1), A + + # A is a column + elif n == 1: + i0 = 0 + + for i in range(m): + if abs(A[i, 0]) > abs(A[i0, 0]): i0 = i + + # P permutes the 0-th and i0-th basis vectors + P = np.eye(m) + P[0,0], P[i0,i0] = 0, 0 + P[i0,0], P[0,i0] = 1, 1 + + if A[i0, 0] != 0: + L = P@A / A[i0, 0] + U = A[i0, 0] * np.eye(1) + else: + L = A + U = np.zeros((1, 1)) + + return P, L, U + else: + n1 = min(m, n)//2 + n2 = n - n1 + + # Write + # + # A = [[A11, A12], + # [A21, A22]], + # + # A1 = [[A11, + # A21]], + # + # A2 = [[A12, + # A22]] + # + # where A11 is n1 by n1 and A22 is n2 by n2 + A11, A12 = A[:n1,:n1], A[:n1,n1:] + A21, A22 = A[n1:,:n1], A[n1:,n1:] + A1, A2 = A[:,:n1], A[:,n1:] + + # Solve the A1 block + P1, L1, U11 = getrf(A1) + + # Apply pivots + # A2 is m by n2 + A2 = la.inv(P1) @ A2 + A12, A22 = A2[:n1,:], A2[n1:,:] + + # Solve A12 + L11, L21 = L1[:n1,:], L1[n1:,:] + A12 = la.inv(L11) @ A12 + + # Update A22 + A22 = -L21@A12 + A22 + + # Solve the A22 block + P2, L22, U22 = getrf(A22) + + # Take P = P1 @ P2_ for + # + # P2_ = [[1, 0, + # 0, P2]] + P2_ = np.eye(m) + P2_[n1:,n1:] = P2 + P = P1 @ P2_ + + # Apply interchanges to L21 + L21 = la.inv(P2) @ L21 + + # Take + # + # L = [[L11, 0], + # [L21, L22]], + # + # U = [[U11, A12], + # [0, U22]] + if m >= n: + L, U = np.zeros((m, n)), np.zeros((n, n)) + else: + L, U = np.zeros((m, m)), np.zeros((m, n)) + L[:n1,:n1] = L11 + L[n1:,:n1] = L21 + L[n1:,n1:] = L22 + U[:n1,:n1] = U11 + U[:n1,n1:] = A12 + U[n1:,n1:] = U22 + + return P, L, U
diff --git a/linear-solvers.ipynb b/linear-solvers.ipynb @@ -0,0 +1,196 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "e1791697-e733-4721-9e10-fcdcfbda9064", + "metadata": {}, + "source": [ + "# Solving Linear systems in SciPy\n", + "\n", + "A linear system is a system of equations of the form\n", + "$$ \\left\\{ \\begin{aligned} a_{1 1} x_1 + a_{1 2} x_2 + \\cdots + a_{1 n} x_n &= b_1 \\\\ a_{2 1} x_1 + a_{2 2} x_2 + \\cdots + a_{2 n} x_n &= b_2 \\\\ & \\vdots \\\\ a_{n 1} x_1 + a_{n 2} x_2 + \\cdots + a_{n n} x_n &= b_n\\end{aligned} \\right.$$\n", + "on the variables $x_1, \\ldots, x_n$.\n", + "\n", + "Solving this system is equivalent to solving the equation $A x = b$ on $x$ where $A = (a_{ij})_{ij}$ is a $n\\times n$ matrix and $x = (x_1, \\ldots, x_n) \\; \\& \\; b = (b_1, \\ldots, b_n)$ are vectors, which can always be done provided $A$ is invertible." + ] + }, + { + "cell_type": "markdown", + "id": "26bc6a30-7853-4c78-adfb-320f0a65dd10", + "metadata": {}, + "source": [ + "In SciPy, we can solve linear systems using the `la.solve` function." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "b1ced47f-6783-48ed-a4e4-4f1f4e2b8835", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[-0.12672176],\n", + " [ 0.1046832 ],\n", + " [ 1.19008264]])" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "import numpy as np\n", + "import scipy.linalg as la\n", + "\n", + "A, b = np.array([[6,15,1],[8,7,12],[2,7,8]]), np.array([[2], [14], [10]])\n", + "la.solve(A, b)" + ] + }, + { + "cell_type": "markdown", + "id": "bfdacdbf-150f-4b4d-8072-ee18758d3b60", + "metadata": {}, + "source": [ + "## But how does `la.solve` work???\n", + "\n", + "Internally, the `la.solve` function uses the the [LAPACK library](https://netlib.org/lapack), a Fortran package for numerical linear algebra. The LAPACK generic linear solver algorithm goes something like the following:\n", + "\n", + "1. Decompose $A$ as $A = PL U$, where $P$ is permutation matrix, $L$ is a lower triangular matrix with $1$ in the diagonal and $U$ is an upper triangular matrix.\n", + "3. Solve $P b' = b$ for $b'$, i.e. compute $b' = P^{-1} b$. Since $P$ is a permutation matrix, this operation is $O(n)$.\n", + "4. Solve $L b'' = b'$ for $b''$, i.e. compute $b'' = L^{-1} P^{-1} b$. Since $L$ is known to be lower triangular, this operation is $O(n^2)$.\n", + "3. Solve $U x = b''$ for $x$, i.e. compute $x = U^{-1} L^{-1} P^{-1} b = A^{-1} b$. Since $U$ is known to be upper triangular, this operation is $O(n^2)$.\n", + "\n", + "This is implemented in the `GETRS` family of subroutines." + ] + }, + { + "cell_type": "markdown", + "id": "5b7ad45f-ea81-4d46-9097-735cc159cf1a", + "metadata": {}, + "source": [ + "As for the decomposition of $A$ in the first step, LAPACK uses a method called [_partial pivoting_](https://en.wikipedia.org/wiki/LU_decomposition#LU_factorization_with_partial_pivoting). A simple simple recurssive algorithm using such method might look something like the following:\n", + "\n", + "1. If $A = a_{11}$ is $1 \\times 1$ then take $P = L = 1$ and $U = a_{11}$.\n", + "2. If $A$ is $n \\times n$ for $n > 1$, choose $i_0$ that maximizes $|a_{i_0, 1}|$ and consider the $n \\times n$ permutation matrix $S_{i_0}$ that swaps the first and $i_0$-th basis vectors.\n", + "3. Write\n", + " $$S_{i_0} A = \\left( \\begin{array}{c|c} a_{i_0} & A_{12}' \\\\ \\hline A_{21}' & A_{22}' \\end{array} \\right), $$\n", + " where $A_{22}'$ is $(n - 1) \\times (n - 1)$. Since $A$ is invertible, $a_{i_0} \\ne 0$.\n", + "4. We want to solve the equation\n", + " $$S_{i_0} A = \\left( \\begin{array}{c|c} 1 & 0 \\\\ \\hline 0 & P_{22} \\end{array} \\right) \\left( \\begin{array}{c|c} 1 & 0 \\\\ \\hline L_{21} & L_{22} \\end{array} \\right) \\cdot \\left( \\begin{array}{c|c} u_{11} & U_{12} \\\\ \\hline 0 & U_{22} \\end{array} \\right),$$\n", + " where $P_{22}$ is a permutation matrix, $L_{22}$ is lower triangular with $1$ in the diagonal entries and $U_{22}$ is upper triangular. In other words, we want to solve the equations\n", + " $$\n", + " \\begin{aligned}\n", + " a_{i_0} &= u_{11} & A_{12}' &= U_{12} \\\\\n", + " A_{21}' &= u_{11} P_{22} L_{21} & A_{22}' &= P_{22} L_{21} U_{12} + P_{22} L_{22} U_{22}.\n", + " \\end{aligned}\n", + " $$\n", + " We must take $u_{11} = a_{i_0}$, $U_{12} = A_{12}'$ and $L_{21} = a_{i_0}^{-1} P_{22}^{-1} A_{12}$, so it remains to solve the bottom-right equation.\n", + "5. Write $(A_{22}' - a_{i_0}^{-1} A_{21}' A_{12}') = P_{22} L_{22} U_{22}$, where $P_{22}$ is a permutation matrix, $L_{22}$ is lower triangular with $1$ in the diagonals and $U_{22}$ is upper triangular.\n", + "6. Take\n", + " $$\n", + " \\begin{aligned}\n", + " L &= \\begin{pmatrix} 1 & 0 \\\\ L_{21} & L_{22} \\end{pmatrix} &\n", + " U &= \\begin{pmatrix} u_{11} & U_{12} \\\\ 0 & U_{22} \\end{pmatrix}\n", + " \\end{aligned}\n", + " $$\n", + " for $L_{21}, L_{22}, u_{11}, U_{12}, U_{22}$ as above, so that\n", + " $$\n", + " S_{i_0} A = \\begin{pmatrix} 1 & 0 \\\\ 0 & P_{22} \\end{pmatrix} L U.\n", + " $$\n", + "7. Hence by taking\n", + " $$\n", + " P = S_{i_0} \\begin{pmatrix} 1 & 0 \\\\ 0 & P_{22} \\end{pmatrix}\n", + " $$\n", + " we get $A = P L U$ as desired!" + ] + }, + { + "cell_type": "markdown", + "id": "0e1e9c6b-5e65-4cf1-aa53-a7202abff788", + "metadata": { + "jp-MarkdownHeadingCollapsed": true + }, + "source": [ + "## Scrap" + ] + }, + { + "cell_type": "markdown", + "id": "4dde0b0b-08cc-4bca-9dd5-c2bced871cef", + "metadata": {}, + "source": [ + "As for the decomposition of $A$ in the first step, given some $m \\times n$ matrix $A$, LAPACK uses a method called [_partial pivoting_](https://en.wikipedia.org/wiki/LU_decomposition#LU_factorization_with_partial_pivoting) to write $A = P L U$, where $P$ is a $m \\times m$ permutation matrix, $L$ is a $m \\times n$ lower trapezoidal matrix with $1$ in the diagona entries and $U$ is a $n \\times n$ upper triangular matrix. Their algorithm goes something like the following:\n", + "\n", + "1. If $A = 0$ then take $P = 1$, $L$ the $m \\times n$ matrix with $1$ in the diagonals and zero elsewhere and $U = 0$.\n", + "2. If $m = 1$, so that $A = \\begin{pmatrix} a_1 & \\cdots & a_n \\end{pmatrix}$, *DIE*\n", + "3. If $n = 1$, so that\n", + " $$ A = \\begin{pmatrix} a_1 \\\\ \\vdots \\\\ a_m \\end{pmatrix} $$\n", + " choose $i_0$ that maximizes $|a_{i_0}|$. Since $A \\ne 0$, $a_{i_0} \\ne 0$. Return $P = \\sigma_{1, i_0}$, $L = a_{i_0}^{-1} P^{-1} A$ and $U = 1$.\n", + "4. Otherwise do the following series of steps.\n", + " 1. Take $n_1 = \\left\\lfloor \\frac{\\min \\{n, m\\}}{2} \\right \\rfloor$ and $n_2 = n - n1$.\n", + " 2. Write\n", + " $$A = \\left( \\begin{array}{c|c} A_{11} & A_{12} \\\\ \\hline A_{21} & A_{22} \\end{array} \\right), $$\n", + " where $A_{11}$ is $n_1 \\times n_2$ and $A_{22}$ is $n_2 \\times n_2$.\n", + " 3. Write\n", + " $$\\begin{pmatrix} A_{11} \\\\ A_{21} \\end{pmatrix} = P_1 L_1 U_1 \\quad L_1 = \\begin{pmatrix} L_1' \\\\ L_1'' \\end{pmatrix} ,$$\n", + " where $P_1$ is a $m \\times m$ permutation matrix, $L_1$ is a $m \\times n_1$ lower trapezoidal matrix, $L_1'$ is $n_1 \\times n_1$ and $U_1$ is a $n_1 \\times n_1$ upper triangular matrix.\n", + " 4. Write\n", + " $$\\begin{pmatrix} A_{12}' \\\\ A_{22}' \\end{pmatrix} = P_1^{-1} \\begin{pmatrix} A_{12} \\\\ A_{22} \\end{pmatrix}$$\n", + " 5. Take\n", + " $$A_{12}'' = L_1'^{-1} A_{12}' \\quad A_{22}'' = -L_1'1A_{12}'' + A_{22}'$$\n", + " 6. Write $A_{12}'' = P_2 L_2 U_2$, where $P_2$ is a $n_1 \\times n_1$ permutation matrix, $L_2$ is a $n_1 \\times n_2$ lower trapezoidal matrix and $U_2$ is a $n_2 \\times n_2$ upper triangular matrix.\n", + " 7. Take\n", + " $$\n", + " \\begin{aligned}\n", + " P &= \\begin{pmatrix} P_1 & 0 \\\\ 0 & P_2 \\end{pmatrix} &\n", + " L &= \\begin{pmatrix} L_1 & 0 \\\\ A_{21}'' & L_2 \\end{pmatrix} &\n", + " U &= \\begin{pmatrix} U_1 & A_{12}'' \\\\ 0 & U_2 \\end{pmatrix},\n", + " \\end{aligned}\n", + " $$\n", + " where $A_{21}'' = P_2^{-1} L_1'1$." + ] + }, + { + "cell_type": "markdown", + "id": "93106465-de78-418d-8c7b-9774b97f6b07", + "metadata": {}, + "source": [ + "Althought similar to the naive method of computing $A^{-1}$ and then applying it to $b$, this algorithm has better performance characteristics since it only computes $A^{-1} b$ (without calculating the product $A^{-1} = U^{-1} L^{-1} P^{-1}$). The `la.solve` function also uses optimized linear solvers when $A$ is known to be symmetric or Hermitian.\n", + "\n", + "For more details please refer to the implementations of the `GETRS`, `SYSV` and `HESV` families of subroutines in LAPACK." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6d77288f-56a8-4037-965d-56f2c443a73c", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.6" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +}
diff --git a/test_getrf.py b/test_getrf.py @@ -0,0 +1,78 @@ +import numpy as np +import scipy.linalg as la +from getrf import getrf +import json + +N_TESTS = 1000 +EPS = 1e-12 + +def matrix_eq(A: np.array, B: np.array) -> bool: + return A.shape == B.shape and np.all(np.abs(A - B) < EPS) + +def test(M: int, N: int, n_tests = N_TESTS): + print(f"Comparing getrf and la.lu (running {n_tests} tests)") + print("=" * 20) + + fails = [] + + for i in range(n_tests): + m = np.random.randint(1, M+1) + n = np.random.randint(1, N+1) + + A = np.random.rand(m, n) + P, L, U = la.lu(A) + + print(f"{i:05d}: Testing a {m:03d}x{n:03d} matrix: ", end="") + + try: + my_P, my_L, my_U = getrf(A) + except Exception as e: + d = {"A": A.tolist(), "runtime error": str(e)} + fails.append(d) + + print(f"\033[91mfailed!\033[0m") + print(" > getrf raised an error!") + continue + + P_test = matrix_eq(P, my_P) + L_test = matrix_eq(L, my_L) + U_test = matrix_eq(U, my_U) + + if (not P_test) or (not L_test) or (not U_test): + d = {"A": A.tolist(),} + + if not P_test: + d["expected P"] = P.tolist() + d["actual P"] = my_P.tolist() + + if not L_test: + d["expected L"] = L.tolist() + d["actual L"] = my_L.tolist() + + if not U_test: + d["expected U"] = U.tolist() + d["actual U"] = my_U.tolist() + + fails.append(d) + + print(f"\033[91mfailed!\033[0m") + print(" > getrf returned wrong values") + continue + + print(f"\033[32mpassed!\033[0m") + + print("\n" + "=" * 20) + if len(fails) == 0: + print(f"All {n_tests} tests passed!") + else: + testlogs = "./testlogs.json" + with open(testlogs, "w") as f: + json.dump(fails, f) + + print(f"{n_tests-len(fails)} tests passed ({len(fails)} fails)") + print(f"Check {testlogs} for the failing inputs") + +m, n = 200, 200 + +test(m, n, 10000) +