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Wikiversity:Community Review/CR process discussion/Resolutions

=Community Review process discussion= Decisions made using Community Review have the weight of policy, and thus Community Review is an important and useful tool for improving Wikiversity overall. The resolutions listed below are aimed at reforming Community Review so that it becomes more focused, and perhaps more welcoming to contributors who have stayed away from past Reviews. After discussion at the main page, the resolutions below are proposed for adoption. Please restrict your comments on each resolution to one or two lines, making sure to include language that makes your preference clear.

Resolution 2011-1: Amendment to Community Review Policy
Community Reviews should be started in the userspace, then moved to the Wikiversity namespace once it becomes clear that the review is a collaborative effort involving two or more established contributors.
 * A proposed review should demonstrate some support before becoming a full community discussion. --mikeu talk 21:05, 11 February 2011 (UTC)
 * I would also support the inclusion of ideas suggested by Moulton and Jtneill --mikeu talk 16:27, 12 February 2011 (UTC)

 * I would also suggest that the parties must have made a good faith effort to directly resolve their concerns before escalating to a community-wide review. —Moulton 21:12, 11 February 2011 (UTC)
 * I support Moulton's position. Some of the reviews I've seen appear to be grudge related, where there is no demonstration that reviews are good faith efforts to improve content and community for the whole project. --LauraHale 21:23, 11 February 2011 (UTC)
 * Could support, if the parties must have made a good faith effort to directly address issues through compromise, deescalation, and disengagement first before escalating to a community-wise review, if reviews must remain respectful, and if reviews in user space must allow people to respond freely. -- dark lama  21:44, 11 February 2011 (UTC)
 * I understand and agree with the concerns of those "commenting", but this is much better than the status quo (and see #6!). --SB_Johnny talk 23:37, 11 February 2011 (UTC)

 * - all CRs, no matter how unfounded, should be up for all in a formal manner. Otherwise, there will be wars over if something is proper or not. We don't need to waste our time with that. Ottava Rima (talk) 23:50, 11 February 2011 (UTC)
 * Pre-reviews sitting in user space can do a lot of damage, I have seen this frequently on en:Wikipedia. It gives a user a free pass to say bad things about another user while denying the other user a reply. Guido den Broeder 00:50, 12 February 2011 (UTC)
 * s In cases where a user is under review, there are always two established users: the initiator and the reviewed. Demanding a second initiator, if that is what is meant here, is something that I oppose. Often, seconds are easy to find but may have their own motives which can be a hindrance in conducting a proper review. If someone cannot find a second, that by itself would be a reason to investigate. Per Moulton, reviews should not be the first resort in case of a conflict. Users should try and work out their differences first, and a mediation attempt can also happen before the community gets drawn in. Guido den Broeder 00:06, 12 February 2011 (UTC)
 * . "Community Review" is a misnomer and a failed system that was imported from another wiki website. After the Hostile Takeover of Wikiversity Community Review was forced upon the Wikiversity community. "Community Review" and all the other disruptive influences of SB Johnny and his buddies should be removed from Wikiversity and nullified. The currently active bureaucrats should resign, along with all policy-violating sysops. The honest Wikiversity community members who were driven away by the Hostile Takeover should be brought back and those who had their tools removed by bogus emergency desysop procedures when there was no emergency or who resigned their tools in protest/disgust after the Hostile Takeover should be given back their tools. Wikiversity should return to holding important discussions at the Colloquium. If someone has something on their mind they should be free to take it to the relevant Wikiversity page or the Colloquium, not have to first start a page in their user space. --JWSchmidt 00:59, 12 February 2011 (UTC)
 * I support the intent of the proposed resolution but suggest redrafting as follows (soften to recommendation; encourage and allow development of policy review in userspace or policy talk pages and ensure policy reviews are switched to policy review sub-pages prior to the resolution stage): "It is recommended that reviews be initially developed in the user namespace or policy talk pages. Good faith, collaborative reviews involving two or more established contributors should be moved to a Policy Review sub-page before progressing to the Resolution stage." -- Jtneill - Talk - c 02:49, 12 February 2011 (UTC)
 * I am a user who tries to stay clear of the politics in Wikiversity, and regret the major events that have caused such disruption in our community. I appreciate other's efforts to make it work however. I'm neither here nor there on this as it seems to be a bandaid on a major hemorrhage. Leighblackall 04:01, 12 February 2011 (UTC)
 * , strongly. Improving CR is only a small part of the solution to some obvious problems. CR is necessarily a relatively slow process, if it's done carefully. And if it's not done carefully, it can wreck the place. The proposals here were designed to improve the process, but we will need better Recusal policy that allows custodians to handle perceived emergencies pending CR. CR, then, can cover the formation of long term consensus, cleaning up afterwards. The suggestion re prior effort to resolve disputes, is already policy with respect to desysopping (which was ignored in some CRs recently), and what do we do when policy is ignored? Policy without clear enforcement mechanism is useless for dealing with serious disruption. The requirement of two users to certify a Request for comment is from Wikipedia, and it avoids a fair number of useless RfCs there. This is really the old process principle of not debating a motion unless it is seconded. Obviously, there remain details to be resolved, but it was also a strong consensus in the discussion on the main CR page (except for Guido).
 * Comment. As to starting in user space, this allows a user to work to develop a cogent presentation, and attract a second. Sensible handling of conflict here can allow users to respond even if the user hosting the CR finds participation disruptive. A user may, for example, create a counter-RfC in their own user space. May the best evidence and arguments win! I would find it offensive if a user was prevented from placing a link on the Talk page attached to the draft CR, that they have responded elsewhere. Common sense, folks. It doesn't need to be complicated. And the right to develop a CR in user space isn't a carte blanche for personal attacks. If anything, personal attacks in a draft CR should be strictly covered by civility policy, but I fail to see how a personal attack there is worse than, say, on WV:RCA, soliciting immediate custodian action! Or, to cite a recent example, than a topic ban proposal on the Colloquium that was then selectively copied to a new CR on desysopping, leaving out 3/4 of the discussion! --Abd 22:53, 12 February 2011 (UTC)
 * - I tend to overlook Wikiversity infighting. I simply don't have time to follow the drama. As Leigh stated, I appreciate other's efforts to make it work. These endless debates are burning up precious resources, people's time, and sullying Wikiversity's reputation. Dramaversity, ugh. --CQ 06:40, 14 February 2011 (UTC)
 * I agree with Leighblackall and CQ just above. This will do nothing to solve the current problems. --Bduke 21:29, 20 February 2011 (UTC)
 * It seems that the most expeditious means of developing meaningful collaboration is to simply put the matter out there in "Wikiversity namespace".Geofferybard 04:18, 22 February 2011 (UTC)
 * I like the idea of a CR needing a little support, but I do not like the idea of beginning them in the user space. I feel this would lead to people "getting their cannons in line."  In short I feel that by some method, all sides of an issue should have an equal chance of shaping a community review.  For better or worse a persons user space is treated like their domain, and people some people will be discourage in editing in the initial phases of the conversation. Thenub314 19:42, 20 April 2011 (UTC)
 * Although this would serve to keep the community review pages "clean" and ensure the quality of community reviews, a few problems and questions appear:
 * 1. Who decides which reviews are "proper"? Popular vote? Consensus?
 * 2. What will the criteria for "proper" reviews be? (My suggestion would be a small number of votes required before moving to community review page. However, this really is no different from voting on the community review site and only restricts less well-versed users of wikiversity from having their say. )
 * 3. How will newly hatched review proposals attract enough attention to make it here?
 * 4. Like previously said, this gives the creator too much power over the review because it is in his/her Userspace.
 * 5. Reviews started by more respected users or by users with more influence would get preference over others.
 * 6. And finally, are we really having any problems with the current policy on these matters?
 * Richardofoakshire 13:38, 27 April 2011 (UTC)
 * 1. Consensus, in the end. Custodians, ad hoc, ad-interim. It takes a custodian to put a CR in the site notice. That should not be automatic, a custodian should agree that either discussion is worthwhile, or that the community wants the discussion. The listing custodian should not be a filer of the CR, unless a request to list has consensus and no other custodian shows up to list it. Not an emergency.
 * 2. In theory, anyone can start a CR. The suggestion of a second is standard deliberative process for avoiding useless debates. If a user can't get a second, the debate is likely to be a waste of time. The user can discuss a CR privately, and this is part of why developing a CR in user space is suggested, not as a requirement, but simply as a possibility. If a CR is "opened" without a second, what happens is that people start to argue against it, and then others see those arguments and disagree with them, perhaps they go too far, etc. It can go on a long time before anyone notices that there is no support for the thesis of the CR. This is a rat-hole that wikis often fall in.
 * 3. As I've suggested, it only takes one additional user. (Jtneill says "two established users," but we might set up several reasonable alternatives.) A "private CR," i.e., in user space, is not an official process and canvassing rules should not apply. (We don't have a policy, but canvassing is still disliked, for good reasons.) So a user, concerned about some situation, could write up a page about it, not having to satisfy formal rules, and then ask users, personally and individually, to look at it and comment or -- I don't suggest this as a first step -- on community pages like the Colloqium.
 * 4. It isn't a Community Review yet, in user space. The fact is that, already, someone who drafts a community review has complete control over the draft, if it's done off-wiki. People can also collaborate and communicate off-wiki, and they do. Doing it in user space is simply more transparent.
 * 5. TANSTAAFL. Usually there is a reason why some users have more influence than others. All wiki process favors, in fact, "respected users." It's not avoidable. However, this process doesn't give special privilege to respected users, beyond "established" -- and "established" doesn't mean "respected." As long as the claim of "personal attack" is not successfully abused, any user can develop a CR draft, under the same civility restraints as apply anywhere. As it's not a formal CR, not certified by two established users, others don't need to respond to it, even they are even aware of it. It's moot, until and unless it gets that sign-off and is moved to WV space as a formal CR. Then if a custodian considers it worthy of whole-wiki attention, it goes in the site notice. Do we imagine that something is going to obtain consensus if it can't, less formally, obtain the support of two established users and at least one custodian who agrees on the site notice?
 * 6. Yes. Notice that this CR resolution was started by a 'crat, and enjoyed the support (if conditional) of all active 'crats. The apparent occasion was a series of highly disruptive CRs filed by Ottava Rima, who retired under a cloud, plus there were older CRs filed by JWSchmidt (currently blocked) that likewise generated far more heat than light. The comment about user space pre-reviews on Wikipedia is misleading. Draft user RfCs and even evidence pages that were submitted to ArbComm are often MfD'd on Wikipedia, sometimes successfully, whereas similar or worse evidence pages created by highly-placed administrators survive. Wikiversity has high preference for keep of pages in user space, and the reverse result (deletion of pages) has only taken place where serious cross-wiki issues arise, and massive disruption is resulting, or "outing" is involved. If a CR is developed as a process in user space, it can be developed over time, there is no rush, necessarily. Yes, it can also serve as a kind of warning, though not one that a user can be held responsible for ignoring.
 * The point of developing a CR in user space is precisely that the user has control over it. The user is not necessarily the user who originally writes a review, rather, it can be a user who facilitates the process and uses it to try to negotiate early consensus or compromise, possibly resolving a dispute before it even comes to Wikiversity space. (This will really work if two users in a conflict both accept the hosting user as a mediator.) If a user abuses their right to control the process, to exclude relevant contributions from other users, they are shooting themselves in the foot. They should have the right to do that. Anyone else could host an independent draft or mediation process!
 * Attack pages with uncivil comment should not be allowed anywhere. "Draft CR" is not a protection against deletion or other custodial supervision. I suggest reading Jtneill's comment. It's quite sane. This isn't about "current problems," really, it's about process that is scalable and more practical, for the future. Had this been in place as a policy, some disruption might have been avoided; possibly, though, all that would have changed is that what took longer might have taken less time. That's a gain, though. --Abd 16:35, 27 April 2011 (UTC)
 * Alright, thanks for the explanation. Richardofoakshire 19:06, 27 April 2011 (UTC)

Resolution 2011-2: Amendment to Community Review Policy
Community Reviews should be narrow in focus, and aimed at clarifying, refining, or changing a Wikiversity policy or guideline.

 * in current form. I think participants and rules should be left out of it, and must not be used to define CR's purpose. CRs should be narrow in focus, be aimed at addressing specific concerns or practices, be based on realistic expectations, and must be about finding common ground. -- dark lama  21:32, 11 February 2011 (UTC)
 * I would like to see Community Reviews that aim to correct, revise, and improve problematic practices with better practices, enroute to ethical best practices as the long-term goal. —Moulton 21:37, 11 February 2011 (UTC)
 * small, narrowly focused changes are easy to agree upon, and are in the best spirit of the "wiki way". Broad changes are disruptive and generally never get consensus. --SB_Johnny talk 23:42, 11 February 2011 (UTC)

 * - they are supposed to be discussions, and discussions that are forced to be focused aren't discussions at all. Ottava Rima (talk) 23:52, 11 February 2011 (UTC)
 * Agree with narrow focus, but this should be open for consideration on a case by case basis. There will be exceptions where we want a broad focus. Guido den Broeder 00:16, 12 February 2011 (UTC)
 * Disagree with the proposed aim. Reviews are held to find a common ground to solve an apparent problem. While this may often come down to a discussion of policy, that is not by definition their purpose. Guido den Broeder 00:16, 12 February 2011 (UTC)
 * . "Community Review" is a misnomer and a failed system that was imported from another wiki website. After the Hostile Takeover of Wikiversity Community Review was forced upon the Wikiversity community. "Community Review" and all the other disruptive influences of SB Johnny and his buddies should be removed from Wikiversity and nullified. The currently active bureaucrats should resign, along with all policy-violating sysops. The honest Wikiversity community members who were driven away by the Hostile Takeover should be brought back and those who had their tools removed by bogus emergency desysop procedures when there was no emergency or who resigned their tools in protest/disgust after the Hostile Takeover should be given back their tools. Wikiversity should return to holding important discussions at the Colloquium. If "Community Review" needs to be artificially limited to narrow topics then that is just one more example of its failings. "Community Review" should not be used as a substitute for the normal process by which the community develops needed policies. --JWSchmidt 01:06, 12 February 2011 (UTC)
 * I support the intent of the resolution and suggest:
 * Redrafting to: "Community Reviews should aim to clarify, refine, or change a Wikiversity policy, proposed policy or guideline." (i.e., simplify intent - no need for narrowness, just productive, collaborative intent)
 * WV:Community Review should renamed to WV:Policy Review (or WV:Policies/Review. (to help reinforce the purpose and make a clear distinction that this is a different approach to WV:CR.)
 * I am mainly supporting this because CR has more or less become WV:User Review. I think its time to build more focus on playing the ball rather than the man. (I would have said (wo)man, but WV:User Review seems to be most popular with blokes. -- Jtneill - Talk - c 01:47, 12 February 2011 (UTC)
 * - noting support for JWSchmidt and Jtneill's remarks on the largely inappropriate use of CV to date. Just not sure if CV is a useful process given its tendency to be abused Leighblackall 04:07, 12 February 2011 (UTC)
 * While there have been a number of interesting suggestions for amending the wording, I feel that the current workding covers a number of the concerns mentioned above (and shared by me) quite well. --mikeu talk 16:34, 12 February 2011 (UTC)
 * This presentation of resolutions was a poor example of process. The wording of matters to be presented for vote is crucial, and the resolution, as stated, is defective. It's correct, but it implies CRs only about policy, neglecting "User CRs". It could be easy to fix. The resolutions should have been negotiated before being presented for community vote in the site notice! --Abd 22:56, 12 February 2011 (UTC) Changed !vote per Jtneill argument. We would simply create a User Review process, similar. Maybe even the same. --Abd 22:58, 12 February 2011 (UTC)
 * A narrower focus may assist in cutting down the drama, although the precise wording may need some fine tuning in future. The present wording is worth trying now. --Bduke 21:32, 20 February 2011 (UTC)
 * The last sentence should end "or addressing a community wide problem." Not all problems will be covered by policies/guidelines nor should they be, but some problems are serious enough to warrant everyones attention.  It might be nice if we provided enough flexibility to include such issues.  To give an example of what I mean by a discussion which might warrant a community review but not be specifically about a guideline/policy, consider a suggestion to seriously reconfigure the mediawiki software running WV, in some way that effects all learning resources. Probably not the only possibility, but if we are going to be formal about what a community review can be, we need to leave a little elbow room.  Thenub314 19:58, 20 April 2011 (UTC)
 * -- I agree with Ottava Rima's sentiment.

Resolution 2011-3: Amendment to Community Review Policy
The main page of a community review will:
 * 1) Begin with a brief statement which explains the purpose of the review and the policies or guidelines to be modified.
 * 2) Have a list of questions with yes/no answers to be addressed, with questions added or removed as indicated by the statements of the participants.
 * 3) Have statements by individual contributors about the topic, under their own headings, not to exceed 700 words. Threaded and long-form discussions will take place on the talk page. Statements may of course contain links to other discussions or longer statements on another page.

Note: An amended version of this has been proposed as a result of this discussion.

 * This will go a long way to preventing the confusion and disorganization that we have seen in some reviews. --mikeu talk 21:12, 11 February 2011 (UTC)
 * If "and the policies or guidelines to be modified" part was removed, I could fully support this. I support 2, and 3. -- dark lama  21:35, 11 February 2011 (UTC)
 * Keep it readable. Not averse to Darklama's proposed subtraction. --SB_Johnny talk 23:43, 11 February 2011 (UTC)
 * amended version suggested by Jtniell below. --SB_Johnny talk 14:47, 12 February 2011 (UTC)

 * sounds more like an ArbCom request than a discussion. Ottava Rima (talk) 23:52, 11 February 2011 (UTC)
 * Per Mikeu. Guido den Broeder 00:25, 12 February 2011 (UTC)
 * . "Community Review" is a misnomer and a failed system that was imported from another wiki website. After the Hostile Takeover of Wikiversity Community Review was forced upon the Wikiversity community. "Community Review" and all the other disruptive influences of SB Johnny and his buddies should be removed from Wikiversity and nullified. The currently active bureaucrats should resign, along with all policy-violating sysops. The honest Wikiversity community members who were driven away by the Hostile Takeover should be brought back and those who had their tools removed by bogus emergency desysop procedures when there was no emergency or who resigned their tools in protest/disgust after the Hostile Takeover should be given back their tools. Wikiversity should return to holding important discussions at the Colloquium. If "Community Review" needs to be artificially structured as proposed in this "resolution" then that is just one more example of its failings. "Community Review" should not be used as a substitute for the normal process by which the Wikiversity community develops needed policies. --JWSchmidt 01:12, 12 February 2011 (UTC)
 * . I support the intent of the resolution and suggest:
 * . Remove "and the policies or guidelines to be modified" (per Darklama)
 * . Remove "to be addressed". Break into two sentences. Do not allow questions to be removed: "Have a list of questions with yes/no answers. Questions can be added as indicated by the statements of the participants."
 * Remove "of course" and change "will" to "should": "Have statements by individual contributors about the topic, under their own headings, not to exceed 700 words. Threaded and long-form discussions should take place on the talk page. Statements may contain links to other discussions or longer statements on another page." -- Jtneill - Talk - c 01:59, 12 February 2011 (UTC)
 * I think Jtneill's amendment is good. Leighblackall 04:10, 12 February 2011 (UTC)
 * Agree that questions should not be removed. Questions should be numbered, with amended questions sub-numbered. Thus an amended version of this question, intended to replace it, would be "3.1", etc. If a question is amended, all those who have !voted on the question should be notified. This could affect closing process, that should be examined separately. Amended questions should have a new subsection underneath the original. --Abd 16:12, 13 February 2011 (UTC)
 * but with Jtneill's amendments to (2) and (3), but not (1) which differs from my support of resolution 2. --Bduke 21:39, 20 February 2011 (UTC)

Resolution 2011-3(a) Amendment to Community Review Policy
The main page of a community review will: ✅ --SB_Johnny  talk 14:05, 26 February 2011 (UTC)
 * 1) Begin with a brief statement which explains the purpose of the review.
 * 2) Have a list of questions with yes/no answers. Questions can be added as indicated by the statements of the participants.
 * 3) Have statements by individual contributors about the topic, under their own headings, not to exceed 700 words. Threaded and long-form discussions should take place on the talk page. Statements may contain links to other discussions or longer statements on another page.
 * amended version. --SB_Johnny talk 15:43, 14 February 2011 (UTC)
 * Guido den Broeder 00:40, 15 February 2011 (UTC)
 * -- Jtneill - Talk - c 04:14, 16 February 2011 (UTC)
 * --mikeu talk 15:03, 16 February 2011 (UTC)
 * only (2) and (3), but not (1) as above. --Bduke 21:39, 20 February 2011 (UTC)

Resolution 2011-4: Amendment to Community Review Policy, Sitenotice Policy
The goal of Community Review is to improve Wikiversity by clarifying or amending policies when they are unclear or are in some way interfering with the Wikiversity mission. In cases where it seems likely that there will be a consensus for passing one or more resolutions, a "Resolutions" subpage should be created to allow for voting to take place. Each resolution should be listed under its own heading, link to the policy it modifies, and be open only for non-threaded comments indicating support or opposition. A resolution will be considered adopted 1 week after its proposal if it achieves at least 70% support. Since resolutions can have a strong effect on policy, they will be advertised on the sitenotice. In keeping with the spirit of narrow focus, a maximum of 7 resolutions will be permitted for any particular review. -- Jtneill - Talk - c 02:07, 12 February 2011 (UTC)
 * This particular resolution should probably be broken up and made the subject of another review, but for now it provides some (badly needed) structure, and maybe will allow us to solve some issues in the meantime. We need a "support provisionally" template. --SB_Johnny talk 23:46, 11 February 2011 (UTC)
 * Ask and you shall receive. maybe -> Maybe -- dark lama  23:55, 11 February 2011 (UTC)
 * I have no idea what this is trying to fix/accomplish. It seems unnecessarily complicated. Ottava Rima (talk) 23:54, 11 February 2011 (UTC)
 * Maybe If the goal of Community Review was to find ways to help improve the Wikiversity mission, the aim of Resolutions were not limited to modifying policies, and consensus rather than voting was encouraged. I am not sure I see the need for this amendment though. -- dark lama  00:05, 12 February 2011 (UTC)
 * Disagree with aim, see 2, and with timeframe. One week is too short, I propose 2 weeks. Guido den Broeder 00:30, 12 February 2011 (UTC)
 * . "Community Review" is a misnomer and a failed system that was imported from another wiki website. After the Hostile Takeover of Wikiversity Community Review was forced upon the Wikiversity community. "Community Review" and all the other disruptive influences of SB Johnny and his buddies should be removed from Wikiversity and nullified. The currently active bureaucrats should resign, along with all policy-violating sysops. The honest Wikiversity community members who were driven away by the Hostile Takeover should be brought back and those who had their tools removed by bogus emergency desysop procedures when there was no emergency or who resigned their tools in protest/disgust after the Hostile Takeover should be given back their tools. Wikiversity should return to holding important discussions at the Colloquium. If "Community Review" needs to be artificially structured as proposed in this "resolution" then that is just one more example of its failings. "Community Review" should not be used as a substitute for the normal process by which the Wikiversity community develops needed policies. Simpler alternative proposal: Anyone like SB Johnny who has vastly disrupted the Wikiversity community should stop trying to force on Wikiversity failed ideas like "Community Review". --JWSchmidt 01:19, 12 February 2011 (UTC)
 * . I think this is too complex and should be split into possibly three resolutions:
 * Purpose
 * Resolutions
 * Community notification
 * yeah, pretty confused by it too Leighblackall 04:12, 12 February 2011 (UTC)
 * I think this will help improve the review process greatly. I'm not set on any one particular scheme or wording for this (see proposal #6 below, ie. it is a wiki and we can modify later if we see ways to improve things) But, in general I feel that we urgently need some modification to the process that we have seen in recent reviews. This is about as good as any other method that I can think of. --mikeu talk 16:41, 12 February 2011 (UTC)
 * as drafted. Setting site notice policy, good thing. Automatic anything is impractical on a wiki, and the attention of the community is a scarce resource, careful process should ensure that only resolutions reasonably likely to obtain consensus go to the site message, or, alternately, that there is consensus that wider attention and comment is required (in which case it is the CR itself which should be announced, not a Resolution). Lower levels of attention already exist in that anyone may point to a discussion on the Colloquium. "1 week after proposal" is not right, it should be one week -- or maybe two! -- after a resolution goes into the site message. Proposal might be a long time earlier. --Abd 22:35, 12 February 2011 (UTC)

Resolution 2011-4(a): Amendment to Community Review Policy
Resolutions to address any actions or changes that seem appropriate in light of the discussions on the main page and talk page should be listed on a subpage titled "Resolutions". Comments on the resolutions page should be kept brief (at most three lines), be on-topic, and clearly indicate the contributor's desire to, , , , or.
 * - this pretty much describes what we're doing on this page already, and it seems to be working reasonably well. --SB_Johnny talk 10:42, 10 March 2011 (UTC)
 * - --mikeu talk 12:01, 21 March 2011 (UTC)
 * - Per SBJ. Thenub314 19:30, 20 April 2011 (UTC)
 * - TeleComNasSprVen 17:17, 21 April 2011 (UTC)
 * - Richardofoakshire 13:54, 27 April 2011 (UTC)
 * violated on this page, but, hey, you gotta start somewhere. Extended comment can be on the attached Talk page and linked, if the user wants to do that. Resolutions from long-term community reviews that are in the site message are where we do something more like voting. Still not exactly, but ... --Abd 16:49, 27 April 2011 (UTC)

Resolution 2011-5: Amendment to Community Review Policy, Policies
Resolutions approved using community review will be noted on the policy pages affected by the resolution under a new heading, to be later incorporated into the policy's main text through the normal editing process. In cases where a resolution modifies a proposed policy, the resolution heading should be left in place until the policy is approved.
 * this will make it easier for newcomers to understand our policies, and also make the "popular" bits of "proposed" policies official (we have a bad track record of passing policies, after all). --SB_Johnny talk 23:47, 11 February 2011 (UTC)

 * Sure, why not. It would preserve original policy and also show related community decisions. Ottava Rima (talk) 23:55, 11 February 2011 (UTC)

✅ --SB_Johnny  talk 14:05, 26 February 2011 (UTC)
 * Maybe I think a link to a resolution ought to be enough, and incorporation into a proposal shouldn't be restricted. In fact, I think holding off until after a proposal has consensus, can have the negative affect of giving the appearance of ignoring the consensus that was just established. People usually value the most current consensus over any consensus that was established by an older discussion. -- dark lama  00:21, 12 February 2011 (UTC)
 * Amend Missing is a timeframe. I suggest that resolutions will be incorporated one week after approval. That allows a short time to discuss the manner of the incorporation, and ensures that incorporation will in fact take place. Guido den Broeder 00:35, 12 February 2011 (UTC)
 * . "Community Review" is a misnomer and a failed system that was imported from another wiki website. After the Hostile Takeover of Wikiversity Community Review was forced upon the Wikiversity community. "Community Review" and all the other disruptive influences of SB Johnny and his buddies should be removed from Wikiversity and nullified. The currently active bureaucrats should resign, along with all policy-violating sysops. The honest Wikiversity community members who were driven away by the Hostile Takeover should be brought back and those who had their tools removed by bogus emergency desysop procedures when there was no emergency or who resigned their tools in protest/disgust after the Hostile Takeover should be given back their tools. Wikiversity should return to holding important discussions at the Colloquium. "Community Review" should not be used as a substitute for the normal process by which the Wikiversity community develops needed policies. "we have a bad track record of passing policies" <-- The problems with Wikiversity policy started when SB Johnny decided that he did not have to follow Wikiversity policy and he started forcing upon Wikiversity policies from other websites. SB Johnny's schemes for ever more absurd rules for "Community Review" are not a solution, they will just continue the problem that he created. --JWSchmidt 01:26, 12 February 2011 (UTC)
 * . I support the intent of the resolution but suggest simplifying to: "If a resolution to change policy receives sufficient support, then the proposed changes should be incorporated into policy as soon as possible". -- Jtneill - Talk - c 02:19, 12 February 2011 (UTC)
 * Leighblackall 04:14, 12 February 2011 (UTC)
 * Very straightforward. --mikeu talk 16:43, 12 February 2011 (UTC)
 * . Jtneill's proposal could be added. This resolution merely gives initial process. --Abd 16:06, 13 February 2011 (UTC)
 * --Bduke 21:42, 20 February 2011 (UTC)

Resolution 2011-6: Amendment to Community Review Policy
Individual resolutions should not be seen as "written in stone", and should be revisited if they are thought to have unintended negative effects that harm Wikiversity's mission.

 * . -- dark lama  21:49, 11 February 2011 (UTC)
 * . --LauraHale 23:37, 11 February 2011 (UTC)
 * completely. --SB_Johnny talk 23:38, 11 February 2011 (UTC)
 * Policies should be ethical guidelines, tempered with common sense, not hard and fast rules to be wielded as weapons in some kind of bare-knuckled political game. —Moulton 23:49, 11 February 2011 (UTC)

✅ --SB_Johnny  talk 14:05, 26 February 2011 (UTC)
 * As long as people feel comfortable enough and safe enough to pursue changes. Ottava Rima (talk) 23:55, 11 February 2011 (UTC)
 * Amend The part after the comma should be dropped. There can be many reasons to revisit a resolution. Guido den Broeder 00:40, 12 February 2011 (UTC)
 * . "Community Review" is a misnomer and a failed system that was imported from another wiki website. After the Hostile Takeover of Wikiversity Community Review was forced upon the Wikiversity community. "Community Review" and all the other disruptive influences of SB Johnny and his buddies should be removed from Wikiversity and nullified. The currently active bureaucrats should resign, along with all policy-violating sysops. The honest Wikiversity community members who were driven away by the Hostile Takeover should be brought back and those who had their tools removed by bogus emergency desysop procedures when there was no emergency or who resigned their tools in protest/disgust after the Hostile Takeover should be given back their tools. Wikiversity should return to holding important discussions at the Colloquium. "Community Review" should not be used as a substitute for the normal process by which the Wikiversity community develops needed policies. I'm tempted to ask for a definition of what a "resolution" is and what resolutions have to do with the mission of Wikiversity, but it would be far easier for everyone to simply return to having normal wiki community discussions and stop trying to salvage the failed "Community Review" idea. --JWSchmidt 01:32, 12 February 2011 (UTC)
 * . This resolution seems to be moot. Nothing on wiki is written in stone. -- Jtneill - Talk - c 02:23, 12 February 2011 (UTC)
 * For same reason Jtneill says - but I recognise the value of advising or reminding people not to turn resolutions into big sticks Leighblackall 04:16, 12 February 2011 (UTC)
 * I disagree with Jtneill that this is moot. I think we need to make an explicit statement about this. --mikeu talk 16:46, 12 February 2011 (UTC)
 * While what Jtneill and Leighblackall say is true, some people have the idea that written policy is strict law, to be strictly enforced. We don't have structure for strict enforcement, we have a structure which depends on voluntary compliance with community consensus, and resolutions and written policies are simply expressions of that, and cannot cover every contingency. In particular, these resolutions will form the basis for changes to policy pages, and those are ordinary editorial changes, with controversies resolved in the ordinary manner. Serious deviation from these resolutions (those with substantial consensus) without working consensus would be a problem, but if there is substantial consensus among those attending to rewriting the policy, that can prevail, and conflicts that remain can be the topic of a new, more specific CR. --Abd 16:03, 13 February 2011 (UTC)
 * I agree with mikeu. --Bduke 21:44, 20 February 2011 (UTC)

Resolution 2011-7: Amendment to Community Review Policy
Community reviews will be closed if they have had no changes for a month, or that have already produced 7 proposed resolutions. Unresolved issues should be addressed in a new Community Review.

 * If we applied this resolution as stated, this review could be closed right now. -- dark lama  21:52, 11 February 2011 (UTC)
 * Not everyone can get things done in a month and things can change during that period. I don't think things should be open indefinitely but on the other hand, closing them at set dates can just encourage grudge wankers to refile every month until the community gets tired and just gives in to shut the person up. Some better middle ground is needed. --LauraHale 23:40, 11 February 2011 (UTC)
 * There should definitely be a minimum time to allow people a reasonable chance to find time in their busy schedules to attend to issues here. I am not opposed to a sunset, but it should probably be more like three months, especially if we are talking about spanning summer breaks or holiday seasons.  —Moulton 23:45, 11 February 2011 (UTC)
 * Closure of CRs (and RfDs, and whatev) are probably worth a separate review. --SB_Johnny talk 23:49, 11 February 2011 (UTC)
 * Make that Amend, per darklama and Guido. --SB_Johnny talk 12:46, 12 February 2011 (UTC)
 * I'd support closed after no discussion in a week and I don't know about the second half. By the way, the "closed" would not mean anything is passed, and closing shouldn't be seen as determining an outcome. Individual outcomes on proposals within a CR and the CR itself should be seen as two different things. Ottava Rima (talk) 23:56, 11 February 2011 (UTC)
 * Amend This can too easily be circumvented by adding a single line once a month. Instead, I suggest closing if it has not produced a new resolution or other kind of proposal for 1 month. Guido den Broeder 00:45, 12 February 2011 (UTC)
 * Amend' I suggest closing if no new ground is covered for 1 month. -- dark lama  00:55, 12 February 2011 (UTC)
 * . "Community Review" is a misnomer and a failed system that was imported from another wiki website. After the Hostile Takeover of Wikiversity Community Review was forced upon the Wikiversity community. "Community Review" and all the other disruptive influences of SB Johnny and his buddies should be removed from Wikiversity and nullified. The currently active bureaucrats should resign, along with all policy-violating sysops. The honest Wikiversity community members who were driven away by the Hostile Takeover should be brought back and those who had their tools removed by bogus emergency desysop procedures when there was no emergency or who resigned their tools in protest/disgust after the Hostile Takeover should be given back their tools. Wikiversity should return to holding important discussions at the Colloquium. Alternative proposal: stop wasting the time of Wikiversity community members by forcing upon them failed ideas like "Community Review" and systems of rules for "resolutions" and other bureaucratic structures. --JWSchmidt 01:37, 12 February 2011 (UTC)
 * per Darklama -- Jtneill - Talk - c 02:26, 12 February 2011 (UTC)
 * per Darklama Leighblackall 04:20, 12 February 2011 (UTC)
 * but amendment should read "may" be closed, not "will be closed." Future CRs may reference a closed one, it is not necessary to re-invent the wheel. --Abd 15:56, 13 February 2011 (UTC)
 * @Moulton: Process that sits for a long time fails to attract new attention. If a CR has a clear result, it should be closed with a conclusion. If not, after a decent time, it should be closed without conclusion, and that close is without prejudice. A new CR on the same or similar topic can be prepared and opened, citing the old one and incorporating it by reference, summarizing it, and new comment solicited per the CR process. Thus deliberation is built. --Abd 16:21, 13 February 2011 (UTC)
 * General comment: I'm uncomfortable with a 7-question limit, particularly since the method by which a question is proposed to the community is left vague. Could I toss up 7 questions on my own, and thereby prevent anyone else from proposing questions? I understand the purpose of the limit, but I'd address that purpose very differently.
 * We should have, for starters, a process for proposing questions that requires a second on any question. No question should go for general vote unless it has at least that level of support, and, for efficiency, we really need the pre-site-notice participants to function as a committee, to find some level of consensus on presentation of questions to the community, with reference to that process (it could be a subpage), so that the community can review it if desired. Understanding how committees function with standard deliberative bodies is very important. They do not control, in theory. They advise. The full assembly is sovereign, but no King can examine every topic in detail. --Abd 16:28, 13 February 2011 (UTC)

 * per darklama. --mikeu talk 14:43, 14 February 2011 (UTC)
 * per darklama. --Bduke 21:46, 20 February 2011 (UTC)

Resolution 2011-7(a): Amendment to Community Review Policy
Community Reviews will be closed if no new ground is covered for 1 month. Unresolved issues may be addressed in a new Community Review. Old issues may be raised in a new Community Review if there is something new to be discussed, or if the resolutions of an older review have caused unexpected problems.
 * - much better to close CRs that have gone inactive. If the subject needs to be revisited later, there would likely be different concerns, a different focus, etc. Also people do tend to come and go, so this would prevent the logjam of trying to balance "legacy votes" when seeking consensus for an additional change. --SB_Johnny talk 10:46, 10 March 2011 (UTC)
 * - --mikeu talk 12:02, 21 March 2011 (UTC)
 * People may want to propose new solutions for resolved issues too, if they feel a previous solution isn't working out. I think people should be allowed to raise any previous issues in a new Community Review, if new ground is covered. I also think if a month manages to go by without new ground being covered a gap before issues can be brought up again should be required to prevent burn out, give resolutions a chance to work, and to give people a chance to see whether issues will resolve themselves. I propose replacing everything after the first sentence with "You may raise issues discussed in previous community reviews, if new ground will be covered, and 2 months have passed since the last discussion of the raised issues." -- dark lama  12:40, 21 March 2011 (UTC)
 * KillerChihuahua 23:25, 24 March 2011 (UTC)
 * Per DL. A little time between CR's seems like a good idea. (The preceding unsigned comment was added by Thenub314 (talk • contribs) .) 19:33, 20 April 2011
 * . Oppose DL amendment, because this resolution is about closing inactive CRs. If an issue remains from an inactive CR, closed for that reason, there should be no prejudice against starting a new one. Otherwise the closing for inactivity would prejudice unresolved issues. The matter about "old issues" is here mixed up with "unresolved issues," and that should be worked out in the actual policy amendment language. Basically, bringing up resolved issues, where consensus was found, shouldn't be done quickly, unless there are new arguments that were not considered, or possibly some abusive or improper close (which amounts to a new argument, simply one over the close). If the requirement for agreement from two established users, or the like, is implemented, then there is adequate safeguard against tendentious re-filing. (Or there is obviously some issue needing attention!) --Abd 16:44, 27 April 2011 (UTC)


text:
User:Yongtao-Li/HW3.4

=Problem 3.4 "Area of Bifolium"=

Given
The classic curve bifolium is shown as following

 * egm6341.s11.team2.hw3.4.1.png

The exact area of this leaf which can be calculated using WolframAlpha is π/16.

Find
1) Do literature search to find history and applications (if any) of this classic curve. First described by Johannes Kepler in 1609, a bifolium is a curve that has the respective Cartesian and polor form,
 * {| style="width:100%" border="0"

$$ \displaystyle (x^2+y^2)(y^2+x(x+b))=4axy^2 $$    (4.0a) and
 * style="width:98%" |
 * style="width:98%" |
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle r=-bcos(\theta)+4acos(\theta)sin^2(\theta) $$    (4.0b) The etymology of bifolium stems from the Latin prefix [bi-] meaning "two" and the Latin root word [folium] meaning "leaved", hence the meaning "two-leaved". The of Descartes] was first discovered by Rene Descartes in 1638 and is represented by the equation,
 * style="width:98%" |
 * style="width:98%" |
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle (x^3+y^3)=3axy $$    (4.0c) Unique to the folium of Descartes is the asymptote located at
 * style="width:98%" |
 * style="width:98%" |
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle x+y+a=0 $$    (4.0d)
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

2) Find area in one leaf to $$ {10^{ - 6}} $$ accuracy using following methods.

2.1) Complex Trapezoidal Rule.

2.2) Complex Simpson's Rule.

2.3) Sum the area of triangles which composite the leaf. If we know two edges a and b and the angle t between them, the area of one triangle is

Formulation
2.1) and 2.2)

First we have to divide the curve into two parts like the red and green curves shown in Figure 4.1. Since we can not explicit show the function relationship in terms x, we can write the function relation in terms of y as following:

Because equation (4.4) and (4.5) are describe the green and red curve separately, we can use equation (4.1) and (4.2) to integrate the area between these two curve and y axis and the area difference is just the area of this leaf.

2.3)

When using summation of triangles that constitute the leaf, we can divide [0,π/2] into several equal angles and the parametric equations are needed to calculate the edge length of each triangles:

Fortran Code for these three methods
In order to simplify the procedure and make it clear, Fortran codes which are written to calculate the leaf area using the above formulation are as following:

Result
Using the above codes, we can have the following output on the screen:

From the above result, the integrated results are calculated exactly. Since we use $$ {10^{ - 6}} $$ accuracy compared to the given exact solution, we also generate the error convergence as figure 4.2 and figure 4.3.

 * egm6341.s11.team2.hw3.4.2.png

 * egm6341.s11.team2.hw3.4.3.png

From figure 4.2 we can find both Complex Trapezoidal and Simpson's method can convergent at the same time, but Complex Trapezoidal integration gives less error for equal block number.

From figure 4.3 we notice that using summation of triangles is very easy to find convergence when just cut a leaf in to more than 1000 triangles which are very easy to calculate its area.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Yongtao-Li/HW3.4


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Category:Bible

Pages related to Bible projects

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User talk:Peter Damian

 Hello Peter Damian, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --mikeu talk 21:58, 11 February 2011 (UTC)

Greetings
Thrilled you are here. Can you convince Milton Roe to join us, too? —Moulton 22:03, 11 February 2011 (UTC)

 * Hi Peter. Nice to see you here. Please pay no attention the madmen in front of the curtains ;-). --SB_Johnny talk 17:59, 13 February 2011 (UTC)

The Sisyphus Paradox
On W-R you wrote:

You are arguing (with some Wikipedian) that experts aren't fairly supported or represented on Wikipedia. The Wikipedian replies "But look at Citizendium - that had experts and it failed". You reply that Citizendium had other problems - no Google attractor, not many actual experts, Larry etc. The Wikipedian then points out a perfectly good article that an expert had sorted out, and the expert was you

So the Sisyphus Paradox is this. The very experts who are complaining about low academic standards on Wikipedia, and the problems of editing there, are the ones who are keeping the standards as high as they are.

An example. I have often complained about Neurolinguistic Programming on Wikipedia, as being junk science disproportionately represented. I mentioned this to an academic, who looked at the introduction to the main article about it and said that it was in fact pretty neutral. I realised to my horror that the introduction was written by me. How can I complain about junk science on Wikipedia when I am one of the culprits who are, often successfully, keeping it out?

There was a similar problem with the article on Existence. I have long used this as evidence of poor standards, and foolishly mentioned this to Connolley who, apart from some minor errors, has generally cleaned it up.

Poor Connolley and poor the other writers on science and academic subjects. They are going through the labours of Sisyphus. Yet they only complain after they have successfully got the boulder up the hill. If only they had waited until the gods of the Wiki had pushed it back again.

Peter Damian Sat Feb 12, 2011, 10:14AM (UTC) From the Wikipedia article on Sisyphus:

As a punishment from the gods for his trickery, Sisyphus was made to roll a huge boulder up a steep hill, but before he could reach the top of the hill, the rock would always roll back down, forcing him to begin again. The maddening nature of the punishment was reserved for Sisyphus due to his hubristic belief that his cleverness surpassed that of Zeus. Somehow, we must devise a way to share our expertise without sounding like we know what we're talking about. Perhaps the Peter Falk character, Columbo, had the right idea here. We need to migrate from writing authoritative articles to crafting entertaining dramas. Wikipedians do seem to have an insatiable appetite for drama.

Moulton 12:44, 12 February 2011 (UTC)
 * Yes but no song parodies please. Peter Damian 13:25, 12 February 2011 (UTC)

 * How the devil can we have a Comic Opera without song parodies?!? —Moulton 13:30, 12 February 2011 (UTC)

Welcome!
If you decide to do more than poke about, I can use my library to get books on medieval philosophy to help with anything. Ottava Rima (talk) 14:16, 12 February 2011 (UTC)
 * Wow. Such an affectionate greeting for one of the Sith from over on WR... Is that your double standard showing? Dinsdale 18:43, 12 February 2011 (UTC)
 * Who are you and why are you trolling on a new name instead of having the guts to use your real name? And Peter knows how I feel about him - I worked on a few articles of his to try and save them from being deleted and I also tried to have him unbanned. Ottava Rima (talk) 14:16, 13 February 2011 (UTC)

 * Are you trying to out me? You are certainly not AGFing by calling me a troll. Please try to AGF in the future Ottava. If you are unable to AGF, then I will take these comments to the appropriate venue for review. Some people are immune to being bullied. If anyone attempts, they will find out that I am in that set. Dinsdale 17:12, 13 February 2011 (UTC)
 * Your first posts show that you have intimate knowledge of WR, the WMF, and other material which reveals this is not your only account and you aren't up to any good. Ottava Rima (talk) 17:23, 13 February 2011 (UTC)

I had to look up what a 'sith' was. Presumably a not very nice person. Gentlemen, please take this elsewhere. Ottava, thanks for the welcome. Peter Damian 17:25, 13 February 2011 (UTC)
 * As you wish, though the Sith remark was tongue in cheek. Dinsdale 17:33, 13 February 2011 (UTC)

Let me remind everyone that WV:CIVIL and WV:AGF apply here, and also that if anyone has a problem with someone from another (non-WV) website then they should discuss those problems there. Please don't import drama here, we have enough of our own, thank you very much. Peter, feel free to create User talk:Peter Damian/Archive and move any off-topic messages there. --mikeu talk 19:01, 13 February 2011 (UTC)

Experts
So true. The problem between what is and what can be as people stress the potential of the masses yet it never amounts to anything while those who have put out stuff are ignored. JWSchmidt and some others have been discussing problems related to his concern as we try to find an identity that is more academic friendly (but probably failed). Ottava Rima (talk) 01:01, 14 February 2011 (UTC)

Corrupt Governance
Hi Peter,

On Wikipedia Review, you wrote a few minutes ago about how Wikipedia is really run:

Peter, is it also your opinion that the above corrupt processes are undermining the ethical governance of Wikiversity, as well?

Moulton 11:21, 20 February 2011 (UTC)

 * Update: Peter Damian has added this summary to the above:

 * To what extent have the corrupt governance practices of Wikipedia seeped into Meta and Wikiversity? —Moulton 12:40, 24 February 2011 (UTC)

Exasperation?
Today on W-R, you wrote:

W-R probably isn't the place for the kind of reasoned discussion you have in mind. This place may not be either. But there are some alternatives. Do you know about Michael Bauwens and the Peer-to-Peer Foundation? There is also NetKnowledge, which has both a Wiki and a BBS forum. Would you like to explore any of those? —Moulton 01:44, 7 March 2011 (UTC)

title: User talk:Peter Damian


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User:Wi2g

Hello! This is the Wikiversity user page for Wi2g.

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User talk:Wi2g

Hello! This is the Wikiversity talk page for user Wi2g.

title: User talk:Wi2g


title: File:Nm1.s11.mtg18.djvu


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User:Egm6341.s11.team2/hwk3

=Problem 3.1 Compare Taylor Series Approximation with Lagrangian Interpolation=

Problem Statement: Taylor Series and Lagrangian Interpolation
Use two methods to approximate the function $$\displaystyle f(x)=\sin x$$,

1. Taylor series expaned with respect to $$\displaystyle \hat{x}=\frac{3\pi }{8}$$ to the order $$\displaystyle n$$,

2. Lagrangian interpolation to the order 4

Compare the error by two methods at $$\displaystyle x=\frac{7\pi }{8}$$, find $$\displaystyle n$$ such that the error of Taylor series expansion is less than 4th order Lagrangian interpolation. And then give the maximum value of Lagrangian interpolation error.

1. Taylor series expansion
The Taylor expansion to $$\displaystyle f(x)=\sin x$$ is

2. Lagrangian interpolation
Use equidistance discretion, the 4th order Lagrangian interpolation requires 5 nodes: While the Lagrangian Interpolation evaluated at $$\displaystyle x=\frac{7\pi }{8}$$ is,

where,

Plug them back into the Eq 1.4 we have,

Comparison of the Taylor and Lagrange Methods
The exact value of $$\displaystyle f(\frac{7\pi }{8})$$ is,

Then we have the error of Lagrangian interpolation,

To find the order n which makes the Taylor series produce small error, we need to try different order.

From the calculation above, the 6th order of Taylor expansion can give better result than Lagrangian interpolation.

The Maximum Value of Lagrangian Interpolation Error
By Lagrangian interpolation error theorem,

Thus the maximum error is contained by $$\displaystyle \left| {{q}_{5}}(t) \right|$$. If $$\displaystyle t$$ is confined within $$\displaystyle (0,\pi )$$, the maximal error is given by

=Problem 3.2 Proof That G(x) Is Zero At All Nodes=

Given: The Definition of G(x)
Three functions $$ \displaystyle G(x)$$, $$ \displaystyle e_n^L(f;x)$$ and $$ \displaystyle q_{n+1}(x)$$ that are $$ \displaystyle (n + 1) $$ times continuously differentiable on the the interval $$ \displaystyle I \in [a,b]$$. Then for some $$ \displaystyle t \in I $$ such that $$ \displaystyle t $$ is NOT equal to any of the nodes $$ \displaystyle x_i $$

 * {| style="width:100%" border="0"

$$ \displaystyle i.e \quad t \neq x_i \quad i=0,1,...,n $$
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

Then let $$ \displaystyle G(x) $$ be defined as,
 * {| style="width:100%" border="0"

$$ \displaystyle G(x):=e_{n}^{L}(f;x) - \frac{q_{n+1}(x)}{q_{n+1}(t)}e_{n}^{L}(f;t) $$     (2.1)
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

Show: G(x)=0 for all n
That $$ \displaystyle G(x_i)=0 $$ for $$ \displaystyle i=0,1,...,n $$

Solution
Let us rewrite (2.1) using a shorthand notation where the subscripts, superscripts and functional dependence on $$ \displaystyle x $$ and $$ \displaystyle t $$ are implicitly considered yet not explicitly written. Doing so we arrive at an equivalent expression for (2.1) as shown here,

 * {| style="width:100%" border="0"

$$ \displaystyle G(x)=e(x) - \frac{q_{n+1}(x)}{q_{n+1}(t)}e(t) $$     (2.2)
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

For conceptual convenience let us write (2.2) as a ratio of the error at t, $$ \displaystyle e(t) $$ and $$ \displaystyle q_{n+1} $$ evaluated at t.
 * {| style="width:100%" border="0"

$$ \displaystyle G(x)=e(x) - q_{n+1}(x) {\color{blue}\left[ \frac{e(t)}{q_{n+1}(t)}\right] } $$     (2.3)
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

Let us next attempt to find the relation shown in (2.3) by considering how one may construct it from scratch. Doing so will aid the understanding of why $$ \displaystyle G(x) =0$$, which we will inevitably show. Suppose we wish to scale the error $$ \displaystyle e(\cdot) $$ relative to $$ \displaystyle q_{n+1}(\cdot) $$ for both $$ \displaystyle x $$ and $$ \displaystyle t $$. Mathematically, this is expressed as follows
 * {| style="width:100%" border="0"

$$ \displaystyle {\color{red}\alpha(x)} = \frac{e(x)}{q_{n+1}(x)} $$     (2.4) And
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$$ \displaystyle {\color{red}\beta(t)} =\frac{e(t)}{q_{n+1}(t)} $$     (2.5)
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 *
 * }

Next we will define $$ \displaystyle d(x) $$ as the difference between $$ {\color{red}\alpha(x)} $$ and $$\displaystyle {\color{red}\beta(t)} $$ as shown in (2.4) and (2.5).
 * {| style="width:100%" border="0"

$$ \displaystyle \begin{align} & d(x) = {\color{red}\alpha(x)} & - & {\color{red}\beta(t)} \\ \implies \quad & d(x) = \frac{e(x)}{q_{n+1}(x)} & - & \frac{e(t)}{q_{n+1}(t)} \end{align} $$     (2.6)
 * style="width:98%" |
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 *
 * }

Multiplying (2.6) by $$ \displaystyle q_{n+1}(x) $$ results in the following equation for $$ \displaystyle G(x) $$ that was previously introduced in (2.3).
 * {| style="width:100%" border="0"

$$ \displaystyle \begin{align} q_{n+1}(x)\cdot d(x) &= \cancel{q_{n+1}(x)}\cdot \frac{e(x)}{\cancel{q_{n+1}(x)}} -  q_{n+1}(x)\cdot\frac{e(t)}{q_{n+1}(t)} \\ \implies \quad G(x)&= e(x) - q_{n+1}(x)\frac{e(t)}{q_{n+1}(t)} \end{align}$$ (2.7)
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

Next we must consider two important questions in order to show that $$ \displaystyle G(x) $$ is zero.

1. What is the value of $$ e(x) $$ at $$ x_i $$?
The value of $$ e(x) $$ at $$ x_i $$ must be zero since there is no error when the point being consider is a node.

2. What is the value of $$ q_{n+1}(x) $$ at $$ x_i $$?
The value of $$ q_{n+1}(x) $$ at $$ x_i $$ is also zero at all node $$ i.e. \ x_i \ \ni \ i=0,1,..,n $$.

Mathematically, one can see that 1 and 2 have the following consequences.
 * {| style="width:100%" border="0"

$$ \displaystyle G(x_i)= {\canceltoe(x_i)} - {\canceltoq_{n+1}(x_i)}\frac{e(t)}{q_{n+1}(t)} $$     (2.8)
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

=Problem 3.3 - Lagrange Interpolation and the Runge Phenomenon=

Given: A Function f(x)
- The function f(x) defined on [-1,1]

 * $$f(x)=\frac{e^{x}-1}{x}$$

- Uniform discretization with n = 4, so the nodes are $$\left \{ x_0,...,x_4 \right \} = \left \{ -1,...,1 \right \}$$

- Fixed points: t = 5, x = 0.75

Find: $$e_{4}^{L}(f,t)$$ and $$e_{4}^{L}(f,x)$$
- Generate plots as in p.14-2

- For middle plot, use i = 2 and generate $$l_{2,4}(.)$$

- Find $$e_{4}^{L}(f,t)$$ and $$e_{4}^{L}(f,x)$$

Solution
Figure 3.3-1 shows the requested plots on the interval [-1,1], and Figure 3.3-2 shows the plots on the interval [-1,5]. The decision was made to generate two separate sets of plots because expanding the domain to [-1,5] to show the evaluation point t (which is desired in the problem statement) greatly distorts the shape of the plots. In order to properly observe and discuss the characteristics of the plotted functions, two sets of graphs were made and will be discussed separately.

 * Egm6341.s11.hw3_3a.jpg]

On the interval [-1,1], the approximation using Lagrange interpolation has a small error as shown in the top graph of Figure 3.3-1. The error at the point x=0.75, defined as

is -1.6274 x 10-4. As can be seen in the top graph of Figure 3.3-2, the error grows if the domain is expanded to [-1,5] to include point t. The error at point t using Equation 3.3.1 was calculated to be e4L(f,t)=11.0239. This is an unacceptably large error, and thus it is recommended that approximations using Lagrange interpolation be used only in the domain in which the nodes are contained. The third plots in FIgures 3.3-1 and 3.3-2 show the values of qn+1(.). According to Equation 3.3.1, the error at a point x is proportional to qn+1(x). The error is zero at the nodes and attains its largest values between the nodes.

 * Egm6341.s11.hw3_3b.jpg]

Given:A Function f(x)
- The function f(x) defined on [-5,5]

 * $$f(x)=\frac{1}{1+4x^2}$$

- Uniform discretization with n = 8, so the nodes are $$\left \{ x_0,...,x_8 \right \} = \left \{ -5,...,5 \right \}$$

- Fixed points: t = 4.5, x = 3

Find: $$e_{8}^{L}(f,t)$$ and $$ e_{8}^{L}(f,x) $$
- Generate plots as in p.14-2

- For middle plot, use i = 3 and generate $$l_{3,8}(.)$$

- Find $$e_{8}^{L}(f,t)$$ and $$e_{8}^{L}(f,x)$$

Solution
In this problem, both of the evaluation points t and x are in the domain of the nodes [-5,5], so there are no large errors caused like in Problem 1. However, this problem is an example of larger errors caused by the Runge phenomenon. In brief, the Runge phenomenon is an error that occurs when approximating some functions using high-order polynomials. The error is worse as higher-order polynomials are used, and increases near the edges of the interval.

 * Egm6341.s11.hw3_3c.jpg]

The red and blue lines in the top plot of Figure 3.3-3 show the differences between the actual function (blue) and the 8th-order polynomial approximation (red). As shown in the third plot of Figure 3.3-3, the qn+1(.) term in the error equation (3.3.1) is on the order of 104 between the first and last pairs of nodes. The error at point x was found to be e8L(f,x)=-0.3786 and the error at point t was found to be e8L(f,x)=1.7840. In order to avoid the Runge phenomenon, non-uniformly distributed nodes must be selected.

MATLAB Code
=Problem 3.4 "Area of Bifolium"=

Given: A Classical Bifolium Curve
The classic curve bifolium is shown as following

 * egm6341.s11.team2.hw3.4.1.png

The exact area of this leaf which can be calculated using WolframAlpha is π/16.

Find: Historical Reference/Application and the Area of One Bifolium Pedal Using 3 Different Methods
1. Do literature search to find history and applications (if any) of this classic curve.

2. Find area in one leaf to $$ {10^{ - 6}} $$ accuracy using following methods.

2.1. Complex Trapezoidal Rule.

2.2. Complex Simpson's Rule.

2.3) Sum the area of triangles which composite the leaf. If we know two edges a and b and the angle t between them, the area of one triangle is

1. Literature Search Containing Historical Importance of the Bifolium
First described by Johannes Kepler in 1609, a bifolium is a curve that has the respective Cartesian and polor form,
 * {| style="width:100%" border="0"

$$ \displaystyle (x^2+y^2)(y^2+x(x+b))=4axy^2 $$    (4.0a) and
 * style="width:98%" |
 * style="width:98%" |
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle r=-bcos(\theta)+4acos(\theta)sin^2(\theta) $$    (4.0b) The etymology of bifolium stems from the Latin prefix [bi-] meaning "two" and the Latin root word [folium] meaning "leaved", hence the meaning "two-leaved". The of Descartes] was first discovered by Rene Descartes in 1638 and is represented by the equation,
 * style="width:98%" |
 * style="width:98%" |
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle (x^3+y^3)=3axy $$    (4.0c) Unique to the folium of Descartes is the asymptote located at
 * style="width:98%" |
 * style="width:98%" |
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle x+y+a=0 $$    (4.0d)
 * style="width:98%" |
 * style="width:98%" |
 *
 * }

2.1 & 2.2: Determination of the Area Using The Complex Trapezoid and Complex Simpson's Rule
First we have to divide the curve into two parts like the red and green curves shown in Figure 4.1. Since we can not explicit show the function relationship in terms x, we can write the function relation in terms of y as following:

Because equation (4.4) and (4.5) are describe the green and red curve separately, we can use equation (4.1) and (4.2) to integrate the area between these two curve and y axis and the area difference is just the area of this leaf.

2.3 Determination of the Area By Summing The Triangles
When using summation of triangles that constitute the leaf, we can divide [0,π/2] into several equal angles and the parametric equations are needed to calculate the edge length of each triangles:

Fortran Code for these three methods
In order to simplify the procedure and make it clear, Fortran codes which are written to calculate the leaf area using the above formulation are as following:

Result
Using the above codes, we can have the following output on the screen:

From the above result, the integrated results are calculated exactly. Since we use $$ {10^{ - 6}} $$ accuracy compared to the given exact solution, we also generate the error convergence as figure 4.2 and figure 4.3.

 * egm6341.s11.team2.hw3.4.2.png

 * egm6341.s11.team2.hw3.4.3.png

From figure 4.2 we can find both Complex Trapezoidal and Simpson's method can convergent at the same time, but Complex Trapezoidal integration gives less error for equal block number.

From figure 4.3 we notice that using summation of triangles is very easy to find convergence when just cut a leaf in to more than 1000 triangles which are very easy to calculate its area.

= Problem 3.5: Evaluate the $$\displaystyle (n+1)^{th}$$ Derivative of Lagrange Interpolation Error =

''' This problem was solved without referring to S10 homework. '''

From the lecture slide Mtg 16-3

Given: The Error for the Lagrangian Interpolation
The Lagrange Interpolation error can be expressed as
 * {| style="width:100%" border="0"

E(x)=f(x)-f_n(x) , $$     (5.1)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

Find: The Error at the (n+1)-th term
Verify the following:

 * {| style="width:100%" border="0"

E^{(n+1)}(x)=f^{(n+1)}(x)-0. $$     (5.2)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

Solution
The Lagrange Interpolation error can be expressed as
 * {| style="width:100%" border="0"

E(x)=f(x)-f_n(x) , $$     (5.3)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }

where

 * {| style="width:100%" border="0"

f_n(x) \epsilon \mathcal{P}_n. $$     (5.4)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }

Because $$\displaystyle f_n(x) \epsilon \mathcal{P}_n $$, it is of the form

 * {| style="width:100%" border="0"

f_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0. $$     (5.5)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }

Differentiating $$ \displaystyle f_n $$ n times gives

 * {| style="width:100%" border="0"

f^{(n)}_n(x)=(n!)a_n , $$     (5.6)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }

which is a constant.

Differentiating the expression $$\displaystyle (n+1)$$ times gives

 * {| style="width:100%" border="0"

E^{(n+1)}(x)=f^{(n+1)}(x)-f_n^{(n+1)}(x) , $$
 * $$\displaystyle
 * $$\displaystyle
 *
(5.7)
 * }

where

 * {| style="width:100%" border="0"

f_n^{(n+1)}(x) = 0. $$
 * $$\displaystyle
 * $$\displaystyle
 *
(5.8)
 * }

Therefore,

= Problem 3.6: Error In a Lagrange Interpolation= From the lecture slide Mtg 16-3

Ref:problem 7, Team 4, S10

Given: A Polynomial q(x)
Defined as
 * {| style="width:100%" border="0"

$$\displaystyle {q_{n + 1}}(x) = \prod\limits_{j = 0}^n {(x - {x_j})} = (x - {x_0})(x - {x_1})...(x - {x_{n - 1}})(x - {x_n}) $$    (6.1)
 *

 * }

Verify: The Error in the Lagrange Term

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q_{n + 1}^{n + 1}(x) = (n + 1)! $$     (6.2)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }

Solution
We know that $$ \displaystyle{q_{n + 1}}(x) \in {p_{n + 1}}$$, so (6.1) can be written as:

 * {| style="width:100%" border="0"

{q_{n + 1}}(x) = \underbrace _{f(x)} + \underbrace {{a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_1}{x^1} + {a_0}{x^0}}_{{p_n}(x)} $$     (6.3)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }

note that in (6.3), $$ {a_{n + 1}} = 1\displaystyle $$.

 * {| style="width:100%" border="0"

q_{n + 1}^{n + 1}(x) = {f^{n + 1}}(x) + p_n^{n + 1}(x) $$     (6.4)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }

where $$\displaystyle p_n^{n + 1}(x) = 0 $$ and $$\displaystyle {f^{n + 1}}(x) $$ can be calculated as follows:

 * {| style="width:100%" border="0"

$$\displaystyle\begin{align} f(x)        &= {x^{n + 1}} \\ f^1(x)      &= (n + 1){x^n} \\ {f^2}(x)      &= (n + 1)(n){x^{n - 1}} \\ {f^3}(x)      &= (n + 1)(n)(n - 1){x^{n - 2}} \\ & \vdots \\ {f^{n - 1}}(x) &= (n + 1)(n)(n - 1) \cdots (2){x^{n - (n - 1)}} \\ {f^n}(x)      &= (n + 1)(n)(n - 1)\cdots (2)(1){x^{n - n}} \\ {f^{n + 1}}(x) &= (n + 1)! \end{align} $$
 *

 * }

and consequently

=Problem 3.7: Deriving the Error in Newton-Cotes=

''' This problem was solved without referring to S10 homework. '''

Given: The Convergence of Numerical Integration
The convergence of numerical integration when applied to the error associated with the Newton-Cotes formula as seen in [[media:nm1.s11.mtg17.djvu|Lecture 17]] is given by

 * {| style="width:100%" border="0"

$$ \displaystyle $$    (7.1)
 * style="width:98%" |
 * style="width:98%" |
 * E_n| \leq \frac{M_{n+1}}{(n+1)!} \int_a^b{|q_{n+1}(x)|dx}
 *

 * }

Where $$ \displaystyle M_{n+1}(x):= $$ be defined as,
 * {| style="width:100%" border="0"

$$ \displaystyle M_{n+1}(x):=\underset{a\le x\le b}{\mathop{\max }}\,f^{(n+1)}(x)$$ (7.2)
 * style="width:98%" |
 * style="width:98%" |
 *

 * }

Derive: The Error in the Simple Trapezoid Rule
Derive the error $$ \displaystyle |E_1| $$ for the simple trapezoid rule such that $$ \displaystyle n=1 $$ and where
 * {| style="width:100%" border="0"

$$\displaystyle n=1 \quad \implies \quad q_2(x)=(x-x_0)(x-x_1)=(x-a)(x-b) $$     (7.3)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Then go on to show that
 * {| style="width:100%" border="0"

$$ \displaystyle $$    (7.4)
 * style="width:98%" |
 * style="width:98%" |
 * E_1| \leq \frac{M_{2}}{2!} \int_a^b{|q_{2}(x)|dx} = \frac{(b-a)^3}{12}M_2
 *

 * }

Solution: By Integration
To find the bounded error for $$ \displaystyle |E_1| $$ on can simply expand the quadratic in the integrand of (7.4) and integrate, as shown here.
 * {| style="width:100%" border="0"

$$\displaystyle \begin{align} &= \frac{M_{2}}{2!} \int_a^b{(x-a)(b-x)dx}\\ &= \frac{M_{2}}{2!} \int_a^b{(-x^2+ax+bx-ab)dx}\\ &= \frac{M_{2}}{2!}\left[-\frac{x^3}{3}+\frac{a}{2}x^2+\frac{b}{2}x^2-abx \right]\bigg|_{x = a}^{x=b} \\ &= \frac{M_{2}}{2!}\left[\left(-\frac{b^3}{3}+\frac{ab^2}{2}+\frac{b^3}{2}-ab^2 \right) - \left(-\frac{a^3}{3}+\frac{a^3}{2}+\frac{a^2b}{2}-a^2b \right)\right] \\ &= \frac{M_{2}}{12}\left[{\color{red}-2b^3}+{\color{green}3ab^2}+{\color{red}3b^3}-{\color{green}6ab^2} + 2a^3-3a^3-{\color{cyan}3a^2b}+{\color{cyan}6a^2b}\right] \\ &= \frac{M_{2}}{12}\left[{\color{red}b^3}-{\color{green}3ab^2}+{\color{cyan}3a^2b}-a^3\right] \\ &= \frac{M_{2}}{12}\left(b-a\right)^3 \end{align}$$ (7.5)
 * style="width:95%" |
 * style="width:95%" |
 * E_1| &\leq \frac{M_{2}}{2!} \int_a^b{|q_{2}(x)|dx}\\
 *

 * }

The final solution is

=Problem 3.8: Deriving the Error in Newton-Cotes=

''' This problem was solved without referring to S10 homework. '''

Given: The Convergence of Numerical Integration
The convergence of numerical integration when applied to the error associated with the Simple Simpson's formula as seen in [[media:nm1.s11.mtg17.djvu|Lecture 17]] is given by

 * {| style="width:100%" border="0"

$$ \displaystyle $$    (8.1)
 * style="width:98%" |
 * style="width:98%" |
 * E_n| \leq \frac{M_{n+1}}{(n+1)!} \int_a^b{|q_{n+1}(x)|dx}
 *

 * }

Where $$ \displaystyle M_{n+1}(x):= $$ be defined as,

 * {| style="width:100%" border="0"

$$ \displaystyle M_{n+1}(x):=\underset{a\le x\le b}{\mathop{\max }}\,f^{(n+1)}(x)$$ (8.2)
 * style="width:98%" |
 * style="width:98%" |
 *

 * }

Derive: The Error in the Simple Simpson's Rule
Derive the error $$ \displaystyle |E_2| $$ for the simple Simpson's rule such that $$ \displaystyle n=2 $$ and where

 * {| style="width:100%" border="0"

$$\displaystyle n=2 \quad \implies \quad q_3(x)=(x-x_0)(x-x_1)(x-x_2)=(x-a)(x-\frac{a+b}{2})(x-b) $$     (8.3)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Then go on to show that

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$$ \displaystyle (8.4)
 * style="width:98%" |
 * style="width:98%" |
 * E_2| \leq \frac{M_{3}}{3!} \int_a^b{|q_{3}(x)|dx} = \frac{192}{{M}_{3}}$$
 *

 * }

Solution
To find the bounded error for $$\displaystyle \left| {{E}_{2}} \right|$$ one can simply expand the polynominal in the integrand of (8.4) and integrate, as shown here.

Since function $$\displaystyle g(x)=\left| (x-a)(x-\frac{a+b}{2})(x-b) \right|$$ is symmetric with respect to $$\displaystyle x=\frac{a+b}{2}$$(from the fact $$\displaystyle g(\frac{a+b}{2}+t)=g(\frac{a+b}{2}-t)$$), then we use this property to eliminate the absolute value sign.

Then we have ,

The length of the panel is $$\displaystyle h=\frac{b-a}{2}$$, so we can write the solution like,

=References=

=Contributing Members & Referenced Lecture=

title: User:Egm6341.s11.team2/hwk3


text:
User:Egm6341.s11.team4.hylon/hw3

=Problem 3.4 Literature search the history and application of bifolium, compute the area of one leaf= Refer to lecture slide [[media:nm1.s11.mtg15.djvu|15-2]] for the problem statement.

Given
The equation of the bifolium curve in polar coordinates:$$r(t)=2\sin (t){{\cos }^{2}}(t)$$, with $$t\in [0,\pi ]$$.

Objectives
1. Do literature research to find history and application(if any) of this classic curve

2. Compute the area in one leaf to $$10^{-6}$$ accuracy.

History and application of bifolium
The folium curve is first described by Johannes Kepler in 1609. And it has two general equation, with $$({{x}^{2}}+{{y}^{2}})({{x}^{2}}+bx+{{y}^{2}})=4axy$$ in Cartesian form and $$\rho =4a\cos t{{\sin }^{2}}t-b\cos t$$ in polar coordinates. It has the meaning of “leaf-shaped” in Latin. It has three forms, known as simple folium(or single folium), the bifolium(or doublefolium), and the trifolium, corresponding to the cases when $$b=4a$$, $$b=0$$ and $$b=a$$, respectively. In 1638, Rene Descartes first discussed the type with Cartesian equation of $${{x}^{3}}+{{y}^{3}}=3axy$$, which thereafter named the folium of Descartes. Rene Descartes has found the correct shape of the curve in the positive quadrant, but he wrongly view that this leaf shape was repeated in each quadrant like the four petals of a flower. The problem to determinate the tangent to the curve was proposed to Gilles de Roberval who, having made the same incorrect assumption, called the curve fleur de jasmine after the four-petal jasmine bloom, a name that was later dropped. The folium of Descartes has an asymptote $$x+y+a=0$$. See The universal book of mathematics, by David J. Darling, for more information.

The bifolium has been studied by Longchamps (1886) and Henri Brocard(1887).

For more information about bifolium, refer to wolfram_Bifolium and 2Dcurves_Bifolium.

Matlab Code for above plots:

Compute the area of one leaf
The exact value of one leaf: Integrate the curve equation with $$t\in (0,\frac{\pi }{2})$$, we get the exact value of area of one leaf:
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:92%; padding:10px; border:2px solid #ff0000" |
 * style="width:92%; padding:10px; border:2px solid #ff0000" |

Area=\int\limits_{0}^{\frac{\pi }{2}}{2\sin (t){{\cos }^{2}}(t)}dt=\frac{\pi }{16}

$$
 * }

Method 1: Using Composite Trapezoidal rule and Composite Simpson's rule

In this method, we have two approaches to calculate the area of one leaf, which are Composite Trapezoidal rule and Composite Simpson’s rule. In order to compute the area of one leaf, we divide the area between the upper leaf curve and the vertical axis into two parts B and C, as depicted in the figure above. We use the equation in Cartesian coordinates in this method, that is $$({{x}^{2}}+{{y}^{2}})=2{{x}^{2}}y$$. First, we use Matlab to find the right-most and top-most point of the folium curve of one leaf:

The right-most point is(0.52, 0.748410929036851).

The top-most point is (0.495377246516107,0.499957647395650).

Matlab Code: The equation in Cartesian coordinates is implicit. And it’s unable for us to express dependent variable y in terms of independent variable x. But we can express x in terms of y. For the reason of convenience, we choose to project in the y-direction to calculate the area of one leaf. Matlab Code for solving explicit express: Using Matlab, we can get following expressions for x of one leaf:
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & {{x}_{1}}=\sqrt{y+y\sqrt{(1-2y)}-{{y}^{2}}} \\ & {{x}_{2}}=\sqrt{y-y\sqrt{(1-2y)}-{{y}^{2}}} \\ \end{align}

$$ $${{x}_{1}}$$ is the equation for bottom curve, and $${{x}_{2}}$$ is the equation for the top curve. Next, we use two approaches to calculate the area of the two different parts.
 * }

Using Composite Trapezoidal rule:

Matlab Code: The result:

Using Composite Simpson’s rule: Matlab Code: The result:

Method 2: Using sum of area of subdivided triangles

We use the equation in polar coordinates in this method, that is $$r=2\sin (t){{\cos }^{2}}(t)$$. Matlab Code:

The result:

Comment
As seen from above results using three different methods to compute the area of one leaf, we can conclude that:

Composite Simpson's rule converges much faster than Composite Trapezoidal rule;

The Sum of Triangle Area method converges much faster than both Composite Trapezoidal rule and Composite Simpson's rule.

Matlab Code:

Reference
1.Wolfram_Bifolium

2.Composite Trapezoidal rule

3.Composite Simpson's rule

Problem solved by Hailong Chen.

title: User:Egm6341.s11.team4.hylon/hw3