text:
User:Egm6341.s10.team3.ynahn81/HW4-8(c)

(c) Compare Comp Trap, Comp Simp, Romberg for $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$
Ref: Lecture notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Problem Statement
For $$\displaystyle I = \int_{0}^{2\pi}\frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}d\theta$$, compare $$\displaystyle I_{n}$$ to $$\displaystyle O(10^{-10})$$ and computational time for - Composite Trap rule, - Composite Simpson's rule and - Romberg table.

Solution
In what follows, we are using the follow matlab function as a given function $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$.

i) Composit Trap. rule

Recall the formula of comp. Trap. rule.

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

T_{0}(n) = h\left[\frac{1}{2}f(x_{0}) + f(x_{1}) + f(x_{2}) + \cdot\cdot\cdot + f(x_{n-1}) + \frac{1}{2}f(x_{n}) \right] $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, realize the fact that if we want to double the number of sub-intervals, we don't have to calculate the functions at even point, i.e. $$\displaystyle f(x_{0}), f(x_{2}), \cdot\cdot\cdot, f(x_{n-2}), f(x_{n}) $$ because we already calculated the functions at these even point at the previous step. Therefore, all we have to do is evaluating the functions only at the odd points and combine them with the calculated values at the previous step. As a result, we can use the following inductive formula.
 * {| style="width:100%" border="0" align="left"

T_{0}(2n) = T_{0}(n) + h\sum_{i=1}^{n}f(x_{2i-1}). $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,trap} = T_{0}(n) = 6.069090959564874, \quad n = 2^{4} = 16 $$ $$\displaystyle E_{n,trap} = I - I_{n,trap} = 1.039168751049147e-13 $$

ii) Composite Simpson's rule

Recall the formula of comp. Simpson's rule.

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

T_{1}(n) = \frac{h}{3}\left[f(x_{0}) + 4f(x_{1}) + 2f(x_{2}) + 4f(x_{3}) + \cdot\cdot\cdot + 4f(x_{n-3}) + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_{n}) \right] $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, realize the fact that if we want to double the number of sub-intervals, we don't have to calculate the functions at even point, i.e. $$\displaystyle f(x_{0}), f(x_{2}), \cdot\cdot\cdot, f(x_{n-2}), f(x_{n}) $$ because we already calculated the functions at these even point at the previous step. Therefore, all we have to do is evaluating the functions only at the odd points and combine them with the calculated values at the previous step. As a result, we can use the following inductive formula.
 * {| style="width:100%" border="0" align="left"

T_{1}(2n) = T_{1}(n) + h\sum_{i=1}^{n}4f(x_{2i-1}) - \underbrace{h\sum_{i=1}^{\frac{n}{4}}2f(x_{4i-2})}_{Already\,\, obtained\,\, when\,\, T_{1}(n) \,\, was\,\, calculated.}. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,Simpson} = T_{1}(n) = 6.069090959564741, \quad n = 2^{5} = 32 $$ $$\displaystyle E_{n,Simpson} = I - I_{n,Simpson} = 2.930988785010413e-14 $$

iii) Romberg Table

Since the first column of Romberg table corresponds to Trap. rule, we can use the below formulas in order to obtain the first column of Romberg talbe.

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

T_{0}(n) = h\left[\frac{1}{2}f(x_{0}) + f(x_{1}) + f(x_{2}) + \cdot\cdot\cdot + f(x_{n-1}) + \frac{1}{2}f(x_{n}) \right] $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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

T_{0}(2n) = T_{0}(n) + h\sum_{i=1}^{n}f(x_{2i-1}). $$ where $$\displaystyle h = \frac{b-a}{n} $$ and $$\displaystyle x_{0} = a,\quad x_{n} = b $$.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Also, we gradually set up the Romberg table in parallel, as soon as the result for $$\displaystyle T_0(2^j) $$ becomes available using the below formula.

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

T_{k}(2n) = \frac{2^{2k}T_{k-1}(2n) - T_{k-1}(n)}{2^{2k}-1}. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

Matlab code

Result

$$\displaystyle I_{n,Romberg} = T_{0}(n) = 6.069090959564874, \quad n = 2^{4} = 16 $$ $$\displaystyle E_{n,Romberg} = I - I_{n,Romberg} = 1.039168751049147e-13 $$

iv) Compare computational time

We can compare computational times of above three methods using tic/toc commands in Matlab.

Matlab code

As above table shown, Simpson's rule has the shortest computational time. However, we should note that the computational time does not really different from each other for three method. Especially, trap rule has a shorter computational time than Romberg Table (in general, computational time of Romberg Table is much shorter than that of Trap. rule.). Since the given function $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$ is a periodic function, the numerical integration of this function is converged very fast by Trap. rule. Therefore, '''Trap. rule is an efficient way to numerically integrate the given function $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$ because this function is a periodic function.'''

title: User:Egm6341.s10.team3.ynahn81/HW4-8(c)


text:
Pulmonology/Mechanical Ventilation

 * Indications for Mechanical Ventilation
 * Initial Ventilator Settings
 * Adjusting Mechanical Ventilation to patient developments.

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

title: Pulmonology/Mechanical Ventilation


text:
Talk:Introduction to Swedish/Numbers and plurals

Fröer
I'm from Sweden and I have never heard the word fröer.. plural of frö is frön. Ameki 23:46, 1 March 2010 (UTC)
 * Fixed! --83.253.232.103 00:00, 14 November 2010 (UTC)

Okay, if the plural of "frö" is corrected to "frön", is there another word to be given in its place as an example of plural type 3b (-er)? "Frön" doesn't belong there any more, if it doesn't have the "-er" ending. LS99.153.246.4 17:49, 13 January 2011 (UTC)


text:
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User talk:WorldEducator~enwikiversity

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title: File:Hw4-prob5.JPG


title: File:Hw4-prob7.JPG


text:
User:Rjorjorjo/HW4

=Problem 5-Prove the Equality (integration by parts)=

Statement
Given $$ \displaystyle g_{k}(t):=\,f(x(t)):\,\,\,x\,\,\in\, [\,x_k,x_{k+1}\,]\,$$

$$ \displaystyle \int_{-1}^{+1}g^{(1)}(t)dt =\, \int_{-1}^{+1}g(t)dt -\,[\,g(-1)+\,g(+1)\,]\,$$

 * {| style="width:100%" border="1" align="left"

 * Ref: Lecture Notes [[media:Egm6341.s10.mtg21.pdf|P21-2]]
 * }
 * }

Solution
Use Integration by parts [[media:Egm6341.s10.mtg5.pdf|P5-2]] $$ \displaystyle \int_{a}^{b}u^{'}vdt =\, [\,uv\,]_{a}^{b}-\,\int_{a}^{b}uv^{'}dt\,$$

Let $$ \displaystyle g^{(1)}(t)=\,u^{'}\,\,\,(-t)=\,v\,$$

 * $$ \displaystyle

\begin{align} \\\int_{-1}^{+1}g^{(1)}(t)(-t)dt &=\, [\,g(t)(-t)\,]_{-1}^{+1}-\,\int_{-1}^{+1}g(t)(-1)dt\; \\ &=\, -\,g(+1)-\,(g(-1))+\,\int_{-1}^{+1}g(t)dt\; \\ &=\, \int_{-1}^{+1}g(t)dt-\,[\,g(-1)+\,g(+1)\,]\,. \end{align} $$

=Problem 7-Download and Install chebfun package=

Statement
Install chebfun, a Matlab code package wrote by Trefethen et al.(2004)

 * {| style="width:100%" border="1" align="left"

 * Ref: Lecture Notes [[media:Egm6341.s10.mtg25.pdf|P25-1]]
 * }
 * }

Install
Use this website to download Matlab package |chebfun

Test with two quick examples: (1)What is the integral of $$ \displaystyle sin(sin(x)) \,$$ from 0 to 10? (2)What is the maximum of $$ \displaystyle sin(x)+\,sin(x_{2})\,$$ over the same interval?

-> installed successfully

(3)calculate the integration of $$ \int_{0}^{1}\,\frac{e^{x}\,-1}{x}\mathrm{d} x $$

Use chebfun : Matlab code :

Get Ich = 1.317902151454403  Ech = 4.403366560268296e-012 (4) How long for using chebfun to calculate?

Get Elapsed time is 0.462024 seconds.

EGM6341.s10.Team1.Ya-Chiao Chang 14:18, 7 March 2010 (UTC)

title: User:Rjorjorjo/HW4


text:
User:Bolo1910

 * I am an active user of Russian Wikiversity.
 * My home page is Участник:Bolo1910.

ru:Участник:Bolo1910

title: User:Bolo1910


text:
University of Canberra Strategic Plan 2008-2012

(adopted by the University Council at its meeting of 7 December 2007).

Context
In our first forty years as an educational institution the University has achievements of which we can be justly proud. Over 60 000 students have graduated from us, and many have gone on to senior and responsible careers in the public service, commerce and the professions. Others have become prominent writers, journalists, performers and sports men and women. As the period covered by this Plan commences, we rank in the top third of Australian universities under the Commonwealth Government’s Learning and Teaching Performance Fund. Our research strengths in ecology and the modelling of social and economic change have an international reputation.

The year in which this Plan was developed (2007) became seen as a turning point. Declining student load, and to a lesser extent student enrolments, had led to budgetary difficulties and the need to re-think our future. We restructured our academic units and administrative services. We reviewed our curriculum. We thought hard about a distinctive, differentiated mission for the University. We made plans for the development of our campus over the next ten years.

We undoubtedly face some challenges:

 * To attract sufficient students in a period when the proportion of the population of traditional university age has been diminishing, the actual size of the 0-19 population in the ACT has reduced, and the share of the population with an undergraduate or postgraduate degree is already far higher than the national average;
 * To deliver courses in a way that best suits the busy life of modern students;
 * To attract and retain research-active educators who will improve the University's academic performance;
 * To invest in information technologies which can promote our education, research and administration; and
 * To measure, contain and offset our impact on the environment.

Outweighing these challenges are important comparative advantages we enjoy over many universities of our generation.

 * Our Name: We carry the name of the nation's capital, and this in itself is an asset, particularly internationally;
 * Our City: Canberra is the exemplar of a designed, knowledge-based city-state in a service economy, from which we can draw inspiration;
 * Our Location: We are centrally located in the Australian Capital Territory and close to all the public sector and national institutions in the capital;
 * Our People: We have a University community with high levels of professional experience and active professional networks;
 * Our Supporters: Many of our graduates occupy leadership positions in federal and local government, national institutions and the professions: here and overseas. Many others who are not graduates of the University would welcome a university dedicated to supporting professional people who serve the public;
 * Our Disciplines: Many of our disciplines are not offered by the other local tertiary institutions, and there is further inter-disciplinary potential within the University which is also not open to them; and
 * Our Campus: We have a large, under-developed site in an area of appreciating land values which for all practical purposes will never be fully required for direct educational and research purposes.

Along with only a few other capital cities around the world, Canberra is most dominated by public administrations, national institutions and organisational headquarters, and by private businesses and professions which exist to support them and their workforce. We can offer professional education and applied research tailored to the needs of a distinctive capital city, with all its international, national and local dimensions.

Although we have an older competitor university locally - which has been specially funded for research for over 50 years - its national focus should be on theoretical and pure research across many disciplines. Our focus for inquiry should be on the nature and delivery of knowledge-based and creative services for the various communities of a capital city and its regions.

Canberra will benefit from two, strong, differentiated universities.

We can realistically aim to be amongst the world's best universities at understanding, researching and educating for responsible public administration and services: whether those services be governmental, environmental, educational, health-related, creative or in the course of business.

Our discipline strengths enable us to take fully into account the wider context of globalisation, political insecurity and danger to the environment.

To achieve this, we need to re-make ourselves. We need strategies that:

 * strengthen the University's capabilities, responsiveness and competitiveness;
 * respond to demographic and social change by making us more attractive to changing student populations;
 * deliver the best possible service to students, most of whom are paying all or a significant proportion of the cost of their education and who may in the future have greater choice about where to study;
 * enable us to take advantage of projections that international students wanting to study in Australia will nearly double by 2025;
 * ensure we have a well-deserved reputation for high quality education;
 * promote the quality, quantum and impact of our research, in an era where this is publicly measured and contributes significantly to the standing of the University in the world; and
 * give us choices about potential partners for sustainability and growth.

This Plan sets out a ten-year vision and the strategies and steps proposed for the first five years to re-make ourselves as an institution, capitalise on our inheritance and comparative advantages and achieve an ambition to be a world-class university of the capital city.

Purpose
The University of Canberra is the university of the capital city. Our role is to serve Canberra, its surrounding communities and its partners through education and research. In doing so, we reflect the different dimensions of Canberra:

 * the nation's capital and the international implications of that status;
 * the seat of government and public administration;
 * the home of national cultural, scientific and sporting institutions;
 * a symbol of Australian identity;
 * a designed and landscaped city; and
 * a local community committed to a sustainable, educated, healthy, prosperous life.

We share many features of other high quality universities, but the way we reflect the dimensions of a unique city differentiates us from all other Australian universities.

Values
We value:

 * service to our students and our disciplines;
 * education as a transformative experience for all people irrespective of their origins, age and circumstances, to be used for the public good;
 * research which has as its end the improvement of culture, society and the natural environment, for the benefit of generations yet to be born.
 * vocational relevance to longstanding and emerging professions; and
 * entrepreneurial spirit that drives innovation and growth.

Vision
By 2018 the University of Canberra will be internationally recognised for its research-led education in public administration and services; whether those services be governmental, environmental, educational, health-related, creative or in the course of business. We will draw students and researchers from around the world because of our reputation for programs which meet the current needs of modern governments, public services and the professions which support them. Specifically:

 * we will be ranked in the upper half of Australian universities in reputable institutional indices, and in the top third for our educational performance;
 * we will have 10,000 full-time equivalent students, with average funding 20% higher in real terms than in 2008;
 * we will have a campus community with a vibrant cultural and recreational life, famous internationally for its musical events and festivals;
 * we will have excellent sporting facilities and be home to sporting clubs from inside and outside the University;
 * we will be at the heart of the Bruce Precinct, which will be a centre of innovation, education, research and health services, with a sustainable ecological footprint;
 * our main campus will accommodate numerous public, non-profit and private enterprises which contribute to the University's education and research activities and to a thriving university community in an Australian setting;
 * we will use information and communication technologies to their full potential;
 * our courses will be known for the way they prepare professional people professionally, through work-based learning and collaborative teaching with external agencies;
 * our research and creative practices will be applied to making the world a better and more secure place for all its inhabitants.

Progress
We will have reached this position in two stages.

In the period covered by this Plan, 2008-12, we will strengthen the organisation's foundations, increase our student numbers to 9000 full-time equivalents, consolidate our ranked position on educational indicators, improve our research performance and engage more effectively with our constituencies. We will have invested $100 million during this period, in addition to our normal operating revenues, to implement the strategies which follow and thus revitalised the University.

By the beginning of the period 2013-17 we will be in a position to invest massively in people, curriculum and infrastructure, through growth in course fees, consulting and research income, and revenues generated directly and indirectly by the campus. The basic metric of total revenue divided by total student load will be rising, and we will be on track for an increase of 20% in real terms compared with 2008. OUR PLANNING

Ours is a planned future. We have:

 * a ten-year vision for the University and its campus;
 * five strategies covering the first five years;
 * an academic plan which describes how we will implement the academic strategies;
 * a capability plan which describes how we will implement the organisational strategies; and
 * operational plans which describe detailed actions annually and how we will resource them.

This framework, depicted in the pyramid below, is underpinned by a Quality Cycle designed to ensure continuous improvement. UC Planning Pyramid

STRATEGY 1 - STRENGTHEN THE FOUNDATIONS
Step 1: Ensure that respect for Australia's traditional owners and concern for their current circumstances influences our plans and actions.

Step 2: Create a great workplace which attracts, engages and retains excellent, entrepreneurial, diverse staff committed to higher education and research with public purposes, in a collegial environment.

Step 3: Implement a Quality Cycle and Framework to assure and improve continuously the quality of what we do.

Step 4: Streamline our procedures and cost structures, making full use of information technologies and eliminating paper transactions wherever possible.

Step 5: Introduce a Budget Model based on cost allocation, which links resources, strategy and outcomes.

Step 6: Encourage a service-oriented culture designed to meet the needs of our students and other stakeholders.

Step 7: Diversify and extend our sources of revenue by making full use of our campus, commercial opportunities arising from our teaching and research, and the support of our alumni.

Step 8: Create an endowment fund into which proceeds from the development of the campus can be put, for the long-term benefit of the University.

Step 9: Eliminate our deferred maintenance backlog.

Step 10: Begin implementation of the Campus Masterplan 2008.

Step 11: Maintain a clear scheme for the performance development and review of all staff, designed to promote career development and reward high achievement.

Step 12: Improve our domestic and international marketing capacity and impact.

Step 13: Re-position the University in the minds of our communities, stakeholders and markets as Australia's Capital University: the university of and for the capital city.

STRATEGY 2 - INCREASE OUR STUDENT LOAD TO 9000 EFTSL BY 2013
Step 14: Develop and implement an ambitious student equity and access agenda.

Step 15: Provide a great student experience, appropriate to the age, stage, background and circumstances of a diverse student population.

Step 16: Introduce a new curriculum from 2009 comprising courses at which we can be distinctively good, which are in demand and which fit with our position as Australia's Capital University.

Step 17: Review our semester system and modalities of course delivery with a view to being attractive to new kinds of well-qualified students.

Step 18: Make the best use of educational technologies and work-based learning opportunities.

Step 19: Strengthen our relationships with Senior Secondary Colleges and Institutes of Technical and Further Education to promote entry into and articulation with the University's courses.

Step 20: Develop University of Canberra College as a pathway for students who lack the immediate qualifications for direct entry to the University.

Step 21: Develop profitable articulation pathways and transnational education programs with high quality overseas universities, with whom we can also collaborate in research or consultancy.

Step 22: Improve strategies and practices for recruiting domestic and international students.

Step 23: Increase the availability of high quality, affordable residential accommodation on and near the campus.

STRATEGY 3 – PERFORM IN THE TOP THIRD OF UNIVERSITIES ON STANDARD EDUCATIONAL MEASURES BY FOCUSING PARTICULARLY ON SELECTED SCALES WHERE WE CURRENTLY PERFORM BELOW THAT LEVEL
Step 24: Improve students' overall satisfaction with the experience of their course, as measured in the Course Experience Questionnaire.

Step 25: Improve students' satisfaction with our teaching, as measured in the Course Experience Questionnaire.

Step 26: Improve students' satisfaction with the way we help them develop generic skills, as measured in the Course Experience Questionnaire.

Step 27: Improve graduates' rates of further study, as measured in the Graduate Destinations Survey. STRATEGY 4 - PERFORM IN THE TOP HALF OF UNIVERSITIES ON PER CAPITA RESEARCH MEASURES ($10m)

Step 28: Provide research training places and post-doctoral fellowships to increase our research output and develop the next generation of potential academic staff members.

Step 29: Devise a workload model which ensures that all academic staff members engage in high quality and high impact research but which also frees up the time of our most successful researchers.

Step 30: Increase the research degree qualifications of academic staff.

Step 31: Strengthen the University's research culture, through mentoring, staff development and appropriate standards in the selection, confirmation, promotion and time-release of academic staff.

Step 32: Increase the University's research income, and ensure that an appropriate amount of the extra revenue which this generates is returned for the benefit of successful researchers' work.

Step 33: Develop the Institute for Applied Ecology and the National Centre for Social and Economic Modelling into world-class, cross-faculty research entities by 2012, and create at least three further national-best, cross-faculty research groupings.

STRATEGY 5 - ENGAGE EFFECTIVELY WITH THE WORLD AROUND US
Step 34: Internationalise our curriculum, culture, relationships and outlook.

Step 35: Expand our relationships in East Asia and South Asia.

Step 36: Establish The University of Canberra Centre for the Capital City ("UC4"), as an embodiment of our commitment to Canberra, its heritage and future, and as a vehicle for collaboration with key capital cities, including Beijing and New Delhi.

Step 37: Develop strategies in an Engagement Plan to strengthen our relations with alumni, government, the public sector, business and industry, and the cultural, sporting and social welfare communities.

Step 38: Encourage collaboration in teaching and research with the Australian National University, the Australian Catholic University, the Australian Defence Force Academy and the Canberra Institute of Technology.

Step 39: Set and meet ambitious targets and standards, as a signatory to the Talloires Declaration, to reduce our ecological footprint.


text:
Talk:University of Canberra Strategic Plan 2008-2012

The reason I have copied this strategic plan into Wikiversity is to work with UC colleagues in analysing it, and aligning our principles of open education to it. The original publication is here. --Leighblackall 05:37, 2 March 2010 (UTC)


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User talk:Munnerley

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Recent Changes Camp Canberra Aug 11, 2010
I saw your profile on the Wikiversity page for University of Canberra staff, and thought, by chance, you might like to come to this: RecentChangesCamp, Canberra is being held at the University of Canberra, Building 7, Room 7XC37 on 11 August 2010. ABOUT | REGISTRATION | SCHEDULE

Hope we'll see you and friends there. Leighblackall 00:26, 21 July 2010 (UTC)

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 2 March 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 2 March 2010 (UTC)

Welcome
Hi, welcome to Wikiversity. I moved your blog to a user subpage: User:PalmdaleHermit/blog. Please feel free to post about how Wikiversity might be improved to help with your learning interests and needs. Sincerely, James. -- Jtneill - Talk - c 22:44, 2 March 2010 (UTC)

Firefox
Apologies if I'm sharing things you already know, but I find CTRL+ in Firefox very helpful (zooms text). -- Jtneill - Talk - c 22:47, 2 March 2010 (UTC)

Hello
I just noticed you added yourself to Topic:Tibetan. Let me know how you go with it. I have learnt the script and can type Tibetan but am still refining my lexicon and my grammar is poor. But I am more than happy to help with your learning initiative in whatever way possible. Best of luck. B9hummingbirdhoverin'æ•ω•ॐ 13:24, 3 March 2010 (UTC)
 * The only thing I'm exclusive about is being inclusive. Have you enabled your computer to input Tibetan script? What operating system are u running?
 * B9hummingbirdhoverin'æ•ω•ॐ 14:23, 10 March 2010 (UTC)

 * I started using TISE initially. Chris Fynn did a wonderful job with that. But I must say I love Linux and with SCIM I can easily change between many scripts and input systems such as Devanagari which I do as much as Tibetan. I am a touch typist and I learnt the script really quickly by transcribing PDFs of Tibetan such as you can get here and then when I had the Wylie, type the Wylie into Uchen, that is Tibetan script, and compare the Tibetan script, the Uchen, back to the PDF for probity. Making any repairs required. So that way I yielded practical Unicode as well as learnt very quickly. How are you doing? Please forgive me I don't come into Wikiversity alot at the moment as I am working on other projects. I apologize for my tardy response. Do you favour any other communication channels/platforms?
 * B9hummingbirdhoverin'æ•ω•ॐ 10:25, 21 March 2010 (UTC)
 * You mentioned a dictionary and that you are interested in colours. There is a colours category that I started. I started adding Tibetan terms on Wiktionary and that could be something practical you could involve yourself with as you learn if you are interested.
 * http://en.wiktionary.org/wiki/Wiktionary:About_Tibetan
 * http://en.wiktionary.org/wiki/Category:Tibetan_language
 * B9hummingbirdhoverin'æ•ω•ॐ 10:34, 21 March 2010 (UTC)

hello
how is the Tibetan going? B9hummingbirdhoverin'æ•ω•ॐ 01:55, 30 April 2010 (UTC)

title: User talk:PalmdaleHermit


text:
User talk:James.hulse

 Hello James.hulse, 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. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
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 * Give feedback about your initial observations
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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 2 March 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 2 March 2010 (UTC)

title: User talk:James.hulse


text:
User talk:Hhuq

 Hello Hhuq, 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. 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.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 2 March 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 2 March 2010 (UTC)

title: User talk:Hhuq


text:
User talk:Lizzia620

 Hello Lizzia620, 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. 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.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 2 March 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 2 March 2010 (UTC)

title: User talk:Lizzia620


text:
Open license

Open licensing refers to the use of copyright licenses which allow a greater degree of freedom than the default all rights reserved.

Examples of open licensing systems include:
 * Creative Commons
 * GFDL

Governments which have a default is open approach to public service information:
 * New Zealand
 * UK
 * Australia

title: Open license


text: Category:Open management

title: Category:Open management


text: Category:Open academia

title: Category:Open academia


text: Category:Open licensing

title: Category:Open licensing


text:
Teaching EFL Listening via FUN WITH ENGLISH Books/Introductory Lesson

Preface
My first week at this positon started on Wednesday, September 1st 2009. Until that Friday I was basically doing version 0.1 (based this lesson largely off lessons I gave in Japan the year prior) of this plan to 50-ish students in their own classrooms. After that my colleague Mr. Brown arrived and I taught version 2.0 to 25+ kids in the English Salon. Overall they went OK, nobody got killed, but I ran my first class of second semester quite differently. I'd recommend basing your first lesson off the latter, not this. I was too much of a friend here, by round two I introduced myself as their teacher.

Introduction
Learning Outcome: Students and teacher will learn one another's names, and get to know each other.

Plan

 * 1) Introduce Myself
 * 2) * 4 Minutes
 * 3) * Name, country, home city, favorite foods and sports.
 * 4) * Check to see if they understand.
 * 5) Have students make a line.
 * 6) * 5 Minutes
 * 7) * Tell students they have to line up by birthday, and that they can only speak in English.
 * 8) Name and Pictures
 * 9) * 10 Minutes
 * 10) * Tell students to come forward, shake hands with me and tell me their English and Chinese names.
 * 11) * I will then take a picture and they have to write their names down, along with their favorite food.
 * 12) * Afterwards, students have to fill in the desks accordingly, not where they were sitting before.
 * 13) Students Interview One Another
 * 14) * 20 Minutes
 * 15) * Students will interview one another to introduce their partner to the class.
 * 16) * Name? Favorite Food? Least Favorite Food? Hobby? Dream? Why do you want to study English? Fear?
 * 17) ** Pick 3 questions, based on their level.
 * 18) Ask if Students Have Any Questions for Me
 * 19) * 5 Minutes
 * 20) * If they don't move on.

Extra Time

 * 1) 2 Truths, 1 Lie
 * 2) * 15 Minutes
 * 3) * Ask if they know what a lie is?
 * 4) * Explain that we are going to play a game where they will write down 2 truths & 1 lie.
 * 5) * Prompts:
 * 6) ** My favorite food is ...
 * 7) ** My dream job is ...
 * 8) ** I am afraid of ...
 * 9) ** My favorite music is ...
 * 10) ** My favorite subject is ...
 * 11) Hangman
 * 12) * 5 Minutes

Acknowledgments
Idea for negotiation rules with the kids came from my boss at East Meets West China at our 2009 Orientation Meeting in Shanghai.


text:
Emerging scholar

Overview
In a collaborative learning environment, the roles of "student" and "teacher" are blurry. At various times, and often simultaneously, teachers are students and students are teachers. If such roles are artificially separated, problems can ensue, not the least of which is the insidious seduction of power and control by "teachers" over "students".

For active, experiential learning to occur, emerging scholars should be permitted and encouraged to engage in constructive nonconformism; in so doing, they are likely to transcend the constrained, "industrial" passive student role.

This pages explores these concepts by suggesting more emancipatory nomenclature for roles played by those participating in collaboratively organised learning environments.

Alternative terms for student
This page has arisen out of dissatisfaction with common use of the term "student" in institutionalised teaching and learning, and a search for more empowering language. Some alternative terms that have been suggested are "emerging scholar", "learner", "participant", and "future leader".

What other terms can you suggest? When you are a learner, how would like to be described and thought of by others?

Rogers continually lamented contemporary educational practices. He did not like the idea of a “teacher” because he felt that the only learning that really mattered was self-initiated learning (Rogers, 1969). Little of consequence occurs when a teacher gives out heaps of information for students to digest. Instead of “teacher,” Rogers preferred “facilitator,” a term that describes the classroom leader as one who creates and then supports an atmosphere conducive to students’ learning. Learning does not follow teaching. Rather, learning follows having one’s interests, goals, and aspirations identified and supported. Personal initiative and self-evaluation are of prime importance. Thus, education is not something a teacher can give to (or force on) a student. Rather, education must be acquired by the student through an investment of his energies and interests. - Johnmarshall Reeve, 2018, p. 382

Emerging scholar
"Emerging scholar" (or "emerging academic") offers an alternative the student-teacher dichotomy. Emerging scholar is an emancipatory term which emphasises scholarly study as an ongoing act of collaboration and collegiality. An emerging scholar engages in scholarship through knowledge-sharing communities such as learning institutions to help learn about and pursue the tasks of academia.

The notion of an emerging scholar offers a more empowering, equitable, and developmental conception of a learner's role and potential.

The "emerging academic" term was inspired by an Otago Polytechnic researcher, Russell Butson (communicated ~2010 by Leigh Blackall to James Neill).

The "emerging scholar" term was used in this tweet by Kelly Matthews, University of Queensland (https://twitter.com/UQkelly/status/786725549553045504).

Learner
"Learner" is an active term, emphasising the act and process of developing skills and knowledge. A student can sit through school and "learn nothing" - but can a learner? Conceptualising a person's role as learner suggest belief in the person's capacity for development.

Participant
Another useful, non-provocative, term is "participants". Everyone who participates in a learning activity can play different roles at different times in terms of leading, guiding, responding etcetera.

Future leader
Often developing sufficient knowledge and skills to then be able to lead others is an educational goal. In such contexts, "future leaders" can be used to "forward-think" the role and purpose of the educational program. For example, UniJobs' University of Canberra Lecturer of the Year (2011-2012), Michael DePercy explains that: "“The attitude you have towards your students is important, I don’t refer to them as kids, I refer to them as future leaders, because that’s what they can be. I try to build them up to be future leaders of their community.”"

Alternative terms for teacher
The notion of a "student" also implies that there is a "teacher". Used in institutionalised education, these terms tend to be used to set and reinforce distinct hierarchical roles. But in collaborative learning and discovery, the teacher role can usually be better described using alternative terms such as "facilitator". "instructor", "leader", or "academic".

What other terms can you suggest?

title: Emerging scholar


text:
Talk:Going naked - Openism and freedom in academia

Feel free to discuss here or in the Responses section.

Gatekeeper function
Academic institutions are currently charged/trusted with a public gatekeeper function. They give recognition (degrees, etc.) to people who have completed a certain course of study and/or passed examinations which then qualify graduates for future employment responsibilities. I can foresee "openism" leading to a more apprenticeship/guild based approach to employment. Do you see the demise of the ladder of academic awards to be replaced by public/social usefulness?--Harkey 16:05, 2 March 2010 (UTC)
 * Hi Harkey, Thanks for the question/comments. Some thoughts: At least in Australia, we are about to see significant deregulation of the tertiary sector such that the definition and funding of "universities" will be freed up. This will foster relations between universities and vocational education providers (which have traditionally been more job skills focused). However the "gatekeeper" function is probably what largely keeps universities in business - i.e., having a good reputation and granting degrees (its the largest export industry in Australia) - so I imagine universities will be keen to retain this business model. But the part of university funding which comes from governments seems likely to increase the public/social usefulness requirements, although it remains to be seen how the metrics for this evolve. -- Jtneill - Talk - c 21:19, 2 March 2010 (UTC)

The video
Thanks for adding the video. Will you note down the steps on the page Session recording? I notice that it is significantly out of synch however.. I think this happens during the compression stages. Did you keep the files at each step? Has the syncing problem only happened once it is embedded in the wiki, or did it happen steps back in the process? II notice that the Youtube versions are good.. --Leighblackall 21:33, 22 March 2010 (UTC)
 * This .ogv seems to work/not work very much depending on browser/system - in Firefox 3.5 on WinXP its good, but in IE8, Google Chrome etc. it doesn't play very well at all. Maybe the problem was SuperC conversion - maybe its MediaWiki, I'm not sure. I've documented the generic steps: Screencasting. Seems that .ogv video needs experimentation/testing. -- Jtneill - Talk - c 23:15, 22 March 2010 (UTC)

Learning about Open Academia
I blogged about this topic some today. Wikibooks' blog, Wikimedia's tech blog, some recent events, and Leigh's blog got me a bit motivated to work on a learning blog myself. -- dark lama  16:57, 28 March 2010 (UTC)
 * Tx for blogging your initial thoughts (and for the discussion posting). We ran an open academia in practice workshop yesterday - which was as much focused on practical "how to" for university teaching staff as this "going naked" session focused on theory. I'm finding this to be an interesting way of approaching academia conversations locally: theory and practice. -- Jtneill - Talk - c 22:56, 29 March 2010 (UTC)
 * I should have said more probably. The page does a good job of speaking for itself, and I had already on the page itself commented on it. Yesterday on IRC there was a lengthy discussion that could be said to be related to this very subject. I may due a part II related to the ideas discussed. -- dark lama  23:32, 29 March 2010 (UTC)
 * I have done a follow up blog entry about the workshop for open academia in practice. I also briefly mention the idea that was discussed on IRC. -- dark lama  21:13, 31 March 2010 (UTC)


text:
User:Gaidheal1

Welcome!

I am from Edinburgh, Scotland. I have finished my second year studying piano in Birmingham Conservatoire, and I'm looking forward to next year, although with a little trepidation. I am a skilled accompanist and choral singer, and have been both singing and playing piano for over 14 years. From 2007 to 2009 I sang as a choral scholar in St. Philip's Cathedral, Birmingham.

Today I accompanied a lovely Alto in the final round of the English Song Prize at the conservatoire: I was the only undergraduate among the group of singers/accompanists! I didn't win anything, but was very happy with my performance.

To work on

 * Ear training
 * User:Gaidheal1/Organising Musicianship course

Navboxes

 * Musical instruments

Subpages

 * All the 'nowrap' related subpages which are now moved to template space
 * Various other templates/pages
 * Piano and Keyboards department

Pages that look interesting

 * Star Stuff

title: User:Gaidheal1


text:
User:Wardell

Okay... Well I'm Not sure how this goes but I just want to get my account starded to see what's going on in this university...

title: User:Wardell


text:
User:EGM6341.s10.Team1.toddmock/HW4

= Problem 21-1: Verify Change of Variables =

Given
Higher Order Error for Trap. rule equation:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \ E_n^1= \sum_{k=0}^{n-1}[ \int_{x_k}^{x_{k+1}}f(x)dx- \frac{h}{2}(f(x_k)+f(x_{k+1}))]
 * }
 * }

Find
Show that using change of variables this equation becomes

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))]
 * }.
 * }.

Where

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle g_k(t)=f(x(t))
 * }.
 * }.

And

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle x(t)=t\dfrac{h}{2}+\dfrac{x_k+x_{k+1}}{2}
 * }.
 * }.

Solution
Using the function
 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle x(t)=t\dfrac{h}{2}+\dfrac{x_k+x_{k+1}}{2}
 * }.
 * }.

This can be changed into

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle x(t)=\dfrac{h}{2}(t+1)+x_k
 * }.
 * }.

This allows us to evaluate the function $$x(t)$$ at the endpoints to obtain

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle x(-1)=x_k
 * }
 * }

And

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle x(1)=x_{k+1}
 * }.
 * }.

Next using

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle g_k(t)=f(x(t))
 * }.
 * }.

we can transform

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle f(x(t))dx
 * }
 * }

into

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle g_k(t)dt
 * }
 * }

by using the derivative

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \dfrac{d(x(t))}{dt}=\dfrac{h}{2}
 * }
 * }

Thus giving

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle d(x(t))=\dfrac{h}{2}dt
 * }.
 * }.

Placing all terms into the original equation with the new limits gives
 * {| style="width:100%" border="0" align="left"

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle E_n^1= \sum_{k=0}^{n-1} \int_{-1}^{1}g_k(t)\dfrac{h}{2}dt-\dfrac{h}{2}(g_k(-1)+g_k(+1))
 * }.
 * }.

Thus producing

 * {| style="width:20%" border="0"

$$\displaystyle E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))] $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

--Egm6341.s10.team1.toddmock 19:01, 5 March 2010 (UTC)
 * }
 * }

= Problem 25-2(3): integration of ellipse arc length =

Given
For $$\displaystyle I = \int_{0}^{2\pi}\frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}d\theta$$,

the arc length integral is:

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

I= 4\int_0^{\pi/2}\sqrt {1-sin^{2}(\frac{\pi}{12}) \sin^2\theta}\ d\theta $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

compare MATLAB trapezoidal rule, Simpson's rule and Chebfun computational times.

Find
Total value of elliptical arc length (circumference) using built in MATLAB codes

Solution
For the Trapezoidal rule

For the Simpson's rule

And for the Chebfun function

As can be seen from the table above the quad function (simpson's rule) produces a result much faster than any of the other built in MATLAB functions --Egm6341.s10.team1.toddmock 19:02, 5 March 2010 (UTC)

title: User:EGM6341.s10.Team1.toddmock/HW4


text: File:New.ogg

title: File:New.ogg


text:
Boundary Value Problems/BVP-ODE-1st-order

Return to main course page Boundary Value Problems

Video of solving first order (click on the PLAY BAR) ODE


text:
User:Akshay.marwah

hi this akshay marwah from allahabad india.presently i studying at united college of engineering & management allahabad which is affliated to uptu...so i m here to get the most of knowledge and render it to my fellow students who are anxiously studying to grab the best of it!!!!

title: User:Akshay.marwah


text:
User talk:Akshay.marwah

 Hello Akshay.marwah, 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. 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.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. Jtneill - Talk - c 21:41, 2 March 2010 (UTC)

 Hi! My name is WikiversityJack. I am a pumpkin and a Wikiversity mascot. My aunt told me that fun + learning = Wikiversity. Let me know if I can help you out.

title: User talk:Akshay.marwah


text:
User:Egm6341.s10.team3.Min/HW4

6(a) Install Chebfun and practice sum commond
Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-1]]

Problem Statement
Install chebfun and practice sum command to do integration$$\displaystyle I = \int_{0}^{1}\frac{e^{x}-1}{x}dx$$, and record the computational time

Solution
i) Chebfun installation The chebfun can be downloaded from http://www2.maths.ox.ac.uk/chebfun/files.html

We installed the "chebfun" successful based on the instruction which is also availabe in the above webpage. In chebfun, 'sum' command will returns the definite integral of a chebfun over its range of defination.

The integral is normally calculated by an FFT-based variant of Clenshaw-Curtis quadrature.

This formula is applied on each fun and then the results are added up.

The chebfun will be used in the following problem to show comparision of speed among different numerical methods for integration.

For detailed information, one can refer the paper: Battles and Trefethen, An extension of Matlab to continuous functions and operators (SIAM J. Sci. Comp., 2004)

ii) Chebfun, Sum command

We use the command 'sum' in chebfun to compute I with error to $$ 10^{-10}$$  Result:

I_c=1.317902151454404 E_c=4.403588604873221e-012

Elapsed time is 0.139211 seconds.

-- By Min Zhong 12:00, 02 March 2010 (UTC)

6 (d) compare trapz, quad, clencurt
Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Problem Statement
For $$\displaystyle I = \int_{0}^{1}\frac{e^{x}-1}{x}dx$$, compare $$\displaystyle I_{n}$$ to $$\displaystyle O(10^{-10})$$ and computational time by - Matlab : trapz, quad and clencurt

Solution
i) Trapz

We use the command 'trapz' in matlab to compute I with error to $$ 10^{-10}$$   Result

I_t = 1.317902151493208

n_t = 32768

E_t = 3.880429311209355e-011

Elapsed time is 0.753827 seconds

ii) Quad

We use the command 'quad' in matlab to compute I with error to $$ 10^{-10}$$   Result:

I_q=1.317902151454412

Elapsed time is 0.043016 seconds.

iii) Clencurt

We use the command 'clencurt' in matlab to compute I with error to $$ 10^{-10}$$   Result:

I_cl = 1.317902151447634

Elapsed time is 0.001750 seconds.

iv) Compare computational time

As above table shown, clencurt has the shortest computational time and trapz takes longest time.

Therefore,clencurt is the more efficient way to get a numerical integration value for the given function.

 -- By Min Zhong 12:00, 02 March 2010 (UTC)

7 (a) Use chebfun to do integration
Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Problem Statement
For $$\displaystyle I = \int_{-5}^{5}\frac{1}{1+x^2}dx$$, compare $$\displaystyle I_{n}$$ to $$\displaystyle O(10^{-10})$$ and computational time by - chebfun: sum command,

Solution
 Chebfun

We use the command 'sum' in chebfun to compute I with error to $$ 10^{-10}$$

Result:

I_c=2.746801533890032

Elapsed time is 0.141973 seconds.

E=4.440892098500626e-016

7 (d) compare trapz, quad, clencurt
Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Problem Statement
For $$\displaystyle I = \int_{-5}^{5}\frac{1}{1+x^2}dx$$, compare $$\displaystyle I_{n}$$ to $$\displaystyle O(10^{-10})$$ and computational time for - chebfun: sum command, - Matlab : trapz, quad and clencurt

Solution
i) Trapz

We use the command 'trapz' in matlab to compute I with error to $$ 10^{-10}$$

Result:

I_t = 2.746801533832626

n_t = 65536

E_t = 5.740607988968804e-011

Elapsed time is 0.061356 seconds.

ii) Quad

We use the command 'quad' in matlab to compute I with error to $$ 10^{-10}$$

Result:

I_q= 2.746801533869947

Elapsed time is 0.029688 seconds.

iii) Clencurt

We use the command 'clencurt' in matlab to compute I with error to $$ 10^{-10}$$ Result:

Elapsed time is 0.011186 seconds.

I_cl = 2.746801533874087

n = 62

iv) Compare computational time

As above table shown, Quad has the shortest computational time among these commonds.

Therefore, Quad is a more efficient way to get a numerical integration value for the given function.   -- By Min Zhong 2:00, 03 March 2010 (UTC)

8 (a) Apply chebfun to integration
Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Problem Statement
Compute $$\displaystyle I = \int_{0}^{2\pi}\frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}d\theta$$, and computational time - chebfun: sum command,

Solution
 Chebfun

We use the command 'sum' in chebfun to compute I with error to $$ 10^{-10}$$

Result:

I_c= 6.069090959564774

E_c=3.552713678800501e-015

Elapsed time is 0.140592 seconds.

8 (d) compare trapz, quad, clencurt
Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Problem Statement
For $$\displaystyle I = \int_{0}^{2\pi}\frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}d\theta$$,

- Matlab : trapz, quad and clencurt

Solution
i) Trapz

We use the command 'trapz' in matlab to compute I with error to $$ 10^{-10}$$

Result:

I_t =6.069090959564874

n_t =16

E_t =1.039168751049147e-013

Elapsed time is 0.023226 seconds.

ii) Quad

We use the command 'quad' in matlab to compute I with error to $$ 10^{-10}$$

Result:

I_q=6.069090959562212

Elapsed time is 0.023456 seconds.

iii) Clencurt

We use the command 'clencurt' in matlab to compute I with error to $$ 10^{-10}$$ Result:

I_cl = 6.069090959664361

n = 20

Elapsed time is 0.006226 seconds.

iv) Compare computational time

As above table shown, clencurt has the shortest computational time among these commonds .

Therefore, cleancurt is a more efficient way to get a numerical integration value for the given function.

-- By Min Zhong 12:00, 02 March 2010 (UTC)

title: User:Egm6341.s10.team3.Min/HW4


text:
User talk:Gaidheal1

 Hello Gaidheal1, 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. 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.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. Jtneill - Talk - c 21:42, 2 March 2010 (UTC)

 Hi , I am Percy. Nice to meet you. I love the beach, swimming, fishing, and I am currently doing some research at Wikiversity on oceanography. I am also learning to play piano. What do you like? Can I be your mascot?

Thank you
Gaidheal1, thanks for your feedback. It is greatly appreciated for a newbie like me!

--Csmayes 20:16, 14 April 2011 (UTC)Csmayes

welcoming users
Hi, thanks for welcoming users. When I first started doing this, a year ago, it was suggested to me that I not welcome users simply upon account creation, but on first edit, and this is what a number of experienced Wikiversity users do. I can think of quite a few reasons for this, but what do you think? See also Wikiversity talk:Welcoming committee. --Abd 12:49, 2 June 2011 (UTC)

Thanks
By the way, thanks for your help with the administrative side of WV! --Draicone (talk) 16:52, 2 June 2011 (UTC)

 * Welcome and thanks from me too. Sincerely, James -- Jtneill - Talk - c 21:16, 6 June 2011 (UTC)

 Hello , you can call me Jacob Frank. I'd like to be your mascot and welcome you to Wikiversity. Even though I was a plain and untutored man back in 18th-century Europe, that still didn't stop me from contributing my ideas to the world. Neither should it stop you. As a religious leader who developed the doctrine of "purification through transgression," my advice to you is to be bold and to ignore all rules—yes, these are two of Wikiversity's official policies! Happy editing.

Wikiversity UK
I am glad to have seen that there is another active wikiversitan in the UK. I have been trying to get otherwikipedian to engage with wikiversity particularly as regards developing learning resources for the Wikimedia UK GLAM initiative (See GLAMcamp London). Would you ne at all interested in this?Leutha 17:40, 3 July 2011 (UTC)

Minor bug in Template:=
When you added documentation to the Template:=, it ended up adding an extra space to the output, so that "=" passed in a URL from Template:UPC ended up splitting the URL. Please watch when adding noincluded information that you don't add whitespace this way. But still, thanks for upgrading the templates! Wnt 17:55, 5 December 2011 (UTC)

Star Stuff from a Newbie
Hello Gaidheal1.

 * You greeted me shortly after I started contributing to the Wikiversity. Thank you for your support. I noticed that you have an empty section titled "Star Stuff"! I have been working on a large lesson plan for folding and constructing a variety of "starry" ornaments and mathematical models. I hope you can enjoy it!

 * However, I am not yet very good at maintaining Wikiversity pages, so I have a problem I could use your help with. I have created a page titled "RCB_DanceWithTheStars". This page is OBSOLETE, incomplete, and with several serious mistakes. Also, my wife fears that a title like that would attract viewers interested in choreography, instead of those interested in paper crafts. Would you please delete it for me?

 * I tried to replace the above named file with another file, whose INTENDED title was

supposed to be "RCB_FoldingWithTheStars". But, I inadvertantly called it "RCB_DancingWithTheStars", by mistake. Can you also please rename this file with the title "RCB_FoldingWithTheStars", as I really intended?

 * Yes, the most recent file still contains mistakes and missing content, but I think I will be able to remove these from small sections more easily than I could revise the entire work. Your advice will be most welcome.

 * Oh, yes. I have found a quick course on setting up one's User page from User:planotse,

and I'm wondering if computer programming skills can be adapted to the "Wikiversity: the Movie" project. I accept encourgement, if you think any is due.

Ray Calvin Baker (talk) 22:29, 7 August 2012 (UTC)

title: User talk:Gaidheal1


text:
User talk:Aiacciu9

 Hello Aiacciu9, 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. 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.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --AFriedman (talk) 21:56, 2 March 2010 (UTC)

 Hello , we are the Baah and Bahh twins and we'd both like to welcome you to Wikiversity. While you sit back and explore Wikiversity, enjoy some of our fresh goat milk. You can help us learn by creating new learning resources or help us by improving the guided tours and other exploration tools. We look forward to seeing you around. Baaaahbye for now.

title: User talk:Aiacciu9


text:
User talk:Smart58

 Hello Smart58, 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. 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.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --AFriedman (talk) 22:38, 2 March 2010 (UTC)

 Hello , we are the Baah and Bahh twins and we'd both like to welcome you to Wikiversity. While you sit back and explore Wikiversity, enjoy some of our fresh goat milk. You can help us learn by creating new learning resources or help us by improving the guided tours and other exploration tools. We look forward to seeing you around. Baaaahbye for now.

title: User talk:Smart58


text:
User:PalmdaleHermit/blog

Well I'm trying to get Firefox to let me create this blog but with my eyesight it has been difficult to get this started. Well I've finally get this page up and hopefully it will now work.

I truly hope that this developing course on the Tibetan language will fill a spot for those like me on limited fuinds who in spite of this would like to learn to read and perhaps write in Tibetan. My hope is that I'll be able to learn enough to read and translate some of the ancient texts that have survived two onslaughts of the Indian Buddhism.

Before I stumbled unto Vikiversity I was successfully able to download Tise and to get my browser as well as OpenOffice to display the Tibetan script. Now I need to learn how to read and to   write using this tool. PalmdaleHermit 22:40, 2 March 2010 (UTC)

Strange I could not find the update RSS feed icon or target anywhere on my blog page I should skip this section for the time being.

--PalmdaleHermit 23:16, 2 March 2010 (UTC)

While I'm happy to say I've had no trouble at getting the Tibetan Alphabet game to work. I think I'll not only enjoy it but well also benefit from it in due time. I'm not so sure how I'll get well I'll learn how to post with the RSS feed system. Oh well one step at a time I guess --PalmdaleHermit 01:44, 4 March 2010 (UTC)

I continue to practice the u-chen alphabet using the online game found in lesson four. I find that as I expected I am getting a little better at putting the Wylie and the U-chen script together.

I would like to find a dictionary plugin for Openoffice for Tibetan. I would also like to find the basic words like colors and other basic words in the U-chen script so I can practice writing and agonizing these words. --PalmdaleHermit 01:58, 5 March 2010 (UTC)

title: User:PalmdaleHermit/blog


text:
User:Egm6341.s10.Team4/HW4

= Problem 1: Higher Order Corrected Trapezoidal Rule =

Given
Refer Lecture slide 20-1 for problem statement.

Error for Higher Order Error for Trapezoidal rule on slide 18-2, is given by:

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

$$ $$
 * $$\displaystyle E_n^1 := I - I_n
 * $$\displaystyle E_n^1 := I - I_n
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

where $$\displaystyle I_n = T(n) $$ is the Composite Trapezoidal approximation of the Integral $$\displaystyle I = \int_{a}^{b} f(x)dx $$ with $$\displaystyle n $$ intervals and $$\displaystyle n+1 $$ points $$\displaystyle \forall x \in [a,b] $$.

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

T(n)=\frac{h}{2}(f_0+2f_1+2f_2+ \cdots +2f_{n-1}+f_n) $$ $$
 * $$ \displaystyle\
 * $$ \displaystyle\
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

where $$\displaystyle\ f_i=f(x_i)$$.

Equation (1) is a Euler-Maclaurin series given by Refer 18-3:

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

$$ $$
 * $$\displaystyle E_n^1 = \sum_{q=1}^{\infty} a_q h^{2q}
 * $$\displaystyle E_n^1 = \sum_{q=1}^{\infty} a_q h^{2q}
 * $$\displaystyle \longrightarrow (3)
 * }.
 * }.

where $$\displaystyle h = \frac{b-a}{n} $$.

Find
1. To develop a general higher order corrected Trapezoidal Rule.

2. To find $$\displaystyle n, I_n, E_n^1 $$ using $$\displaystyle CT_1, CT_2, CT_3 $$ for which $$\displaystyle I - I_n $$ is of the order $$\displaystyle \Theta(10^{-6}) $$ for $$\displaystyle I = \int_{0}^{1} \frac{e^x-1}{x}dx $$.

Solution
1. To develop a general Higher Order Corrected Trapezoidal Rule:

Equation (3),

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

\begin{align} E_n^1 &= \sum_{q=1}^{\infty} a_q h^{2q} \\ \Rightarrow     E_n^1 &= I - T(n) = \sum_{q=1}^{\infty} a_q h^{2q} &= I - T(n) = a_1h^{2} + \sum_{q=2}^{\infty} a_q h^{2q} \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{T(n) - a_1h^{2}}_{CT_1}\right] = \sum_{q=2}^{\infty} a_q h^{2q} \\ &= I - CT_1(n) = a_2h^{4} + \sum_{q=3}^{\infty} a_q h^{2q} \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_1 + a_2h^{4}}_{CT_2}\right] = \sum_{q=3}^{\infty} a_q h^{2q} \\ &= I - CT_2(n) = a_3h^{6} + \sum_{q=4}^{\infty} a_q h^{2q} \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

\begin{align} \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_2 + a_3h^{6}}_{CT_3}\right] = \sum_{q=4}^{\infty} a_q h^{2q} \\ &= I - CT_3(n) = a_4h^{8} + \sum_{q=5}^{\infty} a_q h^{2q} \\ &.\\ &.\\ &.\\ &.\\ &.\\ \Rightarrow     E_n^1 &= I - \left[\underbrace{CT_{(k-1)} + a_kh^{2k}}_{CT_k}\right] = \sum_{q=(k+1)}^{\infty} a_q h^{2q} \\ &= I - \underbrace{CT_k(n)}_{I_n} = a_{(k+1)}h^{2(k+1)} + \sum_{q=k+2}^{\infty} a_q h^{2q} \\
 * $$\displaystyle
 * $$\displaystyle

\end{align} $$
 * }.
 * }.

Hence the Higher Order Corrected Trapezoidal rule is given by,

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

$$\displaystyle
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

\Rightarrow     I_n = CT_k(n) = CT_{(k-1)}(n) + a_kh^{2k}

$$ $$
 * style= |
 * $$\displaystyle \longrightarrow (4)

 * }.
 * }.

2. To find $$\displaystyle I_n $$ using $$\displaystyle CT_1, CT_2, CT_3 $$:

From Equation (4), $$\displaystyle I_n $$ using $$\displaystyle CT_1, CT_2, CT_3 $$ can be given by:

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

\begin{align} \Rightarrow     I_n &= CT_3 \\ &= CT_2 + a_3h^{6}\\ &= CT_1 + a_2h^{4} + a_3h^{6}\\ &= T(n) + a_1h^{2} + a_2h^{4} + a_3h^{6} \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Here,

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

\begin{align} a_i &= d_i \left[ f^{(2i-1)}(b) - f^{(2i-1)}(a)\right]\\ d_i &= - \frac{B_{2i}}{2i}; \ \ \ B_{2i} \rightarrow Bernoulli\ Numbers;\ \ \ d_1 = \frac{-1}{12}; \ \ \ d_2 = \frac{1}{720}; \ \ \ d_3 = \frac{-1}{30240} \end{align}
 * $$\displaystyle

$$

 * }.
 * }.

At $$\displaystyle n = 8$$, the Error is reduced to an order of $$\displaystyle \Theta(10^{-6}) $$.

 Matlab Code:

Author
Solved and typed by - Egm6341.s10.Team4.andy 08:44, 3 March 2010 (UTC)

= Problem 2: Applying periodic function for higher-order of Trap. rule =

Given
Envisage the higher-order error for Trap. rule which has been introduced as following [[media:Egm6341.s10.mtg19.djvu|19-3]], for a function $$\displaystyle f(x) $$ which is periodic on $$\displaystyle [a,b] $$

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle E_n=a_1h^{2\times 1}+a_2h^{2\times 2}+...+a_nh^{2\times n}=\sum_{i=1}^\infty a_ih^{2i}

 * }
 * }

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle for k=0,1,2,...        f^{(k)}(a)=f^{(k)}(b)

 * }
 * }

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle a_i=d_i[f^{(2i-1)}(b)-f^{(2i-1)}(a)]

 * }
 * }

Find
Demonstrate that for which $$\displaystyle n $$, the $$\displaystyle E_n=0 $$ would be achieved?

Solution
Since $$\displaystyle f(x) $$ is a periodic function on the domain, we can use the following property of periodic functions (refer to first equation on the [[media:Egm6341.s10.mtg20.djvu|20-2]]:

If $$\displaystyle f(x) $$ is periodic on $$\displaystyle [a,b] $$, the odd derivatives of it will be equal at the same interval:

 * $$\displaystyle

f^{(k)}(a)=f^{(k)}(b) $$

k=2i-1; i\in \mathbb{N} $$

\Rightarrow f^{(2i-1)}(a)=f^{(2i-1)}(b) $$

\Rightarrow a_i=d(di=-\dfrac {B_{2i}}{(2i!)}\cancelto{0}{[f^{(2i-1)}(b)-f^{(2i-1)}(a)]}=0 $$

Thus, recalling the higher-order error of Trapezoidal rule;

E_n=\sum_{i=1}^n a_ih^{2i}=0 $$

Author
Solved and typed by - Egm6341.s10.Team4.nimaa&m 21:51, 27 February 2010 (UTC) .

= Problem 3: Pros and cons of different numerical methods =

Find

 * {| style="width:70%" border="0" align="center"

 * Discuss the pros and cons of different numerical methods.
 * 1) Taylor's series.
 * 2) Composite Trapezoidal rule.
 * 3) Composite Simpson's rule.
 * 4) Romberg Table (Richardson Extrapolation).
 * 5) Corrected Trapezoidal rule.
 * }.
 * }.

Solution
The pros and cons of the above mentioned methods are discussed below.

Taylors Series:

Advantages:
 * 1) The solution is highly accurate for a small order compared to Trapezoidal and Simpson's rule.
 * 2) It is one step and explicit.

Disadvantages:
 * 1) It needs the explicit form of derivatives of the function.

Composite Trapezoidal rule:

Advantages:
 * 1) The method is quite simple in implementing, compared to other methods.
 * 2) It takes a piecewise linear approximation of the function and hence execution of the solution is faster.
 * 3) The solution is very rapidly convergent.
 * 4) It consists of the fact that weighting coefficients are nearly equal to each other.

Disadvantages:
 * 1) The error is much higher compared to the other methods.
 * 2) This method is less accurate for non linear functions since it has straight line approximation.

Composite Simpson's rule:

Advantages:
 * 1) It assumes piecewise quadratic interpolation and hence the accuracy is much higher compared to trapezoidal rule.
 * 2) It can integrate polynomials up to third order accurately.
 * 3) The error is less compared to that of trapezoidal rule.
 * 4) Weighting coefficients are simple and do not fluctuate in terms of magnitude.

Disadvantages:
 * 1) A large number of ordinates are needed in between the interval.

Romberg Table (Richardson Extrapolation):

Advantages:
 * 1) Error is reduced subsequently as we go from TK(n) to TK+1(n).
 * 2) It is efficient in the sense that TK+1(n) is calculated from the already computed values of TK(n) and TK(2n).
 * 3) The above condition reduces the time and space requirement. Also, the table format is good for pursuing the history of calculations and making some comparisons.

Disadvantages:
 * 1) In Richardson extrapolation, We are neglecting the higher orders of aih2i.

Corrected Trapezoidal rule:

Advantages:
 * 1) When integrals of periodic functions are approximated numerically, Corrected Trapezoidal functions are the best choice. Even we can reach to the error equal to zero in this condition.

Disadvantages:
 * 1) This method demands derivatives of the function at the end points of the interval.
 * 2) Bernoulli's number should be calculated for each order of derivatives.

Author
Solved and typed by - Egm6341.s10.team4.anandankala 13:37, 3 March 2010 (UTC) Reviewed by - Egm6341.s10.Team4.nimaa&m 13:57, 3 March 2010 (UTC) .

= Problem 4: Proof of Higher Order Error for Trap. rule Theorem =

Given
Higher Order Error for Trap. rule equation (1) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]], which is:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle Eq. (1) \ E_n^1= \sum_{k=0}^{n-1}[ \int_{x_k}^{x_{k+1}}f(x)dx- \frac{h}{2}(f_k(x_k)+f(x_{k+1}))]
 * }
 * }

Show
That eq. (1) can be transformed to the following: equation (5) on slide [[media:Egm6341.s10.mtg21.djvu|21-2]]

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))]

where, equation (2) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]]

h= \frac{(b-a)}{n} \ \ \ x_i=a+ih $$ and, equation (3) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]]

x(t)= t*\frac{h}{2} + \frac{x_i+x_{i+1}}{2} \ \ \ t\in[-1,1] $$
 * style= |
 * }.
 * }.

Solution
Defining: equation (1) on slide [[media:Egm6341.s10.mtg21.djvu|21-2]]

g_i(t):=f(x(t)) \ where \ x\in[x_i,x_{i+1}] $$

For one Panel, we have:

x_i=-1 \ \ and \ \ x_{i+1}=1 $$

f(x)_i=g(-1) \ \ \ f(x)_{i+1}=g(1) $$

Also,

\frac{dx(t)}{dt}=\frac{h}{2} $$

\Rightarrow E_n^1= \sum_{k=0}^{n-1} \int_{-1}^{1}g_k(t)\dfrac{h}{2}dt-\dfrac{h}{2}(g_k(-1)+g_k(+1)) $$

Giving,

 * {| style="width:20%" border="0"

$$\displaystyle E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))] $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

 * }
 * }

Author
Solved and typed by - --Egm6341.s10.Team4.roni 07:39, 6 March 2010 (UTC) Reviewed by - --Egm6341.s10.Team4.roni 07:39, 6 March 2010 (UTC) .

= Problem 5: Theorem of Higher Order Error for Trap. rule Step1 =

Given
Higher Order Error for Trap. rule equation (5) on slide [[media:Egm6341.s10.mtg21.djvu|21-1]], which is:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle Eq. (5)\ E_n^1=\dfrac{h}{2} \sum_{k=0}^{n-1}[ \int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))]
 * }
 * }

Find
That using eq. (1) on slide [[media:Egm6341.s10.mtg21.djvu|21-2]],

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle g_k(t):=f(x(t)) \  such \  that \  x \   \epsilon  \left[ x_k,x_{k+1}\right]

For one panel Step 1 :

Eq.(2) \int_{-1}^{1}(-t) {g}^{(1)}(t)dt = \int_{-1}^{1}g_k(t)dt-[(g_k(-1)+g_k(+1))] $$

 * style= |
 * }
 * }

Solution
Using integration by Parts :

\int u {v}^{'}= uv - \int v {u}^{'}\ \ where \ \ u=t \ \ {v}^{'}=g(t) $$

Writing the left side of Eq. (2):

\int_{-1}^{1}(-t) {g}^{(1)}(t)dt = -\int_{-1}^{1}(t) {g}^{(1)}(t)dt $$

= -\Bigg[(t \, g(t)) \Bigg|^{1}_{-1} -\int_{-1}^{1}g(t)dt \Bigg] $$

= -(1*g(1)-(-1)g(-1)-\int_{-1}^{1}g(t)dt) $$

= \int_{-1}^{1}g(t)dt)-(g(-1)+g(1)) $$

 * {| style="width:20%" border="0"

$$\displaystyle \int_{-1}^{1}(-t) {g}^{(1)}(t)dt = \int_{-1}^{1}g(t)dt)-(g(-1)+g(1)) $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

 * }
 * }

Author
Solved and typed by - --Egm6341.s10.Team4.roni 07:40, 6 March 2010 (UTC) Reviewed by - --Egm6341.s10.Team4.roni 07:40, 6 March 2010 (UTC)

= Problem 6: i-derivative of g function in higher-order of Trap. proof =

Given
Consider the following definitions for these two functions such that $$\displaystyle x\in [x_k,x_{k+1}] $$ interval:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle x(t):=t\dfrac {h}{2}+\dfrac {x_k+x_{k+1}}{2}
 * }
 * }

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle g_k(t):=f(x(t))

 * }
 * }

Find
Show that $$\displaystyle g_k^{(i)}(t)=(\dfrac {h}{2})^i.f^{(i)}(x(t)) $$:

Solution
$$\displaystyle g_k(t)=f(x(t))=f(t\dfrac {h}{2}+\dfrac {x_k+x_{k+1}}{2}) $$

$$\displaystyle \Rightarrow g_k^{(1)}(t)=\dfrac {d[f(\dfrac {th}{2}+\dfrac {x_k+x_{k+1}}{2})]}{dt} $$

Using the Chain rule for derivation:

$$\displaystyle \Rightarrow g_k^{(1)}(t)=\dfrac {d[f(x(t))]}{dx}\times\dfrac {h}{2} $$

For second derivative of $$\displaystyle g_k^{(2)} $$, we have:

$$\displaystyle g_k^{(2)}(t)=\frac {d[g_k^{(1)}(t)]}{dt}=\frac {d[\frac {d[f(x(t))]}{dx}\times\frac {h}{2}]}{dt}=\frac {d[\frac {d[f(\frac{th}{2}+\frac {x_k+x_{k+1}}{2})]}{dx}\times\frac {h}{2}]}{dt} $$

Using the Chain rule for derivation:

$$\displaystyle \Rightarrow g_k^{(2)}(t)=\frac {d^2(f(x(t))}{dx^2}\times (\dfrac {h}{2})\times(\dfrac {h}{2})=\frac {d^2(f(x(t))}{dx^2}\times (\dfrac {h}{2})^2 $$

Similarly, for third derivative of $$\displaystyle g_k^{(3)} $$ we will find:

$$\displaystyle \Rightarrow g_k^{(3)}(t)=\frac {d^3(f(x(t))}{dx^3}\times (\dfrac {h}{2})\times (\dfrac {h}{2})\times (\dfrac {h}{2})=\frac {d^3(f(x(t))}{dx^3}\times (\dfrac {h}{2})^3 $$

Thus, for $$\displaystyle i^{th} $$, $$\displaystyle g_k^{(i)} $$ can be predicted as:

$$\displaystyle g_k^{(i)}(t)=(\dfrac {h}{2})^i.f^{(i)}(x(t)) $$

Author
Solved and typed by - Egm6341.s10.Team4.nimaa&m 00:08, 2 March 2010 (UTC) .

= Problem 7: Theorem of Higher Order Error for Trap. rule [Step:2a - Part2] =

Given
Refer Lecture Slide Refer 21-2 for problem statement.

Higher Order Error for Trap. rule is given by [ Refer 21-1 ]:

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

E_n^1=\frac{h}{2} \sum_{k=0}^{n-1} \left[ \underbrace{\int_{-1}^{1}g_k(t)dt-(g_k(-1)+g_k(+1))}_{E}\right] $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

where $$\displaystyle E $$ can be written as,

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

E =\int_{-1}^{1}(-t)g^{(1)}(t)dt $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

Find
Show that,

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

$$\displaystyle E =\int_{-1}^{1}\underbrace{(-t)}_{p_{1}(t)}g^{(1)}(t)dt = \left[p_2(t)g^{(1)}(t) \right]_{-1}^{+1} - \int_{-1}^{+1}p_2(t)g^{(2)}(t)dt $$

$$
 * $$\displaystyle \longrightarrow (3)
 * }.
 * }.

where $$\displaystyle p_2(t) = \int p_1(t)dt$$.

Solution
Integrating by Parts,

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

\int u dv = uv - \int v du $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where,

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

$$\displaystyle \begin{align} u &= g^{(1)}(t) \\ \Rightarrow du &= g^{(2)}(t)dt \end{align} $$

 * }.
 * }.

and,

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

$$\displaystyle \begin{align} dv &= p_1(t)dt \\ &= -tdt \\ & \\ \Rightarrow v &= \int p_1(t)dt &=p_2(t) \end{align} $$

 * }.
 * }.

Equation (3) is then,

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

$$\displaystyle \begin{align} \Rightarrow E &=\int_{-1}^{1}\underbrace{(-t)}_{p_{1}(t)}g^{(1)}(t)dt \\ &= \left[p_2(t)g^{(1)}(t) \right]_{-1}^{+1} - \int_{-1}^{+1}p_2(t)g^{(2)}(t)dt \end{align} $$

 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.andy 08:44, 3 March 2010 (UTC)

= Problem 8: Chebfun coding for Clenshaw-Curtis quad.=

Given
Codes of the Chebfun project are available on the lecture plan link of the course web site as a zip. file to apply in programming softwares. These codes are written for estimating the Clenshaw-Curtis quadrature by (Trefethen et al.)with a command in which started with (sum).

Find
Install Chebfun codes to see how this set of codes can estimate Clenshaw-Curtis quadrature.

Solution
Installation instructions: Chebfun Version 3 is compatible with MATLAB 7.4 (2007a) and more recent versions. First, contents of the zip file to a directory. The name "chebfun" for this directory is suggested. Then, In MATLAB, the chebfun directory is added to the path. This can be done by selecting "File/Set Path..." from the main or Command window menus. It is recommended to select the "Save" button on this dialog so that chebfuns are on the path automatically in future MATLAB sessions. To check whether the system appears to be working or not, "chebtest" should be typed at the command window prompt. Now, $$\displaystyle >> f=chebfun ('f(x)') $$ can be used for estimating the Clenshaw-Curtis quadrature.

The curve between the data points is the polynomial interpolant, which is evaluated by the barycentric formula introduced by Salzer [Berrut & Trefethen 2004, Salzer 1972]. This method of evaluating polynomial interpolants is stable and efficient even if the degree is in the millions [Higham 2004].

As an example if we want to know what is the integral of f from a to b? we should use this command:

>> sum ('f(x)') $$

This command can be applied for integration of $$\displaystyle f(x) $$ by Clenshaw-Curtis quadrature, as an alternative method for composite trapezoidal rule, composite simpson's rule, some Matlab commands and Richardson extrapolation (Romberg table).

Author
Solved and typed by - Egm6341.s10.Team4.nimaa&m 21:59, 27 February 2010 (UTC) .

= Problem 9 : A Comparison of Integration Methods =

Given
We have been given various integration methods thus far throughout the class, we are now asked to compare their efficiency and accuracy.

Find
We are asked to

1) correct matlab code for Romberg efficiently

2) compare the second column of the Rhomberg Table (T1(n)) to the Composite Simpson's Rule, and derive any relationship that exists.

3) integrate the function $$ \displaystyle\ f(x)=\frac{e^x-1}{x}$$ over the interval $$ \displaystyle\ x \epsilon [0,1]$$

using the following methods and to compare the computational cost as a function of run-time -
 * Composite Trapezoidal Rule(optimized)
 * Composite Simpson's Rule
 * Rhomberg Interpolation
 * chebfun sum command - a function that uses Chebyshev polynomials to integrate
 * the matlab functions trapz, quad, and clencurt.

Solution
1) correct matlab code for Romberg efficiently

The composit Trapozoida rule for $$\displaystyle n=2^j$$ and Romberg table was computed efficiently using this equation. (the modified matlab codes are in below)

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

T_0(2^{j})= \frac{1}{2}T_o(2^{j-1})+ h \sum_{i=1}^{2^{j-1}}f(x_{2i-1}) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)

where
 * }.
 * {| style="width:100%" border="0" align="left"

h = \frac{b-a}{2^{j}} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

[reference] derivation of equation(1)
 * {| style="width:100%" border="0" align="left"

 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

T_o(2^{j})= \frac{b-a}{2^{j}} [ \frac{1}{2}f(x_0)+f(x_1)+f(x_2)+f(x_3)+f(x_4)+......+ \frac{1}{2} f(x_{2^{j}})] $$ For example
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

T_0(64)= \frac{b-a}{64} [ \frac{1}{2}f(x_0)+f(x_1)+f(x_2)+f(x_3)+f(x_4)+......f(x_{63}) + \frac{1}{2} f(x_{64})] $$ the rear part can be arranged like that (decouple to the even number & odd number)
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.
 * {| style="width:100%" border="0" align="left"

T_0(64)= \frac{b-a}{64} [ \underbrace{\frac{1}{2}f(x_0)+f(x_2)+f(x_4)+... \frac{1}{2} f(x_{64})}_{even} ~ + \underbrace{f(x_1) +f(x_3)+f(x_5)+......f(x_{63})}_{odd}] $$ The even number part is exactly same with $$\displaystyle \frac{1}{2}T_0(32) $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

\frac{b-a}{64}[ \frac{1}{2}f(x_0)+f(x_2)+f(x_4)+... \frac{1}{2} f(x_{64})] = \frac{1}{2} * \frac{b-a}{32}[ \frac{1}{2}f(x_0)+f(x_1)+f(x_2)+... \frac{1}{2} f(x_{32})] = \frac{1}{2}T_o(32)$$ So, the orginal $$\displaystyle T_0(64) $$ is like that
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

T_0(64)= \frac{1}{2}T_o(32) + \frac{b-a}{64} [f(x_1) +f(x_3)+f(x_5)+......f(x_{63})] = \frac{1}{2}T_o(32) + \frac{b-a}{64} \sum_{i=1}^{32}f(x_{2i-1}) $$ if this is generalized,
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

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

T_0(2^{j})= \frac{1}{2}T_o(2^{j-1})+ h \sum_{i=1}^{2^{j-1}}f(x_{2i-1}) $$ where
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.
 * {| style="width:100%" border="0" align="left"

h = \frac{b-a}{2^{j}} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * style= |
 * }.
 * }.
 * }.

2) compare the second column of the Rhomberg Table (T1(n)) to the Composite Simpson's Rule, and derive any relationship that exists.

For the first three methods, pre-existing codes written by the authors were used in order to evaluate the computational time required to perform the integration. For the Rhomberg Extrapolation, the computation time was determine by the time required to compute T0(n) plus the time to perform the extrapolation.

When the Simpson's Rule and T1 (Richardsen Extrapolation) are compared to each other, the results are identical out to all of the significant digits calculated, thus we believe that the two are equivalents processes. Furthermore,

$$ \displaystyle T_1(2n)=\frac{4T_0(2n)-T_0(n)}{4-1}$$

Breaking this into $$ \displaystyle T_0(n)=h(\frac{1}{2}f_0+f_1+f_2+...\frac{1}{2}f_n$$ and $$ \displaystyle T_0(n/2)=2h(\frac{1}{2}f_0+f_2+f_4+...\frac{1}{2}f_n$$ we can see that $$ \displaystyle T_1(n)=\frac{h}{3}(f_0+4f_1+2f_2+4f_3+...f_n)$$, which is equal to the Simpson's Rule for n points (where h is twice the spacing between two points).

For the Trapezoidal Rule and Rhomberg Extrapolation

For the Simpson's Rule

For Matlab functions and chebshev

For ClenCurt Integration

Author
Egm6341.s10.Team4.riherd 14:29, 3 March 2010 (UTC) --Egm6341.s10.Team4.roni 07:41, 6 March 2010 (UTC) Proof read by Egm6341.s10.Team4.yunseok 22:45, 7 March 2010 (UTC)

= Problem 10 - Another comparison of integration methods =

Find
We are asked to integrate the function $$ \displaystyle\ f(x)=\frac{1}{1+x^2}$$ over the interval $$ \displaystyle\ x \epsilon [-5,5]$$ using the following methods and to compare the computational cost as a function of run-time -
 * Composite Trapezoidal (optimized)
 * Composite Simpson's Rule
 * Rhomberg Interpolation
 * chebfun sum command - a function that uses Chebyshev polynomials to integrate
 * the matlab functions trapz, quad, and clencurt.

Solution
Operating in the same manner as above, only for a different $$ \displaystyle f(x)$$ and bounds of integration.

For the Trapezoidal and Rhomberg Extrapolation

For the Composite Simpson's Rule

For the Matlab functions

For ClenCurt Integration

Author
Solved and Written by Egm6341.s10.Team4.riherd 16:06, 3 March 2010 (UTC)

ClenCurt Solved and Written by --Egm6341.s10.Team4.roni 07:56, 6 March 2010 (UTC)

Proof read by Egm6341.s10.Team4.yunseok 22:44, 7 March 2010 (UTC)

= Problem 11: Application of Engineering Orbital Mech. =

Given
Refer Lecture slide 25-2 for problem statement

Polar form relative to focus
 * {| style="width:100%" border="0" align="left"

r(\theta) = \frac{1-e^2}{1-e \, cos(\theta)} $$
 * [[Image:Ellipse Polar.svg|200px]]
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where, eccentricity $$\displaystyle e = \sin(\frac{\pi}{12}) $$

Find
Find Arc length of ellipse (orbital)using below integration method.

compute $$\displaystyle I_n$$ to the error $$\displaystyle 10^{-10}$$ and

compute time using tic/toc matlab commend

1) Composit Trapozoidal rule

2) Composit Simpson's rule

3) Romberg Table

4) Chebfun, sum commend

5) matlab commend : Trapz, quad, clencurt

Solution
At first, we tried to calculate the arc length of ellipse using the $$\displaystyle r(\theta) = \frac{1-e^2}{1-e \, cos(\theta)}$$ however, we couldn't find any equation or relationship between arc length of ellipse and $$\displaystyle r(\theta) $$.

so, we found another method to compute the circumference of ellipse in wikipidia.

The circumference $$C$$ of an ellipse is $$\displaystyle 4 a E(e)$$, where the function $$E$$ is the complete elliptic integral of the second kind. Here, a=1:

The complete elliptic integral of the second kind E is defined as

 * $$E(e) = \int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta\!$$

So,

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

I= 4\int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

using the equation (1), the arc length of ellipse was calculated by several numerical method.

The result is summarized.

Ramanujan approximation is 's:
 * $$C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi(3(a+b)-\sqrt{10ab+3(a^2+b^2)})$$

where, a = longest radius, b= shortest radius in ellipse

the resuts of our integration method and Ramanujan approximation were same. $$\displaystyle (c \simeq 6.1766)$$

(1) Composit Trapoziodal rule

 Matlab Code:

 subfunction ff(x)

(2) Composit Simpson's rule

 Matlab Code:

(3) Romberg Table

 Matlab Code:

(4) Chebfun

 Matlab Code:

(5) Trapz (matlab commend)

 Matlab Code:

(6) quad (matlab commend)

 Matlab Code:

(7) ClenCurt Integration)

 Matlab Code:

ClenCurt Integration Results -

.

Author
Solved and typed by - Egm6341.s10.Team4.yunseok 16:38, 3 March 2010 (UTC)

ClenCurt Solved and typed by--Egm6341.s10.Team4.roni 07:58, 6 March 2010 (UTC) .

Proof read by - Egm6341.s10.Team4.riherd 17:51, 3 March 2010 (UTC)

And - Egm6341.s10.Team4.nimaa&m 20:52, 7 March 2010 (UTC)

= Contributing Members = Egm6341.s10.Team4.andy 08:55, 3 March 2010 (UTC)- Author of Problems:1, 7 ; Proof Read: 2,4,5,6

Egm6341.s10.Team4.riherd 16:07, 3 March 2010 (UTC) - Author of Problems 9,10; Proof Read:11

Egm6341.s10.team4.anandankala 13:37, 3 March 2010 (UTC)- Author of problems 3; Proof Read:1,7

Egm6341.s10.Team4.yunseok 16:38, 3 March 2010 (UTC)

Egm6341.s10.Team4.nimaa&m 22:15, 3 March 2010 (UTC)- Author of problems 2, 6, 8; Proof Read:3,11

Egm6341.s10.Team4.roni 07:58, 6 March 2010 (UTC)

title: User:Egm6341.s10.Team4/HW4