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Life Cycle Analysis

Content Summary
Life Cycle Analysis (LCA) is an approach to assessing environmental impacts that looks at the whole life cycle of a product, process or service rather than just a small part, such as the use or the manufacture. Over the last 30 years, life cycle thinking has become accepted as an accurate method for assessing total life impact and enabling engineers, scientists and consumers to make like comparisons of products, processes and services.

This learning module aims to introduce the concept of LCA, show the existing methodologies, and especially the ISO 1404x series and through a case study show how LCA can be applied to an environmental assessment.

Life-cycle assessment (LCA, also known as life-cycle analysis, ecobalance, and cradle-to-grave analysis) is a technique to assess environmental impacts associated with all the stages of a product's life from raw material extraction through materials processing, manufacture, distribution, use, repair and maintenance, and disposal or recycling. Designers use this process to help critique their products. LCAs can help avoid a narrow outlook on environmental concerns by:
 * Compiling an inventory of relevant energy and material inputs and environmental releases;
 * Evaluating the potential impacts associated with identified inputs and releases;
 * Interpreting the results to help make a more informed decision.

Learning Tasks

 * (Sustainable Development Goals) Explain why a Life Cycle Analysis is important for assessing the sustainbility of considered products and services.
 * (Decision Support) Explain why a Life Cycle Analysis is important for a decision support system.

Goals and purpose
The goal of LCA is to compare the full range of environmental effects assignable to products and services by quantifying all inputs and outputs of material flows and assessing how these material flows affect the environment. This information is used to improve processes, support policy and provide a sound basis for informed decisions.

The term life cycle refers to the notion that a fair, holistic assessment requires the assessment of raw-material production, manufacture, distribution, use and disposal including all intervening transportation steps necessary or caused by the product's existence.

There are two main types of LCA. Attributional LCAs seek to establish (or attribute) the burdens associated with the production and use of a product, or with a specific service or process, at a point in time (typically the recent past). Consequential LCAs seek to identify the environmental consequences of a decision or a proposed change in a system under study (oriented to the future), which means that market and economic implications of a decision may have to be taken into account. Social LCA is under development as a different approach to life cycle thinking intended to assess social implications or potential impacts. Social LCA should be considered as an approach that is complementary to environmental LCA.

The procedures of life cycle assessment (LCA) are part of the ISO 14000 environmental management standards: in ISO 14040:2006 and 14044:2006. (ISO 14044 replaced earlier versions of ISO 14041 to ISO 14043.) GHG product life cycle assessments can also comply with specifications such as PAS 2050 and the GHG Protocol Life Cycle Accounting and Reporting Standard.

Four main phases

According to the ISO 14040 and 14044 standards, a Life Cycle Assessment is carried out in four distinct phases as illustrated in the figure shown to the right. The phases are often interdependent in that the results of one phase will inform how other phases are completed.

Goal and scope
An LCA starts with an explicit statement of the goal and scope of the study, which sets out the context of the study and explains how and to whom the results are to be communicated. This is a key step and the ISO standards require that the goal and scope of an LCA be clearly defined and consistent with the intended application. The goal and scope document, therefore, includes technical details that guide subsequent work:
 * the functional unit, which defines what precisely is being studied and quantifies the service delivered by the product system, providing a reference to which the inputs and outputs can be related. Further, the functional unit is an important basis that enables alternative goods, or services, to be compared and analyzed. So to explain this a functional system which is inputs, processes and outputs contains a functional unit, that fulfills a function, for example paint is covering a wall, making a functional unit of 1m² covered for 10 years. The functional flow would be the items necessary for that function, so this would be a brush, tin of paint and the paint itself.
 * the system boundaries; which are delimitations of which processes that should be included in the analysis of a product system.
 * any assumptions and limitations;
 * the allocation methods used to partition an environmental load of a process when several products or functions share the same process; allocation is commonly dealt with in one of three ways: system expansion, substitution, and partition. Doing this is not easy and different methods may give different results

and
 * the impact categories chosen for example human toxicity, smog, global warming, eutrophication.

Life cycle inventory

Life Cycle Inventory (LCI) analysis involves creating an inventory of flows from and to nature for a product system. Inventory flows include inputs of water, energy, and raw materials, and releases to air, land, and water. To develop the inventory, a flow model of the technical system is constructed using data on inputs and outputs. The flow model is typically illustrated with a flow chart that includes the activities that are going to be assessed in the relevant supply chain and gives a clear picture of the technical system boundaries. The input and output data needed for the construction of the model are collected for all activities within the system boundary, including from the supply chain (referred to as inputs from the technosphere).

The data must be related to the functional unit defined in the goal and scope definition. Data can be presented in tables and some interpretations can be made already at this stage. The results of the inventory is an LCI which provides information about all inputs and outputs in the form of elementary flow to and from the environment from all the unit processes involved in the study.

Inventory flows can number in the hundreds depending on the system boundary. For product LCAs at either the generic (i.e., representative industry averages) or brand-specific level, that data is typically collected through survey questionnaires. At an industry level, care has to be taken to ensure that questionnaires are completed by a representative sample of producers, leaning toward neither the best nor the worst, and fully representing any regional differences due to energy use, material sourcing or other factors. The questionnaires cover the full range of inputs and outputs, typically aiming to account for 99% of the mass of a product, 99% of the energy used in its production and any environmentally sensitive flows, even if they fall within the 1% level of inputs.

One area where data access is likely to be difficult is flows from the technosphere. The technosphere is more simply defined as the human-made world. Considered by geologists as secondary resources, these resources are in theory 100% recyclable; however, in a practical sense, the primary goal is salvage. For an LCI, these technosphere products (supply chain products) are those that have been produced by human and unfortunately those completing a questionnaire about a process which uses a human-made product as a means to an end will be unable to specify how much of a given input they use. Typically, they will not have access to data concerning inputs and outputs for previous production processes of the product. The entity undertaking the LCA must then turn to secondary sources if it does not already have that data from its own previous studies. National databases or data sets that come with LCA-practitioner tools, or that can be readily accessed, are the usual sources for that information. Care must then be taken to ensure that the secondary data source properly reflects regional or national conditions.

LCI Methods

 * Process LCA
 * Economic Input Output LCA
 * Hybrid Approach

Life cycle impact assessment
Inventory analysis is followed by impact assessment. This phase of LCA is aimed at evaluating the significance of potential environmental impacts based on the LCI flow results. Classical life cycle impact assessment (LCIA) consists of the following mandatory elements:

 * selection of impact categories, category indicators, and characterization models;
 * the classification stage, where the inventory parameters are sorted and assigned to specific impact categories; and
 * impact measurement, where the categorized LCI flows are characterized, using one of many possible LCIA methodologies, into common equivalence units that are then summed to provide an overall impact category total.

In many LCAs, characterization concludes the LCIA analysis; this is also the last compulsory stage according to ISO 14044:2006. However, in addition to the above mandatory LCIA steps, other optional LCIA elements – normalization, grouping, and weighting – may be conducted depending on the goal and scope of the LCA study. In normalization, the results of the impact categories from the study are usually compared with the total impacts in the region of interest, the U.S. for example. Grouping consists of sorting and possibly ranking the impact categories. During weighting, the different environmental impacts are weighted relative to each other so that they can then be summed to get a single number for the total environmental impact. ISO 14044:2006 generally advises against weighting, stating that “weighting, shall not be used in LCA studies intended to be used in comparative assertions intended to be disclosed to the public”. This advice is often ignored, resulting in comparisons that can reflect a high degree of subjectivity as a result of weighting.

Life cycle impacts can also be categorized under the several phases of the development, production, use, and disposal of a product. Broadly speaking, these impacts can be divided into "First Impacts," use impacts, and end of life impacts. "First Impacts" include extraction of raw materials, manufacturing (conversion of raw materials into a product), transportation of the product to a market or site, construction/installation, and the beginning of the use or occupancy. Use impacts include physical impacts of operating the product or facility (such as energy, water, etc.), maintenance, renovation and repairs (required to continue to use the product or facility). End of life impacts include demolition and processing of waste or recyclable materials.

Interpretation
Life Cycle Interpretation is a systematic technique to identify, quantify, check, and evaluate information from the results of the life cycle inventory and/or the life cycle impact assessment. The results from the inventory analysis and impact assessment are summarized during the interpretation phase. The outcome of the interpretation phase is a set of conclusions and recommendations for the study. According to ISO 14040:2006, the interpretation should include:

 * identification of significant issues based on the results of the LCI and LCIA phases of an LCA;
 * evaluation of the study considering completeness, sensitivity and consistency checks; and
 * conclusions, limitations and recommendations.

A key purpose of performing life cycle interpretation is to determine the level of confidence in the final results and communicate them in a fair, complete, and accurate manner. Interpreting the results of an LCA is not as simple as "3 is better than 2, therefore Alternative A is the best choice"! Interpreting the results of an LCA starts with understanding the accuracy of the results, and ensuring they meet the goal of the study. This is accomplished by identifying the data elements that contribute significantly to each impact category, evaluating the sensitivity of these significant data elements, assessing the completeness and consistency of the study, and drawing conclusions and recommendations based on a clear understanding of how the LCA was conducted and the results were developed.

Reference test
More specifically, the best alternative is the one that the LCA shows to have the least cradle-to-grave environmental negative impact on land, sea, and air resources.

LCA uses
Based on a survey of LCA practitioners carried out in 2006 LCA is mostly used to support business strategy (18%) and R&D (18%), as input to product or process design (15%), in education (13%) and for labeling or product declarations (11%). LCA will be continuously integrated into the built environment as tools such as the European ENSLIC Building project guidelines for buildings or developed and implemented, which provide practitioners guidance on methods to implement LCI data into the planning and design process.

Major corporations all over the world are either undertaking LCA in house or commissioning studies, while governments support the development of national databases to support LCA. Of particular note is the growing use of LCA for ISO Type III labels called Environmental Product Declarations, defined as "quantified environmental data for a product with pre-set categories of parameters based on the ISO 14040 series of standards, but not excluding additional environmental information". These third-party certified LCA-based labels provide an increasingly important basis for assessing the relative environmental merits of competing products. Third-party certification plays a major role in today's industry. Independent certification can show a company's dedication to safer and environmental friendlier products to customers and NGOs.

LCA also has major roles in environmental impact assessment, integrated waste management and pollution studies. A recent study evaluated the LCA of a laboratory scale plant for oxygen enriched air production coupled with its economic evaluation in an holistic eco-design standpoint. LCA has also been used to assess the environmental impacts of pavement maintenance, repair, and rehabilitation activities.

Data analysis
A life cycle analysis is only as valid as its data; therefore, it is crucial that data used for the completion of a life cycle analysis are accurate and current. When comparing different life cycle analyses with one another, it is crucial that equivalent data are available for both products or processes in question. If one product has a much higher availability of data, it cannot be justly compared to another product which has less detailed data.

There are two basic types of LCA data – unit process data and environmental input-output data (EIO), where the latter is based on national economic input-output data. Unit process data are derived from direct surveys of companies or plants producing the product of interest, carried out at a unit process level defined by the system boundaries for the study.

Data validity is an ongoing concern for life cycle analyses. Due to globalization and the rapid pace of research and development, new materials and manufacturing methods are continually being introduced to the market. This makes it both very important and very difficult to use up-to-date information when performing an LCA. If an LCA’s conclusions are to be valid, the data must be recent; however, the data-gathering process takes time. If a product and its related processes have not undergone significant revisions since the last LCA data was collected, data validity is not a problem. However, consumer electronics such as cell phones can be redesigned as often as every 9 to 12 months, creating a need for ongoing data collection.

The life cycle considered usually consists of a number of stages including: materials extraction, processing and manufacturing, product use, and product disposal. If the most environmentally harmful of these stages can be determined, then impact on the environment can be efficiently reduced by focusing on making changes for that particular phase. For example, the most energy-intensive life phase of an airplane or car is during use due to fuel consumption. One of the most effective ways to increase fuel efficiency is to decrease vehicle weight, and thus, car and airplane manufacturers can decrease environmental impact in a significant way by replacing heavier materials with lighter ones such as aluminium or carbon fiber-reinforced elements. The reduction during the use phase should be more than enough to balance additional raw material or manufacturing cost.

Data sources are typically large databases. It is not appropriate to compare two options if different data sources have been used to source the data. Common data sources include:

Calculations for impact can then be done by hand, but it is more usual to streamline the process by using software. This can range from a simple spreadsheet, where the user enters the data manually to a fully automated program, where the user is not aware of the source data.

Cradle-to-grave
Cradle-to-grave is the full Life Cycle Assessment from resource extraction ('cradle') to use phase and disposal phase ('grave'). For example, trees produce paper, which can be recycled into low-energy production cellulose (fiberised paper) insulation, then used as an energy-saving device in the ceiling of a home for 40 years, saving 2,000 times the fossil-fuel energy used in its production. After 40 years the cellulose fibers are replaced and the old fibers are disposed of, possibly incinerated. All inputs and outputs are considered for all the phases of the life cycle.

Cradle-to-gate
Cradle-to-gate is an assessment of a partial product life cycle from resource extraction (cradle) to the factory gate (i.e., before it is transported to the consumer). The use phase and disposal phase of the product are omitted in this case. Cradle-to-gate assessments are sometimes the basis for environmental product declarations (EPD) termed business-to-business EPDs. One of the significant uses of the cradle-to-gate approach compiles the life cycle inventory (LCI) using cradle-to-gate. This allows the LCA to collect all of the impacts leading up to resources being purchased by the facility. They can then add the steps involved in their transport to plant and manufacture process to more easily produce their own cradle-to-gate values for their products.

Cradle-to-cradle or closed loop production
Cradle-to-cradle is a specific kind of cradle-to-grave assessment, where the end-of-life disposal step for the product is a recycling process. It is a method used to minimize the environmental impact of products by employing sustainable production, operation, and disposal practices and aims to incorporate social responsibility into product development. From the recycling process originate new, identical products (e.g., asphalt pavement from discarded asphalt pavement, glass bottles from collected glass bottles), or different products (e.g., glass wool insulation from collected glass bottles).

Allocation of burden for products in open loop production systems presents considerable challenges for LCA. Various methods, such as the avoided burden approach have been proposed to deal with the issues involved.

Gate-to-gate
Gate-to-gate is a partial LCA looking at only one value-added process in the entire production chain. Gate-to-gate modules may also later be linked in their appropriate production chain to form a complete cradle-to-gate evaluation.

Well-to-wheel
Well-to-wheel is the specific LCA used for transport fuels and vehicles. The analysis is often broken down into stages entitled "well-to-station", or "well-to-tank", and "station-to-wheel" or "tank-to-wheel", or "plug-to-wheel". The first stage, which incorporates the feedstock or fuel production and processing and fuel delivery or energy transmission, and is called the "upstream" stage, while the stage that deals with vehicle operation itself is sometimes called the "downstream" stage. The well-to-wheel analysis is commonly used to assess total energy consumption, or the energy conversion efficiency and emissions impact of marine vessels, aircraft and motor vehicles, including their carbon footprint, and the fuels used in each of these transport modes. WtW analysis is useful for reflecting the different efficiencies and emissions of energy technologies and fuels at both the upstream and downstream stages, giving a more complete picture of real emissions.

The well-to-wheel variant has a significant input on a model developed by the Argonne National Laboratory. The Greenhouse gases, Regulated Emissions, and Energy use in Transportation (GREET) model was developed to evaluate the impacts of new fuels and vehicle technologies. The model evaluates the impacts of fuel use using a well-to-wheel evaluation while a traditional cradle-to-grave approach is used to determine the impacts from the vehicle itself. The model reports energy use, greenhouse gas emissions, and six additional pollutants: volatile organic compounds (VOCs), carbon monoxide (CO), nitrogen oxide (NOx), particulate matter with size smaller than 10 micrometre (PM10), particulate matter with size smaller than 2.5 micrometre (PM2.5), and sulfur oxides (SOx).

Quantitative values of greenhouse gas emissions calculated with the WTW or with the LCA method can differ, since the LCA is considering more emission sources. In example, while assessing the GHG emissions of a Battery Electric Vehicle in comparison with a conventional internal combustion engine vehicle, the WTW (accounting only the GHG for manufacturing the fuels) finds out that an electric vehicle can save the 50-60% of GHG, while an hybrid LCA-WTW method, considering also the GHG due to the manufacturing and the end of life of the battery gives GHG emission savings 10-13% lower, compared to the WTW.

Economic input–output life cycle assessment
Economic input–output LCA (EIOLCA) involves use of aggregate sector-level data on how much environmental impact can be attributed to each sector of the economy and how much each sector purchases from other sectors. Such analysis can account for long chains (for example, building an automobile requires energy, but producing energy requires vehicles, and building those vehicles requires energy, etc.), which somewhat alleviates the scoping problem of process LCA; however, EIOLCA relies on sector-level averages that may or may not be representative of the specific subset of the sector relevant to a particular product and therefore is not suitable for evaluating the environmental impacts of products. Additionally the translation of economic quantities into environmental impacts is not validated.

Ecologically based LCA
While a conventional LCA uses many of the same approaches and strategies as an Eco-LCA, the latter considers a much broader range of ecological impacts. It was designed to provide a guide to wise management of human activities by understanding the direct and indirect impacts on ecological resources and surrounding ecosystems. Developed by Ohio State University Center for resilience, Eco-LCA is a methodology that quantitatively takes into account regulating and supporting services during the life cycle of economic goods and products. In this approach services are categorized in four main groups: supporting, regulating, provisioning and cultural services.

Exergy based LCA
Exergy of a system is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir. Wall clearly states the relation between exergy analysis and resource accounting. This intuition confirmed by DeWulf and Sciubba lead to Exergo-economic accounting and to methods specifically dedicated to LCA such as Exergetic material input per unit of service (EMIPS). The concept of material input per unit of service (MIPS) is quantified in terms of the second law of thermodynamics, allowing the calculation of both resource input and service output in exergy terms. This exergetic material input per unit of service (EMIPS) has been elaborated for transport technology. The service not only takes into account the total mass to be transported and the total distance, but also the mass per single transport and the delivery time.

Life cycle energy analysis
Life cycle energy analysis (LCEA) is an approach in which all energy inputs to a product are accounted for, not only direct energy inputs during manufacture, but also all energy inputs needed to produce components, materials and services needed for the manufacturing process. An earlier term for the approach was energy analysis.

With LCEA, the total life cycle energy input is established.

Energy production
It is recognized that much energy is lost in the production of energy commodities themselves, such as nuclear energy, photovoltaic electricity or high-quality petroleum products. Net energy content is the energy content of the product minus energy input used during extraction and conversion, directly or indirectly. A controversial early result of LCEA claimed that manufacturing solar cells requires more energy than can be recovered in using the solar cell. The result was refuted. Another new concept that flows from life cycle assessments is energy cannibalism. Energy cannibalism refers to an effect where rapid growth of an entire energy-intensive industry creates a need for energy that uses (or cannibalizes) the energy of existing power plants. Thus during rapid growth the industry as a whole produces no energy because new energy is used to fuel the embodied energy of future power plants. Work has been undertaken in the UK to determine the life cycle energy (alongside full LCA) impacts of a number of renewable technologies.

Energy recovery
If materials are incinerated during the disposal process, the energy released during burning can be harnessed and used for electricity production. This provides a low-impact energy source, especially when compared with coal and natural gas While incineration produces more greenhouse gas emissions than landfills, the waste plants are well-fitted with filters to minimize this negative impact. A recent study comparing energy consumption and greenhouse gas emissions from landfills (without energy recovery) against incineration (with energy recovery) found incineration to be superior in all cases except for when landfill gas is recovered for electricity production.

Criticism
It has also been argued that energy efficiency is only one consideration in deciding which alternative process to employ, and that it should not be elevated to the only criterion for determining environmental acceptability. For example, simple energy analysis does not take into account the renewability of energy flows or the toxicity of waste products;. Incorporating Dynamic LCAs of renewable energy technologies (using sensitivity analyses to project future improvements in renewable systems and their share of the power grid) may help mitigate this criticism.

In recent years, the literature on life cycle assessment of energy technology has begun to reflect the interactions between the current electrical grid and future energy technology. Some papers have focused on energy life cycle,  while others have focused on carbon dioxide (CO2) and other greenhouse gases. The essential critique given by these sources is that when considering energy technology, the growing nature of the power grid must be taken into consideration. If this is not done, a given class of energy technology may emit more CO2 over its lifetime than it initially thought it would mitigate, with this most well documented in wind energy's case.

A problem the energy analysis method cannot resolve is that different energy forms (heat, electricity, chemical energy etc.) have different quality and value even in natural sciences, as a consequence of the two main laws of thermodynamics. A thermodynamic measure of the quality of energy is exergy. According to the first law of thermodynamics, all energy inputs should be accounted with equal weight, whereas by the second law diverse energy forms should be accounted by different values.

The conflict is resolved in one of these ways:
 * value difference between energy inputs is ignored,
 * a value ratio is arbitrarily assigned (e.g., a joule of electricity is 2.6 times more valuable than a joule of heat or fuel input),
 * the analysis is supplemented by economic (monetary) cost analysis,
 * exergy instead of energy can be the metric used for the life cycle analysis.

Critiques
Life cycle assessment is a powerful tool for analyzing commensurable aspects of quantifiable systems. Not every factor, however, can be reduced to a number and inserted into a model. Rigid system boundaries make accounting for changes in the system difficult. This is sometimes referred to as the boundary critique to systems thinking. The accuracy and availability of data can also contribute to inaccuracy. For instance, data from generic processes may be based on averages, unrepresentative sampling, or outdated results. Additionally, social implications of products are generally lacking in LCAs. Comparative life-cycle analysis is often used to determine a better process or product to use. However, because of aspects like differing system boundaries, different statistical information, different product uses, etc., these studies can easily be swayed in favor of one product or process over another in one study and the opposite in another study based on varying parameters and different available data. There are guidelines to help reduce such conflicts in results but the method still provides a lot of room for the researcher to decide what is important, how the product is typically manufactured, and how it is typically used.

An in-depth review of 13 LCA studies of wood and paper products found a lack of consistency in the methods and assumptions used to track carbon during the product lifecycle. A wide variety of methods and assumptions were used, leading to different and potentially contrary conclusions – particularly with regard to carbon sequestration and methane generation in landfills and with carbon accounting during forest growth and product use.

Page Information
This page was based on the following wikipedia-source page:
 * Life-cycle assessment https://en.wikipedia.org/wiki/Life-cycle%20assessment
 * Datum: 6/1/2019
 * Wikipedia2Wikiversity-Converter: https://niebert.github.com/Wikipedia2Wikiversity

title: Life Cycle Analysis


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HW3

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Talk:Instructional design/Learning objectives

Needs: In many cases I am going to use discussion to stage changes, but as they should be, these are elements for discussion, if that discussion is available.

Introduction
1. This is an excellent spot to mention the pre-analysis process that would presumably lead to the creation of instructional content. In many cases, two major questions will arise.

 * a. Is there a need for this project? Does it serve the learner's needs, the needs of the community that is surrounded by this knowledge, and by society at large?
 * b. Are there the resources needed for this project?

 * It is worth mentioning that in regard to many projects for instructional design on Wikiversity the answer is more likely to be affirmative to both questions than it would be for instructor lead courses at physical schools. The reason for this is as follows:

 * i. Wikiversity, and all her wikisister projects, serve all kinds of needs from major needs to niche requirements. As an embodiment of the diversity of knowledge this will almost always answer as yes. The exception may be negative in regard to the production of projects that reach a critical level of redundancy to what has already been developed. Often adding the elements that distinguish the proposed new redundant project from the preceding one, to the the pre-existing course will be sufficient. On a finer level, underneath the grand visions and purposes of Wikiversity, one can use the comments on Wikiversity, as well perform surveys, to determine more specifically what the typical needs of users that currently visit the website are.

 * ii. Wikiversity is based upon utilizing all of the human resources it can attract with as much time to offer as necessary. The Wikimedia Foundation does have objectives to have reach certain goals of editor participation across all of its child projects, but, in principle at least, the human and time resources far less limited. In practical terms it may be worth recognizing some current limitations. For example, currently there are not many editors on Wikiversity, therefore the creation of an instruction with the expectation of receiving a lot of editor community support may result in disappointment. Additionally there are few people that know how to hack MediaWiki well enough to successfully have code committed, therefore instructional design that relies on new technology that doesn't yet exist on Wikiversity may also suffer from this bottleneck. In theory these resources are available (in the wide world of open source) but currently, in practice they are more sparse.

2. There is the issue of the substantiation tag, that there is controversy about the value of learning objectives.

3. Additional data: The crafting of well design courses is often not rewarded by academic institutions, where as publishing research is. There is, at the very least, a Declining by Degrees quote about this, I am sure.

The Structure of Learning Objectives
1. A graphic would be much more visually palatable than the use of font colors for the describing Robert Mager's definition of "Learning Objectives"


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User:Egm6341.s10.Team4.yunseok/HW3

= Problem 4: The Error & Ratio of Trapezoidal & Simpson's rule =

Given
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'''(1) Compsite Trapozoidal rule $$\displaystyle (I_n) $$ and Error of Trapezoidal$$\displaystyle (E_1) $$ for $$\displaystyle n=2, 4, 8, 16, 32, 64, 128, 256, 512 $$

'''(2) The Ratio of Error when $$\displaystyle n $$ is doubled,

'''(3) Asymptotic Estimates Ẽ

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'''(1) Error of Trapezoidal$$\displaystyle (E_1) $$ and Simpson's Rule $$\displaystyle (E_2) $$ for $$\displaystyle n=2, 4, 8, 16, 32, 64, 128 $$

'''(2) The Ratio of Error when $$\displaystyle n $$ is doubled,

Solution
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'''6)
 * {| style="width:100%" border="0" align="left"

I = \int^{2\pi}_{0} e^{cos(x)}dx $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Matlab code and the Result table is bleow.

 Matlab Code:

 subfunction f6(x)

= about No. 5 =

 * {| 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}} $$ In conclusion, if we know the $$\displaystyle T_0(2^{j-1}) $$, we can calculate more efficiently $$\displaystyle T_0(2^{j}) $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

this is my understanding about problem 5..

title: User:Egm6341.s10.Team4.yunseok/HW3


text:
User:Davidhsjenkins

I have been a huge fan of wikipedia since I first discovered it and frequently use it for my own benefit, mainly to add to my knowledge across the wide spectrum of information offered to me by wikipedia.

I live in Sydney, Australia. I was brought up and educated in Britain. My family home was in Scotland. I finished high school there and some years after that I returned to the educational system, enrolling at a technical college to get a diploma in aquaculture. This sat well with my own interest in fish and fishing, and several years of experience as a seine net fisherman in and around my home in ross-shire.

I then worked for several years as manager of a fish farm in wester ross, before emigrating to Australia. I maintained my involvement in aquaculture for the first two or three years in Australia, but then moved onto other interests.

I am close to retirement now, and much of my free time is spent researching historical information relating to the period 1840 to 1940 and the social and economic impact the crimean and first world wars had on the population in general, and on attitudes towards change in particular within the fishing and crofting community of my part of ross and cromarty.

title: User:Davidhsjenkins


text:
User:Egm6341.s10.Team4.nimaa&m/HW3

= Problem 2: Error for composite Simpson's rule =

Given
Use composite Simpson's equation (3) on slide [[media:Egm6341.s10.mtg7.pdf|7-2]], which is:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle I_n=\dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]

 * }
 * }

Find
To demonstrate error of composite Simpson's rule is equal to the following:

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

$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle 	\left | {E_n}^2\right |\leqslant\dfrac{(b-a)^5}{2880n^4}M_4=\dfrac{(b-a)h^4}{2880}M_4

M_4:=max\left |f^{(4)}(\xi)\right | $$

\xi\in[a,b] $$
 * style= |
 * }.
 * }.

Solution
The error definition can be done as:

{E_n}^2=I-I_n=\int\limits_{a}^{b} f(x)dx-I_n $$

From the composite form of Simpson's rule, we have:

I_n=\dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)] $$

\Rightarrow {E_n}^2=\int\limits_{a}^{b} f(x)dx-\dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]= $$

=\sum_{i=1}^{n/2} [\int\limits_{x_{2i-2}}^{x_{2i}} f(x)dx-\dfrac{h}{3}[f(x_{2i-2})+4f(x_{2i-1})+f(x_{2i})]] $$

where,

h=\frac{b-a}{n} = x_i-x_{i-1} $$

According to Lagrange interpolation error for simple Simpson's rule (refer to [14-1]) for $$\displaystyle [x_{i-1},x_{i+1}] $$ ;

E_2=\dfrac{x_{i+1}-x_{i-1}}{2} f^{(4)}(\xi) $$

\xi\in [x_{i-1},x_{i+1}] $$

\Rightarrow \left |{E_n}^2 \right | \le \sum_{i=1}^{n/2} max \left | \dfrac {(x_{i+1}-x_{i-1})^5}{90\times 2^5} f^{(4)}(\xi)\right | =\dfrac{h^5}{90}\times \sum_{i=1}^{n/2} max \left | f^{(4)}(\xi)\right | $$

h=\dfrac {x_{i+1}-x_{i-1}}{2}=\dfrac {b-a}{n} $$

M_4:=max\left | f^{(4)}(\xi) \right | $$

\xi\in [x_{i-1},x_{i+1}] $$

\overline{M_4}:=\sum_{i=1}^{n/2} max\left |f^{(4)}(\xi) \right | $$

\overline{M_4}\le max\left | \dfrac {n}{2} .M_4 \right | $$

\Rightarrow\left |{E_n}^2\right |\le\dfrac{h^5}{90}\times \dfrac {n}{2}.M_4=\dfrac{h^5}{90}\times \dfrac {\dfrac{b-a}{h}}{2}.M_4 $$

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

$$\displaystyle \Rightarrow\left |{E_n}^2\right | \le \dfrac{(b-a)h^4}{180} M_4 $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

 * style= |

'''This answer is different from the asked question, However, this answer is confirmed in the Atkinson's text book "An Intro, Numerical Analysis" 2nd edition (page 257~258)
 * }
 * }

.

Author
Solved and typed by - Egm6341.s10.Team4.nimaa&m 15:31, 17 February 2010 (UTC) Reviewed by - Egm6341.s10.Team4.andy 16:23, 17 February 2010 (UTC) .

title: User:Egm6341.s10.Team4.nimaa&m/HW3


text:
User talk:JTR5121819

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User talk:Tereensio~enwikiversity

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title: User talk:Tereensio~enwikiversity


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User talk:ErgoSum

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User talk:Mrbailey

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User:B9 hummingbird hovering/Blog/Learning Sanskrit Weblog

Manifesto
I have been chanting and reading about Sanskrit for over 20 years and I am still not proficient in Sanskrit without the assistance of technology. But then I do not need to be. But I feel a tangible appropriateness with my lifepath and spiritual journey in its entirety and now commencing this work on the Avadhuta Gita.

Avadhuta Gita weblog
User:B9 hummingbird hovering/Blog/Learning Sanskrit Weblog/Avadhuta Gita

Uddhava Gita weblog
User:B9 hummingbird hovering/Blog/Learning Sanskrit Weblog/Uddhava Gita

User:B9_hummingbird_hovering/Blog/Learning_Sanskrit_Weblog/Avadhuta Upanishad

śvetāśvataropaniṣat
User:B9_hummingbird_hovering/Blog/Learning_Sanskrit_Weblog/śvetāśvataropaniṣat


text:
User:B9 hummingbird hovering/Blog/Learning Sanskrit Weblog/Avadhuta Gita

Avadhūta Gītā A spontaneous song and philosophical poem attributed to Rishi Dattatreya and reputed to have been transcribed by two of his disciples in Sanskrit according to the Nath tradition. It is a key example of Absolute Non-Dual Philosophy. atha prathamo'dhyāyaḥ || īśvarānugrahādeva puṁsāmadvaitavāsanā | mahadbhayaparitrāṇādviprāṇāmupajāyate ||1|| yenedaṁ pūritaṁ sarvamātmanaivāatmanātmani | nirākāraṁ kathaṁ vande hyabhinnaṁ śivamavyayam ||2|| pañcabhūtātmakaṁ viśvaṁ marīcijalasannibham | kasyāpyaho namaskuryāmahameko nirañjanaḥ ||3|| ātmaiva kevalaṁ sarvaṁ bhedābhedo na vidyate | asti nāsti kathaṁ brūyāṁ vismayaḥ pratibhāti me ||4|| vedāntasārasarvasvaṁ jñānaṁ vijñānameva ca | ahamātmā nirākāraḥ sarvavyāpī svabhāvataḥ ||5|| yo vai sarvātmako devo niṣkalo gaganopamaḥ | svabhāvanirmalaḥ śuddhaḥ sa evāyaṁ na saṁśayaḥ ||6|| ahamevāvyayo'nantaḥ śuddhavijñānavigrahaḥ | sukhaṁ duḥkhaṁ na jānāmi kathaṁ kasyāpi vartate ||7|| na mānasaṁ karma śubhāśubhaṁ me na kāyikaṁ karma śubhāśubhaṁ me | na vācikaṁ karma śubhāśubhaṁ me jñānāmṛtaṁ śuddhamatīndriyo'ham ||8|| mano vai gaganākāraṁ mano vai sarvatomukham | mano'tītaṁ manaḥ sarvaṁ na manaḥ paramārthataḥ ||9|| ahamekamidaṁ sarvaṁ vyomātītaṁ nirantaram | paśyāmi kathamātmānaṁ pratyakśaṁ vā tirohitam ||10|| tvamevamekaṁ hi kathaṁ na budhyase samaṁ hi sarveṣu vimṛṣṭamavyayam | sadodito'si tvamakhaṇḍitaḥ prabho divā ca naktaṁ ca kathaṁ hi manyase ||11|| ātmānaṁ satataṁ viddhi sarvatraikaṁ nirantaram | ahaṁ dhyātā paraṁ dhyeyamakhaṇḍaṁ khaṇḍyate katham ||12|| na jāto na mṛto'si tvaṁ na te dehaḥ kadācana | sarvaṁ brahmeti vikhyātaṁ bravīti bahudhā śrutiḥ ||13|| sa bāhyābhyantaro'si tvaṁ śivaḥ sarvatra sarvadā | itastataḥ kathaṁ bhrāntaḥ pradhāvasi piśācavat ||14|| saṁyogaśca viyogaśca vartate na ca te na me | na tvaṁ nāhaṁ jagannedaṁ sarvamātmaiva kevalam ||15|| śabdādipañcakasyāsya naivāsi tvaṁ na te punaḥ | tvameva paramaṁ tattvamataḥ kiṁ paritapyase ||16|| janma mṛtyurna te cittaṁ bandhamokśau śubhāśubhau | kathaṁ rodiṣi re vatsa nāmarūpaṁ na te na me ||17|| aho citta kathaṁ bhrāntaḥ pradhāvasi piśācavat | abhinnaṁ paśya cātmānaṁ rāgatyāgātsukhī bhava ||18|| tvameva tattvaṁ hi vikāravarjitaṁ niṣkampamekaṁ hi vimokśavigraham | na te ca rāgo hyathavā virāgaḥ kathaṁ hi santapyasi kāmakāmataḥ ||19|| vadanti śrutayaḥ sarvāḥ nirguṇaṁ śuddhamavyayam | aśarīraṁ samaṁ tattvaṁ tanmāṁ viddhi na saṁśayaḥ ||20|| sākāramanṛtaṁ viddhi nirākāraṁ nirantaram | etattattvopadeśena na punarbhavasambhavaḥ ||21|| ekameva samaṁ tattvaṁ vadanti hi vipaścitaḥ | rāgatyāgātpunaścittamekānekaṁ na vidyate ||22|| anātmarūpaṁ ca kathaṁ samādhi- rātmasvarūpaṁ ca kathaṁ samādhiḥ | astīti nāstīti kathaṁ samādhi- rmokśasvarūpaṁ yadi sarvamekam ||23|| viśuddho'si samaṁ tattvaṁ videhastvamajo'vyayaḥ | jānāmīha na jānāmītyātmānaṁ manyase katham ||24|| tattvamasyādivākyena svātmā hi pratipāditaḥ | neti neti śrutirbrūyādanṛtaṁ pāñcabhautikam ||25|| ātmanyevātmanā sarvaṁ tvayā pūrṇaṁ nirantaram | dhyātā dhyānaṁ na te cittaṁ nirlajjaṁ dhyāyate katham ||26|| śivaṁ na jānāmi kathaṁ vadāmi śivaṁ na jānāmi kathaṁ bhajāmi | ahaṁ śivaścetparamārthathatattvaṁ samasvarūpaṁ gaganopamaṁ ca ||27|| nāhaṁ tattvaṁ samaṁ tattvaṁ kalpanāhetuvarjitam | grāhyagrāhakanirmuktaṁ svasaṁvedyaṁ kathaṁ bhavet ||28|| anantarūpaṁ na hi vastu kiṁci- ttattvasvarūpaṁ na hi vastu kiṁcit | ātmaikarūpaṁ paramārthatattvaṁ na hiṁsako vāpi na cāpyahiṁsā ||29|| viśuddho'si samaṁ tattvaṁ videhamajamavyayam | vibhramaṁ kathamātmārthe vibhrānto'haṁ kathaṁ punaḥ ||30|| ghaṭe bhinne ghaṭākāśaṁ sulīnaṁ bhedavarjitam | śivena manasā śuddho na bhedaḥ pratibhāti me ||31|| na ghaṭo na ghaṭākāśo na jīvo na jīvavigrahaḥ | kevalaṁ brahma saṁviddhi vedyavedakavarjitam ||32|| sarvatra sarvadā sarvamātmānaṁ satataṁ dhruvam | sarvaṁ śūnyamaśūnyaṁ ca tanmāṁ viddhi na saṁśayaḥ ||33|| vedā na lokā na surā na yajñā varṇāśramo naiva kulaṁ na jātiḥ | na dhūmamārgo na ca dīptimārgo brahmaikarūpaṁ paramārthatattvam ||34|| vyāpyavyāpakanirmuktaḥ tvamekaḥ saphalaṁ yadi | pratyakśaṁ cāparokśaṁ ca hyātmānaṁ manyase katham ||35|| advaitaṁ kecidicchanti dvaitamicchanti cāpare | samaṁ tattvaṁ na vindanti dvaitādvaitavivarjitam ||36|| śvetādivarṇarahitaṁ śabdādiguṇavarjitam | kathayanti kathaṁ tattvaṁ manovācāmagocaram ||37|| yadā'nṛtamidaṁ sarvaṁ dehādigaganopamam | tadā hi brahma saṁvetti na te dvaitaparamparā ||38|| pareṇa sahajātmāpi hyabhinnaḥ pratibhāti me | vyomākāraṁ tathaivaikaṁ dhyātā dhyānaṁ kathaṁ bhavet ||39|| yatkaromi yadaśnāmi yajjuhomi dadāmi yat | etatsarvaṁ na me kiṁcidviśuddho'hamajo'vyayaḥ ||40|| sarvaṁ jagadviddhi nirākṛtīdaṁ sarvaṁ jagadviddhi vikārahīnam | sarvaṁ jagadviddhi viśuddhadehaṁ sarvaṁ jagadviddhi śivaikarūpam ||41|| tattvaṁ tvaṁ na hi sandehaḥ kiṁ jānāmyathavā punaḥ | asaṁvedyaṁ svasaṁvedyamātmānaṁ manyase katham ||42|| māyā'māyā kathaṁ tāta chāyā'chāyā na vidyate | tattvamekamidaṁ sarvaṁ vyomākāraṁ nirañjanam ||43|| ādimadhyāntamukto'haṁ na baddho'haṁ kadācana | svabhāvanirmalaḥ śuddha iti me niścitā matiḥ ||44|| mahadādi jagatsarvaṁ na kiṁcitpratibhāti me | brahmaiva kevalaṁ sarvaṁ kathaṁ varṇāśramasthitiḥ ||45|| jānāmi sarvathā sarvamahameko nirantaram | nirālambamaśūnyaṁ ca śūnyaṁ vyomādipañcakam ||46|| na ṣaṇḍho na pumānna strī na bodho naiva kalpanā | sānando vā nirānandamātmānaṁ manyase katham ||47|| ṣaḍaṅgayogānna tu naiva śuddhaṁ manovināśānna tu naiva śuddham | gurūpadeśānna tu naiva śuddhaṁ svayaṁ ca tattvaṁ svayameva buddham ||48|| na hi pañcātmako deho videho vartate na hi | ātmaiva kevalaṁ sarvaṁ turīyaṁ ca trayaṁ katham ||49|| na baddho naiva mukto'haṁ na cāhaṁ brahmaṇaḥ pṛthak | na kartā na ca bhoktāhaṁ vyāpyavyāpakavarjitaḥ ||50|| yathā jalaṁ jale nyastaṁ salilaṁ bhedavarjitam | prakṛtiṁ puruṣaṁ tadvadabhinnaṁ pratibhāti me ||51|| yadi nāma na mukto'si na baddho'si kadācana | sākāraṁ ca nirākāramātmānaṁ manyase katham ||52|| jānāmi te paraṁ rūpaṁ pratyakśaṁ gaganopamam | yathā paraṁ hi rūpaṁ yanmarīcijalasannibham ||53|| na gururnopadeśaśca na copādhirna me kriyā | videhaṁ gaganaṁ viddhi viśuddho'haṁ svabhāvataḥ ||54|| viśuddho'sya śarīro'si na te cittaṁ parātparam | ahaṁ cātmā paraṁ tattvamiti vaktuṁ na lajjase ||55|| kathaṁ rodiṣi re citta hyātmaivātmātmanā bhava | piba vatsa kalātītamadvaitaṁ paramāmṛtam ||56|| naiva bodho na cābodho na bodhābodha eva ca | yasyedṛśaḥ sadā bodhaḥ sa bodho nānyathā bhavet ||57|| jñānaṁ na tarko na samādhiyogo na deśakālau na gurūpadeśaḥ | svabhāvasaṁvittarahaṁ ca tattva- mākāśakalpaṁ sahajaṁ dhruvaṁ ca ||58|| na jāto'haṁ mṛto vāpi na me karma śubhāśubham | viśuddhaṁ nirguṇaṁ brahma bandho muktiḥ kathaṁ mama ||59|| yadi sarvagato devaḥ sthiraḥ pūrṇo nirantaraḥ | antaraṁ hi na paśyāmi sa bāhyābhyantaraḥ katham ||60|| sphuratyeva jagatkṛtsnamakhaṇḍitanirantaram | aho māyāmahāmoho dvaitādvaitavikalpanā ||61|| sākāraṁ ca nirākāraṁ neti netīti sarvadā | bhedābhedavinirmukto vartate kevalaḥ śivaḥ ||62|| na te ca mātā ca pitā ca bandhuḥ na te ca patnī na sutaśca mitram | na pakśapātī na vipakśapātaḥ kathaṁ hi saṁtaptiriyaṁ hi citte ||63|| divā naktaṁ na te cittaṁ udayāstamayau na hi | videhasya śarīratvaṁ kalpayanti kathaṁ budhāḥ ||64|| nāvibhaktaṁ vibhaktaṁ ca na hi duḥkhasukhādi ca | na hi sarvamasarvaṁ ca viddhi cātmānamavyayam ||65|| nāhaṁ kartā na bhoktā ca na me karma purā'dhunā | na me deho videho vā nirmameti mameti kim ||66|| na me rāgādiko doṣo duḥkhaṁ dehādikaṁ na me | ātmānaṁ viddhi māmekaṁ viśālaṁ gaganopamam ||67|| sakhe manaḥ kiṁ bahujalpitena sakhe manaḥ sarvamidaṁ vitarkyam | yatsārabhūtaṁ kathitaṁ mayā te tvameva tattvaṁ gaganopamo'si ||68|| yena kenāpi bhāvena yatra kutra mṛtā api | yoginastatra līyante ghaṭākāśamivāmbare ||69|| tīrthe cāntyajagehe vā naṣṭasmṛtirapi tyajan | samakāle tanuṁ muktaḥ kaivalyavyāpako bhavet ||70|| dharmārthakāmamokśāṁśca dvipadādicarācaram | manyante yoginaḥ sarvaṁ marīcijalasannibham ||71|| atītānāgataṁ karma vartamānaṁ tathaiva ca | na karomi na bhuñjāmi iti me niścalā matiḥ ||72|| śūnyāgāre samarasapūta- stiṣṭhannekaḥ sukhamavadhūtaḥ | carati hi nagnastyaktvā garvaṁ vindati kevalamātmani sarvam ||73|| tritayaturīyaṁ nahi nahi yatra vindati kevalamātmani tatra | dharmādharmau nahi nahi yatra baddho muktaḥ kathamiha tatra ||74|| vindati vindati nahi nahi mantraṁ chandolakśaṇaṁ nahi nahi tantram | samarasamagno bhāvitapūtaḥ pralapitametatparamavadhūtaḥ ||75|| sarvaśūnyamaśūnyaṁ ca satyāsatyaṁ na vidyate | svabhāvabhāvataḥ proktaṁ śāstrasaṁvittipūrvakam ||76|| iti prathamo'dhyāyaḥ ||1|| atha dvitīyo'dhyāyaḥ || bālasya vā viṣayabhogaratasya vāpi mūrkhasya sevakajanasya gṛhasthitasya | etadguroḥ kimapi naiva na cintanīyaṁ ratnaṁ kathaṁ tyajati ko'pyaśucau praviṣṭam ||1|| naivātra kāvyaguṇa eva tu cintanīyo grāhyaḥ paraṁ guṇavatā khalu sāra eva | sindūracitrarahitā bhuvi rūpaśūnyā pāraṁ na kiṁ nayati nauriha gantukāmān ||2|| prayatnena vinā yena niścalena calācalam | grastaṁ svabhāvataḥ śāntaṁ caitanyaṁ gaganopamam ||3|| ayatnāchālayedyastu ekameva carācaram | sarvagaṁ tatkathaṁ bhinnamadvaitaṁ vartate mama ||4|| ahameva paraṁ yasmātsārātsārataraṁ śivam | gamāgamavinirmuktaṁ nirvikalpaṁ nirākulam ||5|| sarvāvayavanirmuktaṁ tathāhaṁ tridaśārcitam | saṁpūrṇatvānna gṛhṇāmi vibhāgaṁ tridaśādikam ||6|| pramādena na sandehaḥ kiṁ kariṣyāmi vṛttimān | utpadyante vilīyante budbudāśca yathā jale ||7|| mahadādīni bhūtāni samāpyaivaṁ sadaiva hi | mṛdudravyeṣu tīkśṇeṣu guḍeṣu kaṭukeṣu ca ||8|| kaṭutvaṁ caiva śaityatvaṁ mṛdutvaṁ ca yathā jale | prakṛtiḥ puruṣastadvadabhinnaṁ pratibhāti me ||9|| sarvākhyārahitaṁ yadyatsūkśmātsūkśmataraṁ param | manobuddhīndriyātītamakalaṅkaṁ jagatpatim ||10|| īdṛśaṁ sahajaṁ yatra ahaṁ tatra kathaṁ bhavet | tvameva hi kathaṁ tatra kathaṁ tatra carācaram ||11|| gaganopamaṁ tu yatproktaṁ tadeva gaganopamam | caitanyaṁ doṣahīnaṁ ca sarvajñaṁ pūrṇameva ca ||12|| pṛthivyāṁ caritaṁ naiva mārutena ca vāhitam | variṇā pihitaṁ naiva tejomadhye vyavasthitam ||13|| ākāśaṁ tena saṁvyāptaṁ na tadvyāptaṁ ca kenacit | sa bāhyābhyantaraṁ tiṣṭhatyavacchinnaṁ nirantaram ||14|| sūkśmatvāttadadṛśyatvānnirguṇatvācca yogibhiḥ | ālambanādi yatproktaṁ kramādālambanaṁ bhavet ||15|| satatā'bhyāsayuktastu nirālambo yadā bhavet | tallayāllīyate nāntarguṇadoṣavivarjitaḥ ||16|| viṣaviśvasya raudrasya mohamūrcchāpradasya ca | ekameva vināśāya hyamoghaṁ sahajāmṛtam ||17|| bhāvagamyaṁ nirākāraṁ sākāraṁ dṛṣṭigocaram | bhāvābhāvavinirmuktamantarālaṁ taducyate ||18|| bāhyabhāvaṁ bhavedviśvamantaḥ prakṛtirucyate | antarādantaraṁ jñeyaṁ nārikelaphalāmbuvat ||19|| bhrāntijñānaṁ sthitaṁ bāhyaṁ samyagjñānaṁ ca madhyagam | madhyānmadhyataraṁ jñeyaṁ nārikelaphalāmbuvat ||20|| paurṇamāsyāṁ yathā candra eka evātinirmalaḥ | tena tatsadṛśaṁ paśyeddvidhādṛṣṭirviparyayaḥ ||21|| anenaiva prakāreṇa buddhibhedo na sarvagaḥ | dātā ca dhīratāmeti gīyate nāmakoṭibhiḥ ||22|| guruprajñāprasādena mūrkho vā yadi paṇḍitaḥ | yastu saṁbudhyate tattvaṁ virakto bhavasāgarāt ||23|| rāgadveṣavinirmuktaḥ sarvabhūtahite rataḥ | dṛḍhabodhaśca dhīraśca sa gacchetparamaṁ padam ||24|| ghaṭe bhinne ghaṭākāśa ākāśe līyate yathā | dehābhāve tathā yogī svarūpe paramātmani ||25|| ukteyaṁ karmayuktānāṁ matiryānte'pi sā gatiḥ | na coktā yogayuktānāṁ matiryānte'pi sā gatiḥ ||26|| yā gatiḥ karmayuktānāṁ sā ca vāgindriyādvadet | yogināṁ yā gatiḥ kvāpi hyakathyā bhavatorjitā ||27|| evaṁ jñātvā tvamuṁ mārgaṁ yogināṁ naiva kalpitam | vikalpavarjanaṁ teṣāṁ svayaṁ siddhiḥ pravartate ||28|| tīrthe vāntyajagehe vā yatra kutra mṛto'pi vā | na yogī paśyate garbhaṁ pare brahmaṇi līyate ||29|| sahajamajamacintyaṁ yastu paśyetsvarūpaṁ ghaṭati yadi yatheṣṭaṁ lipyate naiva doṣaiḥ | sakṛdapi tadabhāvātkarma kiṁcinnakuryāt tadapi na ca vibaddhaḥ saṁyamī vā tapasvī ||30|| nirāmayaṁ niṣpratimaṁ nirākṛtiṁ nirāśrayaṁ nirvapuṣaṁ nirāśiṣam | nirdvandvanirmohamaluptaśaktikaṁ tamīśamātmānamupaiti śāśvatam ||31|| vedo na dīkśā na ca muṇḍanakriyā gururna śiṣyo na ca yantrasampadaḥ | mudrādikaṁ cāpi na yatra bhāsate tamīśamātmānamupaiti śāśvatam ||32|| na śāmbhavaṁ śāktikamānavaṁ na vā piṇḍaṁ ca rūpaṁ ca padādikaṁ na vā | ārambhaniṣpattighaṭādikaṁ ca no tamīśamātmānamupaiti śāśvatam ||33|| yasya svarūpātsacarācaraṁ jaga- dutpadyate tiṣṭhati līyate'pi vā | payovikārādiva phenabudbudā- stamīśamātmānamupaiti śāśvatam ||34|| nāsānirodho na ca dṛṣṭirāsanaṁ bodho'pyabodho'pi na yatra bhāsate | nāḍīpracāro'pi na yatra kiñci- ttamīśamātmānamupaiti śāśvatam ||35|| nānātvamekatvamubhatvamanyatā aṇutvadīrghatvamahattvaśūnyatā | mānatvameyatvasamatvavarjitaṁ tamīśamātmānamupaiti śāśvatam ||36|| susaṁyamī vā yadi vā na saṁyamī susaṁgrahī vā yadi vā na saṁgrahī | niṣkarmako vā yadi vā sakarmaka- stamīśamātmānamupaiti śāśvatam || 37|| mano na buddhirna śarīramindriyaṁ tanmātrabhūtāni na bhūtapañcakam | ahaṁkṛtiścāpi viyatsvarūpakaṁ tamīśamātmānamupaiti śāśvatam ||38|| vidhau nirodhe paramātmatāṁ gate na yoginaścetasi bhedavarjite | śaucaṁ na vāśaucamaliṅgabhāvanā sarvaṁ vidheyaṁ yadi vā niṣidhyate || 39|| mano vaco yatra na śaktamīrituṁ nūnaṁ kathaṁ tatra gurūpadeśatā | imāṁ kathāmuktavato gurosta- dyuktasya tattvaṁ hi samaṁ prakāśate ||40|| iti dvitīyo'dhyāyaḥ ||2|| atha tṛtīyo'dhyāyaḥ || guṇaviguṇavibhāgo vartate naiva kiñcit rativirativihīnaṁ nirmalaṁ niṣprapañcam | guṇaviguṇavihīnaṁ vyāpakaṁ viśvarūpaṁ kathamahamiha vande vyomarūpaṁ śivaṁ vai ||1|| śvetādivarṇarahito niyataṁ śivaśca kāryaṁ hi kāraṇamidaṁ hi paraṁ śivaśca | evaṁ vikalparahito'hamalaṁ śivaśca svātmānamātmani sumitra kathaṁ namāmi ||2|| nirmūlamūlarahito hi sadodito'haṁ nirdhūmadhūmarahito hi sadodito'ham | nirdīpadīparahito hi sadodito'haṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||3|| niṣkāmakāmamiha nāma kathaṁ vadāmi niḥsaṅgasaṅgamiha nāma kathaṁ vadāmi | niḥsārasārarahitaṁ ca kathaṁ vadāmi jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||4|| advaitarūpamakhilaṁ hi kathaṁ vadāmi dvaitasvarūpamakhilaṁ hi kathaṁ vadāmi | nityaṁ tvanityamakhilaṁ hi kathaṁ vadāmi jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||5|| sthūlaṁ hi no nahi kṛśaṁ na gatāgataṁ hi ādyantamadhyarahitaṁ na parāparaṁ hi | satyaṁ vadāmi khalu vai paramārthatattvaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||6|| saṁviddhi sarvakaraṇāni nabhonibhāni saṁviddhi sarvaviṣayāṁśca nabhonibhāṁśca | saṁviddhi caikamamalaṁ na hi bandhamuktaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||7|| durbodhabodhagahano na bhavāmi tāta durlakśyalakśyagahano na bhavāmi tāta | āsannarūpagahano na bhavāmi tāta jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||8|| niṣkarmadahano jvalano bhavāmi nirduḥkhaduḥkhadahano jvalano bhavāmi | nirdehadehadahano jvalano bhavāmi jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||9|| niṣpāpapāpadahano hi hutāśano'haṁ nirdharmadharmadahano hi hutāśano'ham | nirbandhabandhadahano hi hutāśano'haṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||10|| nirbhāvabhāvarahito na bhavāmi vatsa niryogayogarahito na bhavāmi vatsa | niścittacittarahito na bhavāmi vatsa jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||11|| nirmohamohapadavīti na me vikalpo niḥśokaśokapadavīti na me vikalpaḥ | nirlobhalobhapadavīti na me vikalpo jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||12|| saṁsārasantatilatā na ca me kadācit santoṣasantatisukho na ca me kadācit | ajñānabandhanamidaṁ na ca me kadācit jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||13|| saṁsārasantatirajo na ca me vikāraḥ santāpasantatitamo na ca me vikāraḥ | sattvaṁ svadharmajanakaṁ na ca me vikāro jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||14|| santāpaduḥkhajanako na vidhiḥ kadācit santāpayogajanitaṁ na manaḥ kadācit | yasmādahaṅkṛtiriyaṁ na ca me kadācit jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||15|| niṣkampakampanidhanaṁ na vikalpakalpaṁ svapnaprabodhanidhanaṁ na hitāhitaṁ hi | niḥsārasāranidhanaṁ na carācaraṁ hi jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||16|| no vedyavedakamidaṁ na ca hetutarkyaṁ vācāmagocaramidaṁ na mano na buddhiḥ | evaṁ kathaṁ hi bhavataḥ kathayāmi tattvaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||17|| nirbhinnabhinnarahitaṁ paramārthatattva- mantarbahirna hi kathaṁ paramārthatattvam | prāksambhavaṁ na ca rataṁ nahi vastu kiñcit jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||18|| rāgādidoṣarahitaṁ tvahameva tattvaṁ daivādidoṣarahitaṁ tvahameva tattvam | saṁsāraśokarahitaṁ tvahameva tattvaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||19|| sthānatrayaṁ yadi ca neti kathaṁ turīyaṁ kālatrayaṁ yadi ca neti kathaṁ diśaśca | śāntaṁ padaṁ hi paramaṁ paramārthatattvaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||20|| dīrgho laghuḥ punaritīha name vibhāgo vistārasaṁkaṭamitīha na me vibhāgaḥ | koṇaṁ hi vartulamitīha na me vibhāgo jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||21|| mātāpitādi tanayādi na me kadācit jātaṁ mṛtaṁ na ca mano na ca me kadācit | nirvyākulaṁ sthiramidaṁ paramārthatattvaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||22|| śuddhaṁ viśuddhamavicāramanantarūpaṁ nirlepalepamavicāramanantarūpam | niṣkhaṇḍakhaṇḍamavicāramanantarūpaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||23|| brahmādayaḥ suragaṇāḥ kathamatra santi svargādayo vasatayaḥ kathamatra santi | yadyekarūpamamalaṁ paramārthatattvaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||24|| nirneti neti vimalo hi kathaṁ vadāmi niḥśeṣaśeṣavimalo hi kathaṁ vadāmi | nirliṅgaliṅgavimalo hi kathaṁ vadāmi jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||25|| niṣkarmakarmaparamaṁ satataṁ karomi niḥsaṅgasaṅgarahitaṁ paramaṁ vinodam | nirdehadeharahitaṁ satataṁ vinodaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||26|| māyāprapañcaracanā na ca me vikāraḥ | kauṭilyadambharacanā na ca me vikāraḥ | satyānṛteti racanā na ca me vikāro jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||27|| sandhyādikālarahitaṁ na ca me viyogo- hyantaḥ prabodharahitaṁ badhiro na mūkaḥ | evaṁ vikalparahitaṁ na ca bhāvaśuddhaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||28|| nirnāthanātharahitaṁ hi nirākulaṁ vai niścittacittavigataṁ hi nirākulaṁ vai | saṁviddhi sarvavigataṁ hi nirākualaṁ vai jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||29|| kāntāramandiramidaṁ hi kathaṁ vadāmi saṁsiddhasaṁśayamidaṁ hi kathaṁ vadāmi | evaṁ nirantarasamaṁ hi nirākulaṁ vai jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||30|| nirjīvajīvarahitaṁ satataṁ vibhāti nirbījabījarahitaṁ satataṁ vibhāti | nirvāṇabandharahitaṁ satataṁ vibhāti jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||31|| sambhūtivarjitamidaṁ satataṁ vibhāti saṁsāravarjitamidaṁ satataṁ vibhāti | saṁhāravarjitamiadaṁ satataṁ vibhāti jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||32|| ullekhamātramapi te na ca nāmarūpaṁ nirbhinnabhinnamapi te na hi vastu kiñcit | nirlajjamānasa karoṣi kathaṁ viṣādaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||33|| kiṁ nāma rodiṣi sakhe na jarā na mṛtyuḥ kiṁ nāma rodiṣi sakhe na ca janma duḥkham | kiṁ nāma rodiṣi sakhe na ca te vikāro jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||34|| kiṁ nāma rodiṣi sakhe na ca te svarūpaṁ kiṁ nāma rodiṣi sakhe na ca te virūpam | kiṁ nāma rodiṣi sakhe na ca te vayāṁsi jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||35|| kiṁ nāma rodiṣi sakhe na ca te vayāṁsi kiṁ nāma rodiṣi sakhe na ca te manāṁsi | kiṁ nāma rodiṣi sakhe na tavendriyāṇi jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||36|| kiṁ nāma rodiṣi sakhe na ca te'sti kāmaḥ kiṁ nāma rodiṣi sakhe na ca te pralobhaḥ | kiṁ nāma rodiṣi sakhe na ca te vimoho jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||37|| aiśvaryamicchasi kathaṁ na ca te dhanāni aiśvaryamicchasi kathaṁ na ca te hi patnī | aiśvaryamicchasi kathaṁ na ca te mameti jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||38|| liṅgaprapañcajanuṣī na ca te na me ca nirlajjamānasamidaṁ ca vibhāti bhinnam | nirbhedabhedarahitaṁ na ca te na me ca jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||39|| no vāṇumātramapi te hi virāgarūpaṁ no vāṇumātramapi te hi sarāgarūpam | no vāṇumātramapi te hi sakāmarūpaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||40|| dhyātā na te hi hṛdaye na ca te samādhi- rdhyānaṁ na te hi hṛdaye na bahiḥ pradeśaḥ | dhyeyaṁ na ceti hṛdaye na hi vastu kālo jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||41|| yatsārabhūtamakhilaṁ kathitaṁ mayā te na tvaṁ na me na mahato na gururna na śiṣyaḥ | svacchandarūpasahajaṁ paramārthatattvaṁ jñānāmṛtaṁ samarasaṁ gaganopamo'ham ||42|| kathamiha paramārthaṁ tattvamānandarūpaṁ kathamiha paramārthaṁ naivamānandarūpam | kathamiha paramārthaṁ jñānavijñānarūpaṁ yadi paramahamekaṁ vartate vyomarūpam ||43|| dahanapavanahīnaṁ viddhi vijñānameka- mavanijalavihīnaṁ viddhi vijñānarūpam | samagamanavihīnaṁ viddhi vijñānamekaṁ gaganamiva viśālaṁ viddhi vijñānamekam ||44|| na śūnyarūpaṁ na viśūnyarūpaṁ na śuddharūpaṁ na viśuddharūpam | rūpaṁ virūpaṁ na bhavāmi kiñcit svarūparūpaṁ paramārthatattvam ||45|| muñca muñca hi saṁsāraṁ tyāgaṁ muñca hi sarvathā | tyāgātyāgaviṣaṁ śuddhamamṛtaṁ sahajaṁ dhruvam ||46|| iti tṛtīyo'dhyāyaḥ ||3|| atha caturtho'dhyāyaḥ || nāvāhanaṁ naiva visarjanaṁ vā puṣpāṇi patrāṇi kathaṁ bhavanti | dhyānāni mantrāṇi kathaṁ bhavanti samāsamaṁ caiva śivārcanaṁ ca ||1|| na kevalaṁ bandhavibandhamukto na kevalaṁ śuddhaviśuddhamuktaḥ | na kevalaṁ yogaviyogamuktaḥ sa vai vimukto gaganopamo'ham ||2|| sañjāyate sarvamidaṁ hi tathyaṁ sañjāyate sarvamidaṁ vitathyam | evaṁ vikalpo mama naiva jātaḥ svarūpanirvāṇamanāmayo'ham ||3|| na sāñjanaṁ caiva nirañjanaṁ vā na cāntaraṁ vāpi nirantaraṁ vā | antarvibhannaṁ na hi me vibhāti svarūpanirvāṇamanāmayo'ham ||4|| abodhabodho mama naiva jāto bodhasvarūpaṁ mama naiva jātam | nirbodhabodhaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||5|| na dharmayukto na ca pāpayukto na bandhayukto na ca mokśayuktaḥ | yuktaṁ tvayuktaṁ na ca me vibhāti svarūpanirvāṇamanāmayo'ham ||6|| parāparaṁ vā na ca me kadācit madhyasthabhāvo hi na cārimitram | hitāhitaṁ cāpi kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||7|| nopāsako naivamupāsyarūpaṁ na copadeśo na ca me kriyā ca | saṁvitsvarūpaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||8|| no vyāpakaṁ vyāpyamihāsti kiñcit na cālayaṁ vāpi nirālayaṁ vā | aśūnyaśūnyaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||9|| na grāhako grāhyakameva kiñcit na kāraṇaṁ vā mama naiva kāryam | acintyacintyaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||10|| na bhedakaṁ vāpi na caiva bhedyaṁ na vedakaṁ vā mama naiva vedyam | gatāgataṁ tāta kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||11|| na cāsti deho na ca me videho buddirmano me na hi cendriyāṇi | rāgo virāgaśca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||12|| ullekhamātraṁ na hi bhinnamuccai- rullekhamātraṁ na tirohitaṁ vai | samāsamaṁ mitra kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||13|| jitendriyo'haṁ tvajitendriyo vā na saṁyamo me niyamo na jātaḥ | jayājayau mitra kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||14|| amūrtamūrtirna ca me kadāci- dādyantamadhyaṁ na ca me kadācit | balābalaṁ mitra kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||15|| mṛtāmṛtaṁ vāpi viṣāviṣaṁ ca sañjāyate tāta na me kadācit | aśuddhaśuddhaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||16|| svapnaḥ prabodho na ca yogamudrā naktaṁ divā vāpi na me kadācit | aturyaturyaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||17|| saṁviddhi māṁ sarvavisarvamuktaṁ māyā vimāyā na ca me kadācit | sandhyādikaṁ karma kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||18|| saṁviddhi māṁ sarvasamādhiyuktaṁ saṁviddhi māṁ lakśyavilakśyamuktam | yogaṁ viyogaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||19|| mūrkho'pi nāhaṁ na ca paṇḍito'haṁ maunaṁ vimaunaṁ na ca me kadācit | tarkaṁ vitarkaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||20|| pitā ca mātā ca kulaṁ na jāti- rjanmādi mṛtyurna ca me kadācit | snehaṁ vimohaṁ ca kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||21|| astaṁ gato naiva sadodito'haṁ tejovitejo na ca me kadācit | sandhyādikaṁ karma kathaṁ vadāmi svarūpanirvāṇamanāmayo'ham ||22|| asaṁśayaṁ viddhi nirākulaṁ māṁ asaṁśayaṁ viddhi nirantaraṁ mām | asaṁśayaṁ viddhi nirañjanaṁ māṁ svarūpanirvāṇamanāmayo'ham ||23|| dhyānāni sarvāṇi parityajanti śubhāśubhaṁ karma parityajanti | tyāgāmṛtaṁ tāta pibanti dhīrāḥ svarūpanirvāṇamanāmayo'ham ||24|| vindati vindati na hi na hi yatra chandolakśaṇaṁ na hi na hi tatra | samarasamagno bhāvitapūtaḥ pralapati tattvaṁ paramavadhūtaḥ ||25|| iti caturtho'dhyāyaḥ ||4|| atha pañcamodhyāyaḥ || om iti gaditaṁ gaganasamaṁ tat na parāparasāravicāra iti | avilāsavilāsanirākaraṇaṁ kathamakśarabindusamuccaraṇam ||1|| iti tattvamasiprabhṛtiśrutibhiḥ pratipāditamātmani tattvamasi | tvamupādhivivarjitasarvasamaṁ kimu rodiṣi mānasi sarvasamam ||2|| adhaūrdhvavivarjitasarvasamaṁ bahirantaravarjitasarvasamam | yadi caikavivarjitasarvasamaṁ kimu rodiṣi mānasi sarvasamam ||3|| na hi kalpitakalpavicāra iti na hi kāraṇakāryavicāra iti | padasandhivivarjitasarvasamaṁ kimu rodiṣi mānasi sarvasamam ||4|| na hi bodhavibodhasamādhiriti na hi deśavideśasamādhiriti | na hi kālavikālasamādhiriti kimu rodiṣi mānasi sarvasamam ||5|| na hi kumbhanabho na hi kumbha iti na hi jīvavapurna hi jīva iti | na hi kāraṇakāryavibhāga iti kimu rodiṣi mānasi sarvasamam ||6|| iha sarvanirantaramokśapadaṁ laghudīrghavicāravihīna iti | na hi vartulakoṇavibhāga iti kimu rodiṣi mānasi sarvasamam ||7|| iha śūnyaviśūnyavihīna iti iha śuddhaviśuddhavihīna iti | iha sarvavisarvavihīna iti kimu rodiṣi mānasi sarvasamam ||8|| na hi bhinnavibhinnavicāra iti bahirantarasandhivicāra iti | arimitravivarjitasarvasamaṁ kimu rodiṣi mānasi sarvasamam ||9|| na hi śiṣyaviśiṣyasvarūpaiti na carācarabhedavicāra iti | iha sarvanirantaramokśapadaṁ kimu rodiṣi mānasi sarvasamam ||10|| nanu rūpavirūpavihīna iti nanu bhinnavibhinnavihīna iti | nanu sargavisargavihīna iti kimu rodiṣi mānasi sarvasamam ||11|| na guṇāguṇapāśanibandha iti mṛtajīvanakarma karomi katham | iti śuddhanirañjanasarvasamaṁ kimu rodiṣi mānasi sarvasamam ||12|| iha bhāvavibhāvavihīna iti iha kāmavikāmavihīna iti | iha bodhatamaṁ khalu mokśasamaṁ kimu rodiṣi mānasi sarvasamam ||13|| iha tattvanirantaratattvamiti na hi sandhivisandhivihīna iti | yadi sarvavivarjitasarvasamaṁ kimu rodiṣi mānasi sarvasamam ||14|| aniketakuṭī parivārasamaṁ ihasaṅgavisaṅgavihīnaparam | iha bodhavibodhavihīnaparaṁ kimu rodiṣi mānasi sarvasamam ||15|| avikāravikāramasatyamiti avilakśavilakśamasatyamiti | yadi kevalamātmani satyamiti kimu rodiṣi mānasi sarvasamam ||16|| iha sarvasamaṁ khalu jīva iti iha sarvanirantarajīva iti | iha kevalaniścalajīva iti kimu rodiṣi mānasi sarvasamam ||17|| avivekavivekamabodha iti avikalpavikalpamabodha iti | yadi caikanirantarabodha iti kimu rodiṣi mānasi sarvasamam ||18|| na hi mokśapadaṁ na hi bandhapadaṁ na hi puṇyapadaṁ na hi pāpapadam | na hi pūrṇapadaṁ na hi riktapadaṁ kimu rodiṣi mānasi sarvasamam ||19|| yadi varṇavivarṇavihīnasamaṁ yadi kāraṇakāryavihīnasamam | yadibhedavibhedavihīnasamaṁ kimu rodiṣi mānasi sarvasamam ||20|| iha sarvanirantarasarvacite iha kevalaniścalasarvacite | dvipadādivivarjitasarvacite kimu rodiṣi mānasi sarvasamam ||21|| atisarvanirantarasarvagataṁ atinirmalaniścalasarvagatam | dinarātrivivarjitasarvagataṁ kimu rodiṣi mānasi sarvasamam ||22|| na hi bandhavibandhasamāgamanaṁ na hi yogaviyogasamāgamanam | na hi tarkavitarkasamāgamanaṁ kimu rodiṣi mānasi sarvasamam ||23|| iha kālavikālanirākaraṇaṁ aṇumātrakṛśānunirākaraṇam | na hi kevalasatyanirākaraṇaṁ kimu rodiṣi mānasi sarvasamam ||24|| iha dehavidehavihīna iti nanu svapnasuṣuptivihīnaparam | abhidhānavidhānavihīnaparaṁ kimu rodiṣi mānasi sarvasamam ||25|| gaganopamaśuddhaviśālasamaṁ atisarvavivarjitasarvasamam | gatasāravisāravikārasamaṁ kimu rodiṣi mānasi sarvasamam ||26|| iha dharmavidharmavirāgatara- miha vastuvivastuvirāgataram | iha kāmavikāmavirāgataraṁ kimu rodiṣi mānasi sarvasamam ||27|| sukhaduḥkhavivarjitasarvasama- miha śokaviśokavihīnaparam | guruśiṣyavivarjitatattvaparaṁ kimu rodiṣi mānasi sarvasamam ||28|| na kilāṅkurasāravisāra iti na calācalasāmyavisāmyamiti | avicāravicāravihīnamiti kimu rodiṣi mānasi sarvasamam ||29|| iha sārasamuccayasāramiti | kathitaṁ nijabhāvavibheda iti | viṣaye karaṇatvamasatyamiti kimu rodiṣi mānasi sarvasamam ||30|| bahudhā śrutayaḥ pravadanti yato viyadādiridaṁ mṛgatoyasamam | yadi caikanirantarasarvasamaṁ kimu rodiṣi mānasi sarvasamam ||31|| vindati vindati na hi na hi yatra chandolakśaṇaṁ na hi na hi tatra | samarasamagno bhāvitapūtaḥ pralapati tattvaṁ paramavadhūtaḥ ||32|| iti pañcamo'dhyāyaḥ ||5|| atha ṣaṣṭhamo'dhyāyaḥ || bahudhā śrutayaḥ pravadanti vayaṁ viyadādiridaṁ mṛgatoyasamam | yadi caikanirantarasarvaśiva- mupameyamathohyupamā ca katham ||1|| avibhaktivibhaktivihīnaparaṁ nanu kāryavikāryavihīnaparam | yadi caikanirantarasarvaśivaṁ yajanaṁ ca kathaṁ tapanaṁ ca katham ||2|| mana eva nirantarasarvagataṁ hyaviśālaviśālavihīnaparam | mana eva nirantarasarvaśivaṁ manasāpi kathaṁ vacasā ca katham ||3|| dinarātrivibhedanirākaraṇa- muditānuditasya nirākaraṇam | yadi caikanirantarasarvaśivaṁ ravicandramasau jvalanaśca katham ||4|| gatakāmavikāmavibheda iti gataceṣṭaviceṣṭavibheda iti | yadi caikanirantarasarvaśivaṁ bahirantarabhinnamatiśca katham ||5|| yadi sāravisāravihīna iti yadi śūnyaviśūnyavihīna iti | yadi caikanirantarasarvaśivaṁ prathamaṁ ca kathaṁ caramaṁ ca katham ||6|| yadibhedavibhedanirākaraṇaṁ yadi vedakavedyanirākaraṇam | yadi caikanirantarasarvaśivaṁ tṛtīyaṁ ca kathaṁ turīyaṁ ca katham ||7|| gaditāviditaṁ na hi satyamiti viditāviditaṁ nahi satyamiti | yadi caikanirantarasarvaśivaṁ viṣayendriyabuddhimanāṁsi katham ||8|| gaganaṁ pavano na hi satyamiti dharaṇī dahano na hi satyamiti | yadi caikanirantarasarvaśivaṁ jaladaśca kathaṁ salilaṁ ca katham ||9|| yadi kalpitalokanirākaraṇaṁ yadi kalpitadevanirākaraṇam | yadi caikanirantarasarvaśivaṁ guṇadoṣavicāramatiśca katham ||10|| maraṇāmaraṇaṁ hi nirākaraṇaṁ karaṇākaraṇaṁ hi nirākaraṇam | yadi caikanirantarasarvaśivaṁ gamanāgamanaṁ hi kathaṁ vadati ||11|| prakṛtiḥ puruṣo na hi bheda iti na hi kāraṇakāryavibheda iti | yadi caikanirantarasarvaśivaṁ puruṣāpuruṣaṁ ca kathaṁ vadati ||12|| tṛtīyaṁ na hi duḥkhasamāgamanaṁ na guṇāddvitīyasya samāgamanam | yadi caikanirantarasarvaśivaṁ sthaviraśca yuvā ca śiśuśca katham ||13|| nanu āśramavarṇavihīnaparaṁ nanu kāraṇakartṛvihīnaparam | yadi caikanirantarasarvaśiva- mavinaṣṭavinaṣṭamatiśca katham ||14|| grasitāgrasitaṁ ca vitathyamiti janitājanitaṁ ca vitathyamiti | yadi caikanirantarasarvaśiva- mavināśi vināśi kathaṁ hi bhavet ||15|| puruṣāpuruṣasya vinaṣṭamiti vanitāvanitasya vinaṣṭamiti | yadi caikanirantarasarvaśiva- mavinodavinodamatiśca katham ||16|| yadi mohaviṣādavihīnaparo yadi saṁśayaśokavihīnaparaḥ | yadi caikanirantarasarvaśiva- mahameti mameti kathaṁ ca punaḥ ||17|| nanu dharmavidharmavināśa iti nanu bandhavibandhavināśa iti | yadi caikanirantarasarvaśivaṁ- mihaduḥkhaviduḥkhamatiśca katham ||18|| na hi yājñikayajñavibhāga iti na hutāśanavastuvibhāga iti | yadi caikanirantarasarvaśivaṁ vada karmaphalāni bhavanti katham ||19|| nanu śokaviśokavimukta iti nanu darpavidarpavimukta iti | yadi caikanirantarasarvaśivaṁ nanu rāgavirāgamatiśca katham ||20|| na hi mohavimohavikāra iti na hi lobhavilobhavikāra iti | yadi caikanirantarasarvaśivaṁ hyavivekavivekamatiśca katham ||21|| tvamahaṁ na hi hanta kadācidapi kulajātivicāramasatyamiti | ahameva śivaḥ paramārtha iti abhivādanamatra karomi katham ||22|| guruśiṣyavicāraviśīrṇa iti upadeśavicāraviśīrṇa iti | ahameva śivaḥ paramārtha iti abhivādanamatra karomi katham ||23|| na hi kalpitadehavibhāga iti na hi kalpitalokavibhāga iti | ahameva śivaḥ paramārtha iti abhivādanamatra karomi katham ||24|| sarajo virajo na kadācidapi nanu nirmalaniścalaśuddha iti | ahameva śivaḥ paramārtha iti abhivādanamatra karomi katham ||25|| na hi dehavidehavikalpa iti anṛtaṁ caritaṁ na hi satyamiti | ahameva śivaḥ paramārtha iti abhivādanamatra karomi katham ||26|| vindati vindati na hi na hi yatra chandolakśaṇaṁ na hi na hi tatra | samarasamagno bhāvitapūtaḥ pralapati tattvaṁ paramavadhūtaḥ ||27|| iti ṣaṣṭhamo'dhyāyaḥ ||6|| atha saptamo'dhyāyaḥ || rathyākarpaṭaviracitakanthaḥ puṇyāpuṇyavivarjitapanthaḥ | śūnyāgāre tiṣṭhati nagno śuddhanirañjanasamarasamagnaḥ ||1|| lakśyālakśyavivarjitalakśyo yuktāyuktavivarjitadakśaḥ | kevalatattvanirañjanapūto vādavivādaḥ kathamavadhūtaḥ ||2|| āśāpāśavibandhanamuktāḥ śaucācāravivarjitayuktāḥ | evaṁ sarvavivarjitaśānta- stattvaṁ śuddhanirañjanavantaḥ ||3|| kathamiha dehavidehavicāraḥ kathamiha rāgavirāgavicāraḥ | nirmalaniścalagaganākāraṁ svayamiha tattvaṁ sahajākāram ||4|| kathamiha tattvaṁ vindati yatra rūpamarūpaṁ kathamiha tatra | gaganākāraḥ paramo yatra viṣayīkaraṇaṁ kathamiha tatra ||5|| gaganākāranirantarahaṁsa- stattvaviśuddhanirañjanahaṁsaḥ | evaṁ kathamiha bhinnavibhinnaṁ bandhavibandhavikāravibhinnam ||6|| kevalatattvanirantarasarvaṁ yogaviyogau kathamiha garvam | evaṁ paramanirantarasarva- mevaṁ kathamiha sāravisāram ||7|| kevalatattvanirañjanasarvaṁ gaganākāranirantaraśuddham | evaṁ kathamiha saṅgavisaṅgaṁ satyaṁ kathamiha raṅgaviraṅgam ||8|| yogaviyogai rahito yogī bhogavibhogai rahito bhogī | evaṁ carati hi mandaṁ mandaṁ manasā kalpitasahajānandam ||9|| bodhavibodhaiḥ satataṁ yukto dvaitādvaitaiḥ kathamiha muktaḥ | sahajo virajaḥ kathamiha yogī śuddhanirañjanasamarasabhogī ||10|| bhagnābhagnavivarjitabhagno lagnālagnavivarjitalagnaḥ | evaṁ kathamiha sāravisāraḥ samarasatattvaṁ gaganākāraḥ ||11|| satataṁ sarvavivarjitayuktaḥ sarvaṁ tattvavivarjitamuktaḥ | evaṁ kathamiha jīvitamaraṇaṁ dhyānādhyānaiḥ kathamiha karaṇam ||12|| indrajālamidaṁ sarvaṁ yathā marumarīcikā | akhaṇḍitamanākāro vartate kevalaḥ śivaḥ ||13|| dharmādau mokśaparyantaṁ nirīhāḥ sarvathā vayam | kathaṁ rāgavirāgaiśca kalpayanti vipaścitaḥ ||14|| vindati vindati na hi na hi yatra chandolakśaṇaṁ na hi na hi tatra | samarasamagno bhāvitapūtaḥ pralapati tattvaṁ paramavadhūtaḥ ||15|| iti saptamo'dhyāyaḥ ||7|| atha aṣṭamo'dhyāyaḥ || tvadyātrayā vyāpakatā hatā te dhyānena cetaḥparatā hatā te | stutyā mayā vākparatā hatā te kśamasva nityaṁ trividhāparādhān ||1|| kāmairahatadhīrdānto mṛduḥ śucirakiñcanaḥ | anīho mitabhuk śāntaḥ sthiro maccharaṇo muniḥ ||2|| apramatto gabhīrātmā dhṛtimān jitaṣaḍguṇaḥ | amānī mānadaḥ kalpo maitraḥ kāruṇikaḥ kaviḥ ||3|| kṛpālurakṛtadrohastitikśuḥ sarvadehinām | satyasāro'navadyātmā samaḥ sarvopakārakaḥ ||4|| avadhūtalakśaṇaṁ varṇairjñātavyaṁ bhagavattamaiḥ | vedavarṇārthatattvajñairvedavedāntavādibhiḥ ||5|| āśāpāśavinirmukta ādimadhyāntanirmalaḥ | ānande vartate nityamakāraṁ tasya lakśaṇam ||6|| vāsanā varjitā yena vaktavyaṁ ca nirāmayam | vartamāneṣu varteta vakāraṁ tasya lakśaṇam ||7|| dhūlidhūsaragātrāṇi dhūtacitto nirāmayaḥ | dhāraṇādhyānanirmukto dhūkārastasya lakśaṇam ||8|| tattvacintā dhṛtā yena cintāceṣṭāvivarjitaḥ | tamo'haṁkāranirmuktastakārastasya lakśaṇam ||9|| dattātreyāvadhūtena nirmitānandarūpiṇā | ye paṭhanti ca śṛṇvanti teṣāṁ naiva punarbhavaḥ ||10|| iti aṣṭamo'dhyāyaḥ ||8|| iti avadhūtagītā samāptā ||


text:
Survey research/Exercise/Openly-available survey-based research study

Openly-available survey-based research study
Find a survey research article which is freely available on the web e.g., via Google search i.e., not behind a paid or university login.

This exercise is a bit unusual in its request to use openly available material, but the intention is to encourage people to access and share open research. Thus, the selected survey study examples should be openly available tor the world, not just to those with logins or able to afford access to closed databases.


text: Category:Survey research

title: Category:Survey research


text: User:Mokoia

title: User:Mokoia


text:
Votian/Transdialectal morpho-phonemic writing for Votic morphology

Transdialectal morpho-phonemic writing (2009.)
I use self-made morpho-phonemic system in this project to facilitate representing morphemes  of this language, which has no oficial standard  but a lot of dialectal alternatives .

Priorities

 * 1) Each  morpheme  should have only one graphic representation.
 * 2) This goal could be easier achieved using  Chinese   logograms.
 * 3) Representation of a  morpheme  should correspond to it's  morphophonemic  qualities.
 * 4)  Obstruents in the beginning of  closed syllables  with short vowels have been weakened due to the  consonant gradation, but this is not easy to explain relying only on contemporary Votic examples, because a weak grade (of  singular inessive :  orgo-za  ('in a valley')) opposed to a strong grade (of singular nominative:  orko- ) may have preserved no visible reason of this opposition any more until we ask advise of  phonological history  and  comparative linguistics . Even though there is a slight possibility, that Votic may have had an inessive morpheme different from other  Finnic  dialects , I have marked it by  Proto-Finnic  -ssa , which clearly explains the riddle above:  orgo-za  <  * orkos-sa .
 * 5) Representation of a  morpheme  should base on it's most complete and least damaged  allomorphs  found in Votian like dialects.
 * 6) E.g. I have marked  translative  by  Proto-Finnic  -kse, because it may be pronounced either -s̄ ,   -ssi ,   -hsi  or even -sse..
 * 7) This writing system should not mark  phonetic  differences caused by  phonetic context, if we don't want to emphasize a local or individual peculiarity.
 * 8) E.g. most of Votian morphemes have a front vowel and a back vowel alternative according to  vowel harmony  (I have preferred the one without  diaeresis  and other  diacritic  marks):
 * 9) uppot+ttu >  upottu  (" drowned ")
 * 10) hüppät+ttu >  üpättü  (" jumped ")

Concordance with phonology
This writing system bases on the phonemic orthography  used in  Dictionary of Votic language by Elna Adler  and  Merle Leppik with few exceptions:
 * Letters which mark lenis   plosives  like  ʙ, ᴅ, ɢ  are represented by their  strong grade  counterparts.
 *  Stem   morphemes  are represented by their  strong grade  if the weak grade is predictable.
 * · which marks a stressed  vowel, is usually ignored, because it has no  morphophonemic  significance.
 *  Anything written down using this morpho-phonemic system should be pronounced according to the individual ( dialectal ) background (or free choice) of a reader.
 *  '  may mark omitted  phonemes (usually in the end of a word    ), if we want to emphasize a local or individual peculiarity . E.g. " saunast' " < " saunass " < " saunass' ", " 'p' òlĺu' 'an'í anńéttava " < " b õllu ani annõttava " < " 'B õllu ani annõttava ".
 * 1)  Grave accent  flags a letter, which in that particular  phonetic context  should not be pronounced the way as usual, if we want to emphasize a local or individual peculiarity. E.g. " ò " in " 'p' òlĺu' 'an'í anńéttava " < " b õllu ani annõttava " < " 'B õllu ani annõttava " . Nevertheless it could be pronounced as usual o according to  Kukkuzi dialect.
 * 2)  Acute accent  flags a letter, which should be pronounced the way as it's written discarding any  phonetic context, if we want to emphasize a local or individual peculiarity. E.g. " ĺ " in " 'p' òlĺu' 'an'í anńéttava " < " b õllu ani annõttava " < " 'B õllu ani annõttava " . Nevertheless we could write it: " 'p' òlǹu' 'an'í anńéttava " presuming, that the reader is familiar with Votian morphophonology. We could write it even: " epi olnut hanhe antattava " and leave readers completely free to choose how to pronounce it.
 * A buffering i, which follows a consonant stem   may be omitted, because it is required there by  phonetic context . E.g. " mer " < " mer/i ".
 * That kind of final i is represented by stem vowel e in a vowel stem  . E.g. " í " in " 'an'í " < " b õllu ani annõttava " < " 'B õllu ani annõttava " could be written: " epi olnut hanhe antattava ", because Votic (Finnic) phonotactics  requires  i  <  * e  in such a position.

 * tš has been replaced by k . (It is a regular allophone  preceding a front wovel.) E.g. " kive " < " tšivi " < " čivi ".
 * ĺ has been replaced by l etc. E.g. " päällen " < " pääĺie " < " pǟl̨ėe ".
 *  ɑ, ɑ̈  have been replaced with  a, ä .
 * õ has been replaced by the vowel, which occurs in that particular stem  in  Kukkuzi  or any closest  Finnic  dialect . I.e. it may be:
 * 1)  o  e.g. noistak ('to rise') < nõis/a.
 * 2)  a  e.g. palatak ('to burn') < põl/õa.
 * 3)  u  e.g. puippu ('chick') < põip/põ.
 * 4)  e  e.g. pelto ('field') < põl/to .   e  and  è  may stand for õ, if there is not enough knowledge to choose another option.
 * yõ has been replaced by ee. E.g. " veeras " < " vyõrɑZ " < " vė̮e̮rɑZ ".
 * y has been preserved. E.g. " gryba " < " gr ba ", " õnnõvɑd by " < " e̮nne̮vɑd  ̮ b  ".
 * χ has been replaced by hh . E.g. " hhott " < " χott " < "  ott ".
 * Ordinary lower case letters represent  voiceless vowels  as well (even in  Kukkuzi dialect) . E.g. " siällä " < " siällÄ ".
 * 1) Upper case letters do not mark beginning of a  sentence.
 * 2) Upper case letters may mark  proper names  and  toponyms   e.g.  Kattila-lla on kaks külan ottsata  <  k a t t i л a - ллa on kahs tšülǟ  e̮ttsā  ('There are two (village-) ends in Kattila.')
 * 3) Upper case letters may mark components of a  compound morpheme  (the first one excluded) like in  Zulu language  (Zulu: isiZulu)  e.g.  akkunAlla ~ akkun'Al'la  <   * akkunalu'sella  <   * akkuna(n) aluksella  ('in the courtyard')
 * 4) Morphemes may be separated by   hyphen  too, like in  Hakka language  (Hakka: Hak-kâ-fa) e.g.  akkun-alla ~ akkun'-al'la  <   * akkunalu'sella  <   * akkuna(n) aluksella  ('in the courtyard')


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User:Egm6341.s2010.Team1.lei/HW3

For HW 3, I am responsible for 1 problem: 19-2.

=Problem 9-1 =

Find
a) Modify matlab code to make the comp. of $$ T_{0}\left ( 2^j \right )\,$$ efficient, i.e,$$ T_{0}\left ( 2^j \right )=T_{0}\left (2^{(j-1)}\right )+...\,$$

b) Romberg table, compare to previous results

Given
Cont'd  [[media:Egm6341.s10.mtg6.pdf|Lec. 6-5]]
 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle f(x)= \frac{e^x-1}{x} $$
 * $$\displaystyle I= \int\limits_{0}^{1}\frac{e^x-1}{x}\, dx $$
 * }
 * }
 * }

Solution
a) From [[media:Egm6341.s10.mtg18.pdf|Lec. 18-3]]

 * $$ T(2n)\,$$ comp. based on $$ T(n)\,$$, and $$ f(x_i), i=1,3,5,...\,$$

 * $$ T(n)=h [1/2f_0+f_1+f_2+...+f_{n-1}+1/2f_n], h=(b-a)/n \,$$

 * $$ T(2n)=\underbrace{\left (h [1/2f_0+f_2+f_4...+f_{2(n-1)}+1/2f_n]\right)}_{T(n)}+h [f_1+f_3+f_5+...+f_{2n-1}], h=(b-a)/2n \,$$

 * $$\Rightarrow T_{0}\left ( 2^j \right )= T_{0}\left (2^{(j-1)}\right )+ (b-a)/2n [f_1+f_3+f_5+...+f_{2(j-1)}] \,$$

Based on this method,modify matlab code to make the comp.of $$ T_{0}\left ( 2^j \right )\,$$ efficient.

Modified Matlab Code:

b) Romberg table display the results as follows:

Previous results:

From the comparation, for a given value of n,the most accurate value is for the largest value of k.

Matlab Code:

Egm6341.s10.team1.lei Egm6341.s10.team1.lei 16:06, 17 February 2010 (UTC)

title: User:Egm6341.s2010.Team1.lei/HW3