InProceedings of the fifth Berkeley symposium on mathematical statistics and probability (Vol. Google Scholar, Cohen AJ, Brauer M, Burnett R, Anderson HR, Frostad J, Estep K, Balakrishnan K, Brunekreef B, Dandona L, Dandona R, Feigin V (2017) Estimates and 25-year trends of the global burden of disease attributable to ambient air pollution: an analysis of data from the Global Burden of Diseases Study 2015. In chapter 1 of this handbook, reference is made to the following five product groups: All food items (SITC codes 0, 1, 22 and 4 . A staff study of the United Nations Research Institute for Social Development (UNRISD) (pp. Authors like Sen (1983), Goossens et al. Sala-i-Martin, X. BMJ(Overseas and retired doctor ed. Geographic regions (3) Upper-Middle Income Economies (UMIE) with GNI between $3996 and $12,375. https://doi.org/10.1073/pnas.95.25.14863, Everitt BS, Landau S, Leese M (2001) Cluster analysis arnold. (1970). Springer. Atmos Environ 75:383392, Shanahan DF, Lin BB, Gaston KJ, Bush R, Fuller RA (2014) Socio-economic inequalities in access to nature on public and private lands: a case study from Brisbane, Australia. All authors have accepted responsibility for the entire content of this manuscript and approved its submission. The classifications are updated each year on July 1 and are based on the GNI per capita of the previous year (2021). Acute multidimensional poverty: A new index for developing countries. Using data mining for learning and clustering FCM. Economies are currently divided into four income groupings: low, lower-middle, upper-middle, and high. United Nations Development Programme (UNDP). New Economic Foundation (NEF). The current unavailability of electricity have further worsened the situation. XI: Journal of Economics Perspectives. All of your personal information, including email address, name, and IP address will be deleted from this site. Consider the classification of high-, middle-, and low-income countries and note what factors are thought of as advantageous. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Country Classification The PPI Projects Database uses a number of variables to characterize a country. This debate crosses political, economic, and social arguments. Rev Environ Health 23(4):243297, CAS Twin peaks: Growth and convergence in models of distribution dynamics. Thank you for asking. Fifty years ago, this number was 57 per cent. 40, 13251352. Economic development in the third world (4th ed.). . The main classifications provided are by geographic region, by income group, and by the operational lending categories of the World Bank Group. United Nations development programme human development report office background paper. Country classifications For analytical purposes, the WESP classifies all countries of the world into one of three broad categories: developed economies, economies in transition and developing . Environ Impact Assess Rev 27(2):161175, dos Santos MA, Damazio JM, Rogerio JP, Amorim MA, Medeiros AM, Abreu JL, Maceira ME, Melo AC, Rosa LP (2017) Estimates of GHG emissions by hydroelectric reservoirs: the Brazilian case. Or it has some other status? Luo, D., & Song, B. Accordingly, the World Development Reports by the World Bank classifies countries into four income groups. your institution. Provided by the Springer Nature SharedIt content-sharing initiative, Environmental Science and Pollution Research, https://doi.org/10.1007/s11356-023-26679-x, access via Grey relational evaluation of innovation competency in an aviation industry cluster. Starting with the April 2021 WEO, real GDP data and forecasts for New Zealand are reported on a production basis rather than an expenditure basis. GeoJournal Having taken each of the dimensions of development independently, our results show that no cluster has performed poorly in all four aspects. Zarandi, S., Shahabi, V., Emami, M., Rezaee, A., & Hassanloo, M. (2014). Bi, K. X., Zheng, Z. W., & Yang, S. F. (2011). World Development, 26(3), 517528. The situation on the ground is that production has gone down due to the unavailability of foreign currency to import critical raw materials, new machinery to retool and the unpredictable economic operating environment. (1970). Department of Economics, Christ University, Bengaluru, Karnataka, 560029, India, Department of Economics, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam, Andhra Pradesh, 515134, India, You can also search for this author in Partners in development: Report of the commission on international development. Classification of goods. Environmental Science and Pollution Research Handbook on constructing composite indicators: Methodology and user guide. SEE ALSO: > Developing Countries in "Understanding the WTO" Measurement and analysis of progress at the local level (Vols. Environment Statistics United Nations Development Programme. RPR: Have made substantial contributions to the conceptionand revising the work. (2010), Schepelmann et al. This list may not reflect recent changes. I'd like to understand the change of classification regarding Comoros. The lancet 383(9919):785795, Article The United Nations Statistics Division is committed to the advancement of the global statistical system. The interior point where the first order conditions are met is therefore a minimum. On the methodological framework of composite indices: A review of the issues of weighting, aggregation, and robustness. Grey relational analysis method of linguistic information and its application in group decision. The classification is updated each year on July 1st. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Article 2 The UNDP country classification system also classifies. European Economic Review, 35(1), 350357. is it developed?? The authors declares that they have no conflict of interests. 20. New York: Praeger. I am looking for an excel sheet showing the historical classification of countries by income, after 2005. Management Science Letters, 4(10), 23192324. Or under development?? Deng, J. (2008). Application of data mining techniques to classify world stock markets. What does it means then to coutry in lower middle income? Journal of Economic Theory, If you continue to navigate this website beyond this page, cookies will be placed on your browser. 3136). A comprehensive review of literature on construction of development index can be found in Basel et al. The measurement of economic performance and social progress revisited: reflections and overview. 509564). Srinivasan, T. N. (1994). Banking Statistical Classifications The income classification is based on a measure of national income per person, or GNI per capita, calculated using the Atlas method. Learn more about Institutional subscriptions, Twenty-three years hence similar language is still in use in the World Bank. USEPA. The areas of the triangles are as follows: Therefore, the sum of the areas of the triangles is. The single indicator often used to classify the countries on the basis of level of development is the per capita gross domestic product expressed in U.S. dollars (PCGDP). High income: $12,616 or more. 106, 10451055. Data Centre. Goossens, Y., Mkip, A., Schepelmann, P., Van de Sand, I., Kuhndt, M., & Herrndorf, M. (2007). These conditions are met because \( L(x) \) is strictly convex. A modified Human Development Index. This site uses cookies to optimize functionality and give you the best possible experience. Lynge Nielsen. Social Indicators Research, 141(1), 6194. Grey Systems: Theory and Application, 4(2), 321327. Hu, R., & Xia, H. (2012). Biol Conserv 141(6):16951703, Mujtaba G, Shahzad SJH (2021) Air pollutants, economic growth and public health: implications for sustainable development in OECD countries. What about east African countries their income categorical level, especially Somalia? Background: Antineoplastic drugs produce serious drug-related problems and their management is challenging. Please review the terms of use for this website. List of social concerns common to most OECD countries, OECD social indicator development programme. Until last year (Fiscal Year 2019), the income classifications had an analytical purpose and did not influence the World Banks lending terms. What is the GNI range for so-called LMIC? A grey decision and prediction model for investment in the core competitiveness of product development. Human development report. Provided by the Springer Nature SharedIt content-sharing initiative, https://doi.org/10.1007/s10708-021-10479-2, access via United Nations Research Institute for Social Development (UNRISD). Wilkinson, R. G., Pickett, K. E., & De Vogli, R. (2010). Pearson Education Limited. Indicators enable . Income is measured using gross national income (GNI) per capita, in U.S. dollars, converted from local currency using the World Bank Atlas method. Provided by the Springer Nature SharedIt content-sharing initiative, $$ \frac{{\partial^{2} E_{3} }}{{\partial x_{0}^{2} }} = \frac{{x_{1} L^{\prime \prime} (x_{0} )}}{2} > 0 $$, $$ \frac{{\partial^{2} E_{3} }}{{\partial x_{1}^{2} }} = \frac{{(1 - x_{0} )L^{\prime \prime} (x_{1} )}}{2} > 0.$$, $$ \left| {\begin{array}{*{20}c} {\frac{{\partial^{2} E_{3} }}{{\partial x_{0}^{2} }}} & {\frac{{\partial^{2} E_{3} }}{{\partial x_{1} \partial x_{0} }}} \\ {\frac{{\partial^{2} E_{3} }}{{\partial x_{0} \partial x_{1} }}} & {\frac{{\partial^{2} E_{3} }}{{\partial x_{1}^{2} }}} \\ \end{array} } \right| = \left| {\begin{array}{*{20}c} {\frac{{x_{1} L^{\prime \prime} (x_{0} )}}{2}} & {\frac{{L^{\prime \prime} (x_{0} ) - L^{\prime \prime} (x_{1} )}}{2}} \\ {\frac{{L^{\prime} (x_{0} ) - L^{\prime} (x_{1} )}}{2}} & {\frac{{(1 - x_{0} )L^{\prime \prime} (x_{1} )}}{2}} \\ \end{array} } \right| \ge^{?} United States Environmental Protection Agency, Research Triangle Park, NC, USA. This is a preview of subscription content, access via Retrieved from http://web.worldbank.org/WBSITE/EXTERNAL/DATASTATISTICS/0,,contentMDK:204870, World Economic Situation and Progress (WESP). The country and lending groups page provides a complete list of economies classified by income, region, and lending status and links to previous years' classifications. IMF Working Paper 11/31. As of July 1, 2019, the new thresholds for classification by income are: The following countries are assigned to new income groups: The country and lending groups page provides a complete list of economies classified by income, region, and lending status and links to previous years classifications. The World Bank classifies the world's economies into four income groups high, upper-middle, lower-middle, and low. The International Monetary Fund 197278: Cooperation on trial. many thanks. Beyond low and middle income countries: What if there were five clusters of developing countries?. They are the basis for international data statistical collections and ensure the comparabiltity of data provided by countries around the world. Economic country classifications (6 C, 41 P) M. Microstates (5 P) S. Small Island Developing States (56 P) T. Types of countries (1 C, 29 P) Pages in category "Country classifications" The following 24 pages are in this category, out of 24 total. Praeger. Journal of Social and Economic Development, 20(2), 308326. Jain, A. K., & Dubes, R. C. (1988). Analysis of the income elasticity of the consumer demand of Chinese rural residents and prediction of its trend. (2023. https://techxplore.com/news/2023-01-leases-significantly-uk-offshore-power.html, Zandpour F, Harich KR (1996) Think and feel country clusters: a new approach to international advertising standardization. Rich and poor countries Categories of development of countries in the world today. 28:7787, Lim S, Kim J, Kim T, Lee K, Yang W, Jun S, Yu S (2012) Personal exposures to PM2. I have reservations on the upgrading of Zimbabwe from a low income to lower- middle income status as from 1 July 2019. Nielsen, L. (2011). Classifications of countries based on their level of development: How it is done and how it could be done. (2012). Environ Health Perspect 115(6):848855, Zhang W, Lu Z, Xu Y, Wang C, Gu Y, Xu H, Streets DG (2018) Black carbon emissions from biomass and coal in rural China. Sci Total Environ 657:234241, Ramanathan V, Feng Y (2009) Air pollution, greenhouse gases and climate change: Global and regional perspectives. Keywords: Country classification systems, developing countries, developed countries Author's E-Mail Address:lnielsen@imf.org . Big data for official statistics Application of metabolic GM (1, 1) model in financial repression approach to the financing difficulty of the small and medium-sized enterprises. Beyond GDP: The need for new measures of progress. Grey Systems: Theory and Application, 2(2), 284298. Applying fuzzy grey relational analysis for ranking the advanced manufacturing systems. The datasets generated and/or analyzed during the current study are available in the World Bank database 2018 (https://databank.worldbank.org/data/source/world-development-indicators#). Econometric Reviews, 32(1), 734. Income per capita: Countries have been categorized into four income level groups (low, lower-middle, upper-middle, and high) consistent with the World Bank classification of economies and its Atlas gross national income per capita estimates. International Monetary Fund, 700 19th Street, N.W., Washington, DC, 20431, USA, You can also search for this author in ), Advances in grey systems research (pp. These systems are found lacking in clarity with regard to their underlying rationale. (2018). Model-based clustering based on sparse finite Gaussian mixtures. (2008). Other variables measure supply of infrastructure services. The variable shows the expected lifetime income of a newborn at a 0% discount rate under the assumption of no future income growth. Classification of Economies Experts debate exactly how to define the level of economic development of a countrywhich criteria to use and, therefore, which countries are truly developed. American Economic Review, Inequality among the world citizens: 18201992. While difficult conceptual issues need to be recognized, the pragmatic starting point of this paper is that there is a need for such (2011). Cameron, A. C., & Miller, D. L. (2015). (1975). Algorithms for clustering data. Writingreviewing and editing: Pallavi Kudal, Amitabh Patnaik, Raj Kumar Satankar, Sunny Dawar, and Prince Dawar. 3,000 new books annually, covering a wide range of subjects including biomedicine and the life sciences, clinical medicine, Any feedback you have provided that others have supported will be attributed to "Anonymous". The paper analyzes how the UNDP, the World Bank, and the IMF classify countries based on their level of development. Developing countries comprise a majority of the WTO membership. IMF (2010) Selected Decisions and Selected Documents of the International Monetary Fund, Thirty-fifth Issue (Washington). People above the age of 64 are projected to make up 16 . the high-income threshold is also a determining factor for lending rates, https://datahelpdesk.worldbank.org/knowledgebase/articles/906519-world-, https://datahelpdesk.worldbank.org/knowledgebase/articles/906519. Grey Systems: Theory and Application, 2(2), 148156. https://doi.org/10.1007/s11356-023-26679-x, DOI: https://doi.org/10.1007/s11356-023-26679-x. This is a preview of subscription content, access via Energy 133:99107, Downs GM, Barnard JM (2002) Clustering methods and their uses in computational chemistry. (2006). Data for GNI, GNI per capita, GDP, GDP PPP, and Population for 2018 are now available on World Bank's Open Data Catalog. Atkinson, A. Most of the products labelled 'made in Zimbabwe'' are only repackaged in Zimbabwe after being imported in bulk such as cooking oil, washing soaps and other basic commodities. your institution, https://indianexpress.com/article/lifestyle/health/air-pollution-severe-hazardous-delhi-ncr-animals-toxic-air-air-quality-index-peta-india-6119128/, https://www.un.org/en/development/desa/news/population/world-urbanization-prospects-2014.html, https://www.who.int/airpollution/data/cities-2016/en/. The IMF Press Center is a password-protected site for working journalists. See note on GNP and other terminology. Open Journal of Applied Sciences, 7(12), 661690. Some methods for classification and analysis of multivariate observations. International Bank for Reconstruction and Development, Gross Domestic Product (GDP)/Gross National Income (GNI), Quarterly External Debt Statistics (QEDS), Financing & Risk On the measurement of inequality. Int J Advert. A countrys GNI per capita, which can change with economic growth, inflation, exchange rates, and population. All of your ideas without support will be deleted. Environ Sci Pollut Res 28:1268612698, Nagendra H, Munroe DK, Southworth J (2004) From pattern to process: landscape fragmentation and the analysis of land use/land cover change. Indicators are clues that helps to detect DRPs within the healthcare organization and minimize overall harm from medications. DRPs are critical, for saving on therapeutic costs, particularly in resource poor settings within low-middle-income countries such as India. Liu, Y., Du, J., & Li, J. Based on commonalities between various variables connected to the global environmental sector, we developed a novel typology of country clusters. I will share it for others.i am interested because I want to know about progress in Pakistan economy. Could you explain me why is there difference between the value here and the actual GNI per capita dataset? Springer Nature or its licensor (e.g. CAS According to the United Nations' stages of economic development for classifying countries with respect to levels of industrialization, _____ consist of industrially developing countries just entering world trade with relatively low per capita income. Decision Science Letters, 3(3), 269274. A novel grey object matrix incidence clustering model for panel data and its application. 5. Elsevier. International Social Science Journal, 47(143), 2537. Nber. Human development: The debate about the index. Pardee Working Paper No.4. Sen, A. 2010/11. 575586). IDS Working Papers,2012(404), pp. (2009) Costanza et al. Basel, S., Gopakumar, K.U. We compile and disseminate global statistical information, develop standards and norms for statistical activities, and support countries' efforts to strengthen their national statistical systems. To learn more about cookies, click here. OECD Publishing. The human development indices: Are they redundant? Costanza, R., Hart, M., Posner, S., & Talberth, J. European Environment Agency, Eisen MB, Spellman PT, Brown PO, Botstein D (1998) Cluster analysis and display of genome-wide expression patterns. Shi, H. X., Liu, S. F., Fang, Z. G., & Zhang, A. A framework for diagnosing changes in evolving data streams. Decancq, K., & Lugo, M. A. 140. Scientia Iranica, 28(1), 371385. UNRISD. Noise filtering of new motion capture markers using modified K-means. (No. Published by Social Progress Imperative can be retrieved from https://www.socialprogress.org/. Journal of Social and Economic Development, 21(1), 182206. New York: Random House. GeoJournal 87, 39153933 (2022). Contents and measurement of socioeconomic development. List of classifications in the UNSD national classifications database (as of 18 June 2021). The Zimbabwean market is mainly dominated by foreign products mainly from South Africa. Operational lending categories (1974). Can be accessed from https://unctad.org/en/pages/publications/World-Economic-Situation-and-Prospects-(Series).aspx. https://doi.org/10.1111/j.0265-0487.1996.00032.pp.x, Zhang J, Smith KR (2007) Household air pollution from coal and biomass fuels in China: measurements, health impacts, and interventions. Lim, D., Anthony, P., & Chong Mun, H. (2012). Ravallion, M. (1997). 0 $$, $$ x_{1} (1 - x_{0} )L^{\prime \prime} (x_{0} )L^{\prime \prime} (x_{1} ) - (L^{\prime} (x_{0} ) - L^\prime (x_{1} ))^{2} $$, $$ L^{\prime \prime} (x_{0} ) = \frac{{L\prime (x_{1} )x_{1} - L(x_{1} )}}{{x_{1}^{2} }} = \frac{{\frac{{1 - L(x_{0} )}}{{1 - x_{0} }}x_{1} - L(x_{1} )}}{{x_{1}^{2} }} = \frac{{1 - L(x_{0} )}}{{(1 - x_{0} )x_{1} }} - \frac{{L(x_{1} )}}{{x_{1}^{2} }} $$, $$ L^{\prime \prime} (x_{1} ) = \frac{{ - L\prime (x_{0} )(1 - x_{0} ) + 1 - L(x_{0} )}}{{(1 - x_{0} )^{2} }} = \frac{{ - \frac{{L(x_{1} )}}{{x_{1} }}(1 - x_{0} ) + 1 - L(x_{0} )}}{{(1 - x_{0} )^{2} }} = - \frac{{L(x_{1} )}}{{(1 - x_{0} )x_{1} }} + \frac{{1 - L(x_{0} )}}{{(1 - x_{0} )^{2} }} $$, $$ \begin{aligned} &(1 - x_{0} )x_{1} L^{\prime \prime} (x_{0} )L^{\prime \prime} (x_{1} ) \\ & \quad = (1 - x_{0} )x_{1} \left\{ {\frac{{1 - L(x_{0} )}}{{(1 - x_{0} )x_{1} }} - \frac{{L(x_{1} )}}{{x_{1}^{2} }}} \right\}\left\{ { - \frac{{L(x_{1} )}}{{(1 - x_{0} )x_{1} }} + \frac{{1 - L(x_{0} )}}{{(1 - x_{0} )^{2} }}} \right\} \\ & \quad = \left\{ {1 - L(x_{0} ) - \frac{{L(x_{1} )(1 - x_{0} )}}{{x_{1} }}} \right\}\left\{ { - \frac{{L(x_{1} )}}{{(1 - x_{0} )x_{1} }} + \frac{{1 - L(x_{0} )}}{{(1 - x_{0} )^{2} }}} \right\} \\ & \quad = - \frac{{L(x_{1} )}}{{(1 - x_{0} )x_{1} }} + \frac{{1 - L(x_{0} )}}{{(1 - x_{0} )^{2} }} + \frac{{L(x_{0} )L(x_{1} )}}{{(1 - x_{0} )x_{1} }} - \frac{{L(x_{0} )[1 - L(x_{0} )]}}{{(1 - x_{0} )^{2} }} + \frac{{L^{2} (x_{1} )}}{{x_{1}^{2} }} - \frac{{L(x_{1} )[1 - L(x_{0} )]}}{{(1 - x_{0} )x_{1} }} \\ & \quad = \frac{{ - L(x_{1} ) + L(x_{0} )L(x_{1} ) - L(x_{1} )[1 - L(x_{0} )]}}{{(1 - x_{0} )x_{1} }} + \frac{{1 - L(x_{0} ) - L(x_{0} )[1 - L(x_{0} )]}}{{(1 - x_{0} )^{2} }} + \frac{{L^{2} (x_{1} )}}{{x_{1}^{2} }} \\ &\quad = \frac{{ - L(x_{1} )[1 - L(x_{0} )] - L(x_{1} )[1 - L(x_{0} )]}}{{(1 - x_{0} )x_{1} }} + \frac{{1 - L(x_{0} ) - L(x_{0} ) + L^{2} (x_{0} )}}{{(1 - x_{0} )^{2} }} + \frac{{L^{2} (x_{1} )}}{{x_{1}^{2} }} \\ & \quad = - 2\frac{{L(x_{1} )}}{{x_{1} }}\frac{{1 - L(x_{0} )}}{{1 - x_{0} }} + \frac{{(1 - L(x_{0} ))^{2} }}{{(1 - x_{0} )^{2} }} + \frac{{L^{2} (x_{1} )}}{{x_{1}^{2} }} \\ & \quad = \left[ {\frac{{L(x_{1} )}}{{x_{1} }} - \frac{{1 - L(x_{0} )}}{{1 - x_{0} }}} \right]^{2} \\ \end{aligned} $$, $$ T_{1} = \frac{1}{2}(x_{1} - x_{0} )(L(x_{1} ) - L(x_{0} )) = \frac{1}{2}x_{1} L(x_{1} ) - \frac{1}{2}x_{1} L(x_{0} ) - \frac{1}{2}x_{0} L(x_{1} ) + \frac{1}{2}x_{0} L(x_{0} ) $$, $$ T_{2} = \frac{1}{2}(x_{2} - x_{1} )(L(x_{2} ) - L(x_{1} )) = \frac{1}{2}x_{2} L(x_{2} ) - \frac{1}{2}x_{2} L(x_{1} ) - \frac{1}{2}x_{1} L(x_{2} ) + \frac{1}{2}x_{1} L(x_{1} ) $$, $$ T_{n - 2} = \frac{1}{2}(x_{n - 2} - x_{n - 3} )(L(x_{n - 2} ) - L(x_{n - 3} )) = \frac{1}{2}x_{n - 2} L(x_{n - 2} ) - \frac{1}{2}x_{n - 2} L(x_{n - 3} ) - \frac{1}{2}x_{n - 3} L(x_{n - 2} ) + \frac{1}{2}x_{n - 3} L(x_{n - 3} ) $$, $$ T_{n - 1} = \frac{1}{2}(1 - x_{n - 2} )(1 - L(x_{n - 2} )) = \frac{1}{2} - \frac{1}{2}L(x_{n - 2} ) - \frac{1}{2}x_{n - 2} + \frac{1}{2}x_{n - 2} L(x_{n - 2} ) $$, $$ \sum\limits_{i = 0}^{n - 1} {T_{i} } = \sum\limits_{i = 0}^{n - 2} {x_{i} L(x_{i} ) - \frac{1}{2}\sum\limits_{i = 0}^{n - 3} {x_{i + 1} L(x_{i} ) - \frac{1}{2}\sum\limits_{i = 0}^{n - 3} {x_{i} L(x_{i + 1} )} } } - \frac{1}{2}L(x_{n - 2} ) - \frac{1}{2}x_{n - 2} + \frac{1}{2} $$, $$ R_{0} = (x_{1} - x_{0} )L(x_{0} ) = x_{1} L(x_{0} ) - x_{0} L(x_{0} ) $$, $$ R_{1} = (x_{2} - x_{1} )L(x_{1} ) = x_{2} L(x_{1} ) - x_{1} L(x_{1} ) $$, $$ R_{2} = (x_{3} - x_{2} )L(x_{2} ) = x_{3} L(x_{2} ) - x_{2} L(x_{2} ) $$, $$ R_{n - 3} = (x_{n - 2} - x_{n - 3} )L(x_{n - 3} ) = x_{n - 2} L(x_{n - 3} ) - x_{n - 2} L(x_{n - 2} ) $$, $$ R_{n - 2} = (1 - x_{n - 2} )L(x_{n - 2} ) = L(x_{n - 2} ) - x_{n - 2} L(x_{n - 2} ) $$, $$ \sum\limits_{i = 0}^{n - 2} {R_{i} } = \sum\limits_{i = 0}^{n - 3} {x_{i + 1} L(x_{i} ) + L(x_{n - 2} ) - \sum\limits_{i = 0}^{n - 2} {x_{i} L(x_{i} )} } $$, $$ \begin{aligned} E_{n} & = \sum\limits_{i = 0}^{n - 1} {T_{i} } + \sum\limits_{i = 0}^{n - 2} {R_{i} } - \int\limits_{0}^{1} {L(x)} \\ & = \sum\limits_{i = 0}^{n - 2} {x_{i} L(x_{i} ) - \frac{1}{2}\sum\limits_{i = 0}^{n - 3} {x_{i + 1} L(x_{i} ) - \frac{1}{2}\sum\limits_{i = 0}^{n - 3} {x_{i} L(x_{i + 1} )} } } - \frac{1}{2}L(x_{n - 2} ) - \frac{1}{2}x_{n - 2} + \frac{1}{2} \\ & \quad + \sum\limits_{i = 0}^{n - 3} {x_{i + 1} L(x_{i} ) + L(x_{n - 2} ) - \sum\limits_{i = 0}^{n - 2} {x_{i} L(x_{i} )} } - \int\limits_{0}^{1} {L(x)} \\ & = \frac{1}{2}\sum\limits_{i = 0}^{n - 3} {x_{i + 1} L(x_{i} ) - \frac{1}{2}\sum\limits_{i = 0}^{n - 3} {x_{i} L(x_{i + 1} )} } + \frac{1}{2}L(x_{n - 2} ) - \frac{1}{2}x_{n - 2} + \frac{1}{2} - \int\limits_{0}^{1} {L(x)} \\ \end{aligned} $$, \( x_{n - 1} \equiv L(x_{n - 1} ) \equiv 1, \), $$ E_{n} = E_{1} + \frac{1}{2}\sum\limits_{i = 0}^{n - 2} {x_{i + 1} L(x_{i} ) - \frac{1}{2}\sum\limits_{i = 0}^{n - 2} {x_{i} L(x_{i + 1} )} } $$, $$ \left( \begin{array} {l} \frac{{\partial E_{n} }}{{\partial x_{0} }} \hfill \\ \frac{{\partial E_{n} }}{{\partial x_{1} }} \hfill \\ \frac{{\partial E_{n} }}{{\partial x_{2} }} \hfill \\ \vdots \hfill \\ \frac{{\partial E_{n} }}{{\partial x_{n - 2} }} \hfill \\ \end{array} \right) = \frac{1}{2}\left( \begin{array} {l} x_{1} L^\prime (x_{0} ) - L(x_{1} ) \hfill \\ L(x_{0} ) + x_{2} L^\prime (x_{1} ) - x_{0} L^\prime (x_{1} ) - L(x_{2} ) \hfill \\ L(x_{1} ) + x_{3} L^\prime (x_{2} ) - x_{1} L^\prime (x_{2} ) - L(x_{3} ) \hfill \\ \vdots \hfill \\ L(x_{n - 3} ) + L^\prime (x_{n - 2} ) - x_{n - 3} L^\prime (x_{n - 2} ) - 1 \hfill \\ \end{array} \right) = 0 \Leftrightarrow \left( \begin{array} {l} L^\prime (x_{0} ) \hfill \\ L^\prime (x_{1} ) \hfill \\ L^\prime (x_{2} ) \hfill \\ \vdots \hfill \\ L^\prime (x_{n - 2} ) \hfill \\ \end{array} \right) = \left( \begin{array} {l} \frac{{L(x_{1} )}}{{x_{1} }} \hfill \\ \frac{{L(x_{2} ) - L(x_{0} )}}{{x_{2} - x_{0} }} \hfill \\ \frac{{L(x_{3} ) - L(x_{1} )}}{{x_{3} - x_{1} }} \hfill \\ \vdots \hfill \\ \frac{{1 - L(x_{n - 3} )}}{{1 - x_{n - 3} }} \hfill \\ \end{array} \right) $$, https://doi.org/10.1007/s11205-012-0191-9, access via
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