User:Cffk/sandbox
From Wikipedia, the free encyclopedia
 | This is the user sandbox of Cffk. A user sandbox is a subpage of the user's user page. It serves as a testing spot and page development space for the user and is not an encyclopedia article. Create or edit your own sandbox here. Other sandboxes: Main sandbox | Template sandbox Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in Wikipedia? Submit your draft for review! |
Extension to compute the divided difference of a polynomial
Horner's method can be modified to compute the dividing difference,
proceed as follows[1]
At completion, we have 
Notes
References
- Fateman, R. J.; Kahan, W. (2000). Improving exact integrals from symbolic algebra systems (PDF) (Report). PAM. University of California, Berkeley: Center for Pure and Applied Mathematics.
Misc
See (Carlson 2010)
- Milne-Thomson, L. M. (1983) [June 1964]. "Chapter 17, eqn 17.6.4". In Abramowitz, Milton; Stegun, Irene Ann (eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 599. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
- Carlson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
- Hofmann-Wellenhof, B.; Moritz, H. Physical Geodesy (2nd ed.). Springer. ISBN 978-3-211-33544-4. § 2.1: “The total force acting on a body at rest on the earth’s surface is the resultant of gravitational force and the centrifugal force of the earth’s rotation and is called gravity.”
{{cite book}}: CS1 maint: postscript (link) - Lee, L. P. (1976). "Conformal Projections Based on Jacobian Elliptic Functions". Cartographica: The International Journal for Geographic Information and Geovisualization. 13: 67–101. doi:10.3138/X687-1574-4325-WM62.
- Vose, M. D. (1991). "A linear algorithm for generating random numbers with a given distribution". IEEE Transactions on Software Engineering. 17 (9): 972–975. doi:10.1109/32.92917.
- Panneton, F. O.; l'Ecuyer, P.; Matsumoto, M. (2006). "Improved long-period generators based on linear recurrences modulo 2" (PDF). ACM Transactions on Mathematical Software. 32 (1): 1–16. doi:10.1145/1132973.1132974.
- Jacobi, C. G. J. (2009) [1866]. Lectures on Dynamics. Translated by K. Balagangadharan. Hindustan Book Agency. ISBN 978-81-85931-91-3. OCLC 440645889. English translation of Vorlesungen über Dynamik (Reimer, Berlin, 1866).
{{cite book}}: External link in|postscript= - Vincenty, T. (1975a). "Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations" (PDF). Survey Review. 23 (176): 88–93. doi:10.1179/sre.1975.23.176.88. Addendum: Survey Review 23 (180): 294 (1976).
{{cite journal}}: CS1 maint: postscript (link) - Borre, K.; Strang, W. G. (2012). Algorithms for Global Positioning. Wellesley-Cambridge Press. ISBN 978-0-9802327-3-8. OCLC 795014501. Chapter 11. Geometry of the Ellipsoid.
{{cite book}}: External link in|postscript= - Lyusternik, L. (1964). Shortest Paths: Variational Problems. Popular Lectures in Mathematics. Vol. 13. New York: Macmillan. MR 0178386. OCLC 1048605. Translation from Russian of Кратчайшие Линии: Вариационные Задачи (Moscow, 1955) by P. Collins and R. B. Brown
{{cite book}}: CS1 maint: postscript (link) - Sjöberg, L. E. (2012c). "Solutions to the direct and inverse navigation problems on the great ellipse". Journal of Geodetic Science. 2 (3): 200–205. doi:10.2478/v10156-011-0040-9.
- Botnev, V.A; Ustinov, S.M. (2014). "Методы решения прямой и обратной геодезических задач с высокой точностью [Metody resheniya pryamoy i obratnoy geodezicheskikh zadach s vysokoy tochnost'yu]" [Methods for direct and inverse geodesic problems solving with high precision] (PDF). St. Petersburg State Polytechnical University Journal (in Russian). 3 (198): 49–58.
- Bowring, B. R. (1984). "The direct and inverse solutions for the great elliptic line on the reference ellipsoid". Bulletin Géodésique. 58: 101–108. doi:10.1007/BF02521760.
- Williams, R. (1996). "The Great Ellipse on the Surface of the Spheroid". Journal of Navigation. 49 (2): 229–234. doi:10.1017/S0373463300013333.
- Walwyn, P. R. (1999). "The Great Ellipse Solution for Distances and Headings to Steer between Waypoints". Journal of Navigation. 52 (3): 421–424. doi:10.1017/S0373463399008516.
- Bennett, G. G. (1996). "Practical Rhumb Line Calculations on the Spheroid". Journal of Navigation. 49: 112–119. doi:10.1017/S0373463300013151.
- Carlton-Wippern, K. C. (1992). "On Loxodromic Navigation". Journal of Navigation. 45 (2): 292–297. doi:10.1017/S0373463300010791.
- Smart, W. M. (1946). "On a Problem in Navigation". Monthly Notices of the Royal Astronomical Society. 106 (2): 124–127. doi:10.1093/mnras/106.2.124.
{{cite journal}}: CS1 maint: unflagged free DOI (link) - Williams, J. E. D. (1950). "Loxodromic Distances on the Terrestrial Spheroid". Journal of Navigation. 3 (2): 133–140. doi:10.1017/S0373463300045549.
- Ehlert, D. (1993). Methoden der ellipsoidischen Dreiecksberechnung [Methods for ellipsoidal triangulatioin] (Technical report). Reihe B: Angewandte Geodäsie, Heft Nr. 292 (in German). Deutsche Geodätische Kommission. OCLC 257615376.
- Mathar, R. J. (2007). "Geodetic line at constant altitude above the ellipsoid". arXiv:0711.0642 [math.MG].
- Karney, C. F. F. (2011). "Geodesics on an ellipsoid of revolution". arXiv:1102.1215v1 [physics.geo-ph]. Errata.
{{cite arXiv}}: External link in|postscript= - Saito, T. (1979). "The computation of long geodesics on the ellipsoid through Gaussian quadrature". Journal of Geodesy. 53 (2): 165–177. Bibcode:1979BGeod..53..165S. doi:10.1007/BF02521087.
- Thomas, C. M.; Featherstone, W. E. (2005). "Validation of Vincenty's formulas for the geodesic using a new fourth-order extension of Kivioja's formula". Journal of Surveying Engineering. 131 (1): 20–26. doi:10.1061/(ASCE)0733-9453(2005)131:1(20). hdl:20.500.11937/36501.
- Bagratuni, G. V. (1967) [1962]. Course in Spheroidal Geodesy (PDF). OCLC 6150611. Translation of Курс сфероидической геодезии [Kurs Sferoidicheskoi Geodezii] by U.S. Air Force (FTD-MT-64-390)
{{cite book}}: External link in|postscript= - Tobey, W. M. (1928), Geodesy, Publication, vol. 11, Ottawa: Geodetic Survey of Canada
- Hammer, E. [in German] (1910). "Gegenazimutale Projektion" [Retro-azimuthal projection]. Petermanns Geographische Mitteilungen (in German). 56 (3): 153–155 + plate.
- Hinks, A. R. (1929). "A retro-azimuthal equidistant projection of the whole sphere". The Geographical Journal. 73 (3): 245–247. doi:10.2307/1784715. JSTOR 1784715.
- Lee, Y. C. (2011). "측지 역 문제 해석기법의 정확도 분석" [The accuracy analysis of methods to solve the geodetic inverse problem]. Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography (in Korean). 29 (4): 329–341. doi:10.7848/ksgpc.2011.29.4.329.
- Klingenberg, W. P. A. (1982). Riemannian Geometry. de Gruyer. ISBN 978-3-11-008673-7. MR 0666697. OCLC 8476832.
- Bessel, F. W. (1844a). "Ueber Veränderlichkeit der eigenen Bewegungen der Fixsterne" [Variations of the proper motions of the fixed stars]. Astronomische Nachrichten (in German). 22 (514): 145–160. Bibcode:1844AN.....22..145B. doi:10.1002/asna.18450221002.
- Bessel, F. W. (1844b). "Ueber Veränderlichkeit der eigenen Bewegungen der Fixsterne (Fortsetzung)" [Variations of the proper motions of the fixed stars (continued)]. Astronomische Nachrichten (in German). 22 (515): 169–184. Bibcode:1844AN.....22..169B. doi:10.1002/asna.18450221202.
- Bessel, F. W. (1844c). "On the variations of the proper motions of Procyon and Sirius". Monthly Notices of the Royal Astronomical Society. 6 (11): 136–141. Bibcode:1844MNRAS...6R.136B. doi:10.1093/mnras/6.11.136a.
{{cite journal}}: CS1 maint: unflagged free DOI (link) - Viik, T. (2006). F.W. Bessel and Geodesy (PDF). Struve Geodetic Arc 2006 International Conference: The Struve Arc and Extensions in Space and Time. August 13–15, 2006. Haparanda and Pajala, Sweden: Lantmäteriet, Gävle, Sweden, 2006. pp. 53–63.
- Bessel, F. W. (1838b). "On the parallax of 61 Cygni". Monthly Notices of the Royal Astronomical Society. 4 (17): 152–161. Bibcode:1838MNRAS...4..152B. doi:10.1093/mnras/4.17.152.
{{cite journal}}: CS1 maint: unflagged free DOI (link) - Bessel, F. W. (1838). "Bestimmung der Entfernung des 61sten Sterns des Schwans" [Determination of the distance to 61 Cygni]. Astronomische Nachrichten (in German). 16 (365–366): 65–96. Bibcode:1838AN.....16...65B. doi:10.1002/asna.18390160502.
- Bessel, Friedrich Wilhelm (1846). "Bessel's Tod" [Bessel's death]. Astronomische Nachrichten (in German). 24 (556): 49–52. Bibcode:1846AN.....24...49B. doi:10.1002/asna.18460240402.
{{cite journal}}: CS1 maint: ref duplicates default (link) - Bessel, F. W. (1837). "Bestimmung der Axen des elliptischen Rotationssphäroids, welches den vorhandenen Messungen von Meridianbögen der Erde am meisten entspricht" [Estimation of the axes of the ellipsoid through measurements of the meridian arc]. Astronomische Nachrichten (in German). 14 (333): 333–346. Bibcode:1837AN.....14..333B. doi:10.1002/asna.18370142301.
- Bessel, F. W. (1841). "Ueber einen Fehler in der Berechnung der französischen Gradmessung und seinen Einfluß auf die Bestimmung der Figur der Erde" [Concerning an error in the calculation of the French survey and its influence on the determination of the figure of the Earth]. Astronomische Nachrichten (in German). 19 (438): 97–116. Bibcode:1841AN.....19...97B. doi:10.1002/asna.18420190702.
- Knörrer, H. (1980). "Geodesics on the ellipsoid". Inventiones Mathematicae. 59 (2): 119–143. Bibcode:1980InMat..59..119K. doi:10.1007/BF01390041.
- Kivioja, L. A. (1971). "Computation of geodetic direct and indirect problems by computers accumulating increments from geodetic line elements". Bulletin Géodésique. 99 (1): 55–63. Bibcode:1971BGeod..45...55K. doi:10.1007/BF02521679.
- Karney, C. F. F. (2013). "Algorithms for geodesics". Journal of Geodesy. 87 (1): 43–42. arXiv:1109.4448. Bibcode:2013JGeod..87...43K. doi:10.1007/s00190-012-0578-z (open access). Addenda.
{{cite journal}}: External link in|postscript= - Itoh, J.-I.; Kiyohara, K. (2004). "The cut loci and the conjugate loci on ellipsoids". Manuscripta Mathematica. 114 (2): 247–264. doi:10.1007/s00229-004-0455-z.
- Carlson, B. C. (1995). "Numerical computation of real or complex elliptic integrals". Numerical Algorithms. 10 (1): 13–98. arXiv:math/9409227. Bibcode:1995NuAlg..10...13C. doi:10.1007/BF02198293.
- Beltrami, E. (1865). "Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette" [Mapping a surface to a plane so that geodesics are represented by straight lines]. Annali di Matematica Pura ed Applicata, Series 1 (in Italian). 7: 185–204. doi:10.1007/BF03198517.
- Saito, T. (1970). "The computation of long geodesics on the ellipsoid by non-series expanding procedure". Bulletin Géodésique. 98 (4): 341–373. Bibcode:1970BGeod..44..341S. doi:10.1007/BF02522166.
- Rainsford, H. F. (1955). "Long geodesics on the ellipsoid". Bulletin géodésique. 37 (3): 12–22. Bibcode:1955BGeod..29...12R. doi:10.1007/BF02527187.
- Oriani, B. (1826). "Auszug aus einem Briefe des Herrn Oriani an den Herausgeber" [Excerpt from a letter to the Editor]. Astronomische Nachrichten (in French). 4 (94): 461–466. Bibcode:1826AN......4..461O. doi:10.1002/asna.18260043201.
- Bessel, F. W. (1827). "Über einen Aufsatz von Ivory im Philosophical Magazine" [Comments on a paper by Ivory in the Philosophical Magazine]. Astronomische Nachrichten (in German). 5 (108): 177–180. Bibcode:1826AN......5..177B. doi:10.1002/asna.18270051202.
- Ivory, J. (1798). "A New Series for the Rectification of the Ellipsis". Transactions of the Royal Society of Edinburgh. 4 (2): 177–190. doi:10.1017/s0080456800030817.
- Ivory, J. (1826a). "On the properties of a line of shortest distance traced on the surface of an oblate spheroid, Part 1". Philosophical Magazine Series 1. 67 (336): 241–249. doi:10.1080/14786442608674051.
- Ivory, J. (1826). "On the properties of a line of shortest distance traced on the surface of an oblate spheroid, Part 2". Philosophical Magazine Series 1. 67 (337): 340–352. doi:10.1080/14786442608674069.
- Munk, W. H.; Forbes, A. M. G. (1989). "Global Ocean Warming: An Acoustic Measure?". Journal of Physical Oceanography. 19 (11): 1765–1778. Bibcode:1989JPO....19.1765M. doi:10.1175/1520-0485(1989)019<1765:GOWAAM>2.0.CO;2.
- Kim, J. J.; Burnside, W. D. (1986). "Simulation and analysis of antennas radiating in a complex environment". IEEE Transactions on Antennas and Propagation. 34 (4): 554–562. Bibcode:1986ITAP...34..554K. doi:10.1109/TAP.1986.1143838.
- Jacobi, C. G. J. (1837). "Zur Theorie der Variations-Rechnung und der Differential-Gleichungen" [The theory of the calculus of variations and of differential equations]. Journal für die reine und angewandte Mathematik (Crelles Journal) (in German). 1837 (17): 68–82. doi:10.1515/crll.1837.17.68.
- Beltrami, E. (1865). "Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette" [Mapping a surface to a plane so that geodesics are represented by straight lines]. Annali di Matematica Pura ed Applicata, Series 1 (in Italian). 7: 185–204. doi:10.1007/BF03198517.
- Bliss, G. A. (1916). "Jacobi's condition for problems of the calculus of variations in parametric form". Transactions of the American Mathematical Society. 17 (2): 195–206. doi:10.1090/S0002-9947-1916-1501037-4 (free access).
{{cite journal}}: CS1 maint: postscript (link) - Jacobi, C. G. J. (1837). "Zur Theorie der Variations-Rechnung und der Differential-Gleichungen" [The theory of the calculus of variations and of differential equations]. Journal für die reine und angewandte Mathematik (Crelles Journal) (in German). 1837 (17): 68–82. doi:10.1515/crll.1837.17.68.
- Jacobi, C. G. J. (1839). "Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution" [The geodesic on an ellipsoid and various applications of a remarkable analytical substitution]. Journal für die reine und angewandte Mathematik (Crelles Journal) (in German). 1839 (19): 309–313. doi:10.1515/crll.1839.19.309. Letter to Bessel, Dec. 28, 1838. French translation (1841).
{{cite journal}}: External link in|postscript= - Jacobi, C. G. J. (1855). "Nouvelles formules de Géodésie" [New formulas for geodesy]. Astronomische Nachrichten (in French). 41 (14). Communicated by E. Luther: 209–217. Bibcode:1855AN.....41..209J. doi:10.1002/asna.18550411401.
- Luther, E. (1855). "Jacobi's Ableitung der in seinem Aufsatze: 'Solution nouvelle d'un problème de Géodésie fondamental' enthaltenen Formeln" [Jacobi's derivation of the formulas in 'New solution of a fundamental problem of geodesy']. Astronomische Nachrichten (in German). 42 (22–23): 337–358. Bibcode:1856AN.....42..337J. doi:10.1002/asna.18550422201.
- Poincaré, H. (1905). "Sur les lignes géodésiques des surfaces convexes" [Geodesics lines on convex surfaces]. Transactions of the American Mathematical Society (in French). 6 (3): 237–274. doi:10.2307/1986219. JSTOR 1986219. PDF.
{{cite journal}}: External link in|postscript= - Sjöberg, L. E. (2007). "Precise determination of the Clairaut constant in ellipsoidal geodesy". Survey Review. 39 (303): 81–86. doi:10.1179/003962607X165014.
- Rollins, C. M. (2010). "An integral for geodesic length" (PDF). Survey Review. 42 (315): 20–26. doi:10.1179/003962609X451663.
- Bachoven von Echt, C. A. H (1865). Die Kürzeste auf dem Erdsphäroid nebst den Hauptaufgaben der Geodäsie in neuer Darstellung [Shortest lines on the spheroid together with the geodesic problems in a new representation] (in German). Coesfeld: Istwann.
- Sjöberg, L. E.; Shirazian, M. (2012). "Solving the direct and inverse geodetic problems on the ellipsoid by numerical integration". Journal of Surveying Engineering. 138 (1): 9–16. doi:10.1061/(ASCE)SU.1943-5428.0000061.
- Panou, G.; Delikaraoglou, D.; Korakitis, R. (2013). "Solving the geodesics on the ellipsoid as a boundary value problem". Journal of Geodetic Science. 3 (1): 40–47. Bibcode:2013JGeoS...3...40P. doi:10.2478/jogs-2013-0007 (open access).
{{cite journal}}: CS1 maint: postscript (link) - Panou, G. (2013). "The geodesic boundary value problem and its solution on a triaxial ellipsoid". Journal of Geodetic Science. 3 (3): 240–249. Bibcode:2013JGeoS...3..240P. doi:10.2478/jogs-2013-0028 (open access).
{{cite journal}}: CS1 maint: postscript (link) - Huang, T. T.; Quan, J. C.; Zhao, X. Y.; Song, X. S.; Xiao, R.; Yu, B. (2013). "The computation of azimuth and the shortest path between two points on the Earth's surface". Applied Mechanics and Materials. 364: 211–215. doi:10.4028/www.scientific.net/AMM.364.211.
- Karney, C. F. F.; Deakin, R. E. (2010). "F.W. Bessel (1825): The calculation of longitude and latitude from geodesic measurements". Astronomische Nachrichten. 331 (8): 852–861. arXiv:0908.1824. Bibcode:2010AN....331..852K. doi:10.1002/asna.201011352. (Errata).
{{cite journal}}: External link in|postscript=
- This is the English translation of Bessel, F. W. (1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen". Astronomische Nachrichten. 4 (16): 241–254. arXiv:0908.1823. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601.
- Bessel, F. W. (1825). "Ueber die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen" [The calculation of longitude and latitude from geodesic measurements]. Astronomische Nachrichten (in German). 4 (16): 241–253. arXiv:0908.1823. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601.
- Karney, C. F. F. (2007). "Quaternions in molecular modeling". Journal of Molecular Graphics and Modelling. 25 (5): 595–604. arXiv:physics/0506177. doi:10.1016/j.jmgm.2006.04.002. PMID 16777449.
- Karney, C. F. F. (2011). "Transverse Mercator with an accuracy of a few nanometers". Journal of Geodesy. 85 (8): 475–476. arXiv:1002.1417. Bibcode:2011JGeod..85..475K. doi:10.1007/s00190-011-0445-3.