Inversion of Matrix by Elementary Column Transformation: Representation of Numerical Data by a Polynomial Curve

Biswajit Das; Dhritikesh Chakrabarty

doi:https://doi.org/10.14445/22315373/IJMTT-V42P507

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 42 | Number 1 | Year 2017 | Article Id. IJMTT-V42P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V42P507

Inversion of Matrix by Elementary Column Transformation: Representation of Numerical Data by a Polynomial Curve

Biswajit Das, Dhritikesh Chakrabarty

Abstract

There is necessity of a method/formula, for representing a given set of numerical data on a pair of variables by a the curve, in interpolation by the approach which consists of the representation of numerical data by a polynomial first and then to compute the value of the dependent variable from the polynomial corresponding to any given value of the independent variable. Due to this necessity, a method has been composed for representing a given set of numerical data on a pair of variables by a polynomial curve. The method has been composed with the help of elementary column transformation of matrix. This paper describes the method composed here with numerical example in order to show the application of the method to numerical data.

Keywords

Numerical data, polynomial curve, matrix inversion, Gauss Jordan method.

References

1. Abramowitz M (1972) : “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”, 9th printing. New York: Dover, p. 880.
2. Bathe K. J. & Wilson E. L. (1976): “Numerical Methods in Finite Element Analysis”, Prentice-Hall, Englewood Cliffs, NJ.
3. Biswajit Das & Dhritikesh Chakrabarty (2016a): “Lagranges Interpolation Formula: Representation of Numerical Data by a Polynomial Curve”, International Journal of Mathematics Trend and Technology (ISSN (online): 2231-5373, ISSN (print): 2349-5758), 34 part-1 (2), 23-31.
4. Biswajit Das & Dhritikesh Chakrabarty (2016b): “Newton’s Divided Difference Interpolation Formula: Representation of Numerical Data by a Polynomial Curve”, International Journal of Mathematics Trend and Technology (ISSN (online): 2231-5373, ISSN (print): 2349-5758), 35 part-1 (3), 26-32.
5. Biswajit Das & Dhritikesh Chakrabarty (2016c): “Newton’s Forward Interpolation Formula: Representation of Numerical Data by a Polynomial Curve”, International Journal of Applied Research (ISSN (online): 2456 - 1452, ISSN (print): 2456 - 1452), 1 (2), 36-41.
6. Biswajit Das & Dhritikesh Chakrabarty (2016d): “Newton’s Backward Interpolation Formula: Representation of Numerical Data by a Polynomial Curve”, International Journal of Statistics and Applied Mathematics (ISSN (online): 2394-5869, ISSN (print): 2394-7500), 2 (10), 513-517.
7. Biswajit Das & Dhritikesh Chakrabarty (2016e): “Matrix Inversion: Representation of Numerical Data by a Polynomial curve”, Aryabhatta Journal of Mathematics & Informatics (ISSN (print): 0975-7139, ISSN (online): 2394-9309), 8(2), 267-276.
8. Biswajit Das & Dhritikesh Chakrabarty (2016f): “Inversion of Matrix by Elementary Transformation: Representation of Numerical Data by a Polynomial Curve”, Journal of Mathematis and System Sciences (ISSN (print): 0975-5454), 12, 27-32.
9. Chutia Chandra & Gogoi Krishna (2013): “NEWTON’S INTERPOLATION FORMULAE IN MS EXCEL WORKSHEET”, “International Journal of Innovative Research in Science Engineering and Technology”, Vol. 2, Issue 12.
10. Chwaiger J. (1994): “On a Characterization of Polynomials by Divided Differences”, “Aequationes Math”, 48, 317-323.
11. C.F. Gerald & P.O. Wheatley (1994): “Applied Numerical Analysis”, fifth ed., Addison-Wesley Pub. Co., MA.
12. Chwaiger J. (1994): “On a Characterization of Polynomials by Divided Differences”, “Aequationes Math”, 48, 317-323.
13. Corliss J. J. (1938): “Note on an Eatension of Lagrange's Formula”, “American Mathematical Monthly”, Jstor, 45(2), 106—107.
14. De Boor C (2003): “A divided difference expansion of a divided difference”, “J. Approx. Theory ”,122 , 10–12.
15. Dokken T & Lyche T (1979): “A divided difference formula for the error in Hermite interpolation” , “ BIT ”, 19, 540–541.
16. David R. Kincard & E. Ward Chaney (1991):” Numerical analysis”, Brooks /Cole, Pacific Grove, CA.
17. Endre S & David Mayers (2003): “An Introduction to Numerical Analysis”, Cambridge, UK.
18. Erdos P. & Turan P. (1938): “On Interpolation II: On the Distribution of the Fundamental Points of Lagrange and Hermite Interpolation”, “The Annals of Mathematics, 2nd Ser”, Jstor, 39(4), 703—724.
19. Echols W. H. (1893): “On Some Forms of Lagrange's Interpolation Formula”, “The Annals of Mathematics”, Jstor, 8(1/6), 22—24.
20. Fred T (1979): “Recurrence Relations for Computing with Modified Divided Differences”, “Mathematics of Computation”, Vol. 33, No. 148, 1265-1271.
21. Floater M (2003): “Error formulas for divided difference expansions and numerical differentiation”, “J. Approx. Theory”, 122, 1–9.
22. Gertrude Blanch (1954): “On Modified Divided Differences”, “Mathematical Tables and Other Aids to Computation”, Vol. 8, No. 45, 1-11.
23. Hummel P. M. (1947): “A Note on Interpolation (in Mathematical Notes)”, “American Mathematical Monthly”, Jstor, 54(4), 218—219.
24. Herbert E. Salzer (1962): “Multi-Point Generalization of Newton's Divided Difference Formula”, “Proceedings of the American Mathematical Society”, Jstor, 13(2), 210—212.
25. Jordan C (1965): “Calculus of Finite Differences”, 3rd ed, New York, Chelsea.
26. Jeffreys H & Jeffreys B.S. (1988) : “Divided Differences” , “ Methods of Mathematical Physics ”, 3rd ed, 260-264
27. Jan K. Wisniewski (1930): “Note on Interpolation (in Notes)”, “Journal of the American Statistical Association”, Jstor 25(170), 203—205.
28. John H. Mathews & Kurtis D.Fink (2004): “Numerical methods using MATLAB”, 4ed, Pearson Education, USA.
29. James B. Scarborough (1996): “Numerical Mathematical Analysis”, 6 Ed, The John Hopkins Press, USA.
30. Kendall E.Atkinson (1989): “An Introduction to Numerical Analysis”, 2 Ed., New York.
31. Lee E.T.Y. (1989): “A Remark on Divided Differences”, “American Mathematical Monthly”, Vol. 96, No 7, 618-622.
32. Mills T. M. (1977): “An introduction to analysis by Lagrange interpolation”, “Austral. Math. Soc. Gaz.”, 4(1), MathSciNet, 10—18.
33. Nasrin Akter Ripa (2010): “Analysis of Newton’s Forward Interpolation Formula”, “International Journal of Computer Science & Emerging Technologies (E-ISSN: 2044-6004)”, 12, Volume 1, Issue 4.
34. Neale E. P. & Sommerville D. M. Y. (1924): “A Shortened Interpolation Formula for Certain Types of Data”, “Journal of the American Statistical Association”, Jstor, 19(148), 515—517.
35. Quadling D. A. (1966): “Lagrange's Interpolation Formula”, “The Mathematical Gazette”, L(374), 372—375.
36. Robert J.Schilling & Sandra L. Harries (2000): “Applied Numerical Methods for Engineers”, Brooks /Cole, Pacific Grove, CA.
37. Revers & Michael Bull (2000): “On Lagrange interpolation with equally spaced nodes”, “Austral. Math. Soc MathSciNet.”, 62(3), 357-368.
38. S.C. Chapra & R.P. Canale (2002): “Numerical Methods for Engineers”, third ed., McGraw-Hill, NewYork.
39. S.D. Conte & Carl de Boor (1980): “Elementary Numerical Analysis”, 3 Ed, McGraw-Hill, New York, USA.
40. Traub J. F. (1964): “On Lagrange-Hermite Interpolation”, “Journal of the Society for Industrial and Applied Mathematics”, Jstor , 12(4), 886—891.
41. Vertesi P (1990): “SIAM Journal on Numerical Analysis”, Vol. 27, No. 5, pp. 1322-1331 , Jstor.
42. Whittaker E. T. and Robinson G (1967): “The Gregory-Newton Formula of Interpolation" and "An Alternative Form of the Gregory-Newton Formula", “The Calculus of Observations: A Treatise on Numerical Mathematics”, 4th ed, pp. 10-15.
43. Whittaker E. T. & Robinson G. (1967): “Divided Differences & Theorems on Divided Differences”, “The Calculus of Observations: A Treatise on Numerical Mathematics”,4th ed., New York, 20-24.
44. Wang X & Yang S (2004): “On divided differences of the remainder of polynomial interpolation”, www.math.uga.edu/-mjlai/pub.html.
45. Wang X & Wang H (2003): “Some results on numerical divided difference formulas”, www.math.uga.edu/-mjlai/pub.html.
46. Whittaker E. T. & Robinson G. (1967): “Lagrange's Formula of Interpolation”, “The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed.”, §17, New York: Dover, 28-30.

Citation :

Biswajit Das, Dhritikesh Chakrabarty, "Inversion of Matrix by Elementary Column Transformation: Representation of Numerical Data by a Polynomial Curve," International Journal of Mathematics Trends and Technology (IJMTT), vol. 42, no. 1, pp. 45-49, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V42P507