In-Between Forward and Central Difference Approximation for Evenly Spaced Data Using the Shift Operator

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2021 by IJMTT Journal
Volume-67 Issue-7
Year of Publication : 2021
Authors : Aishatu Imam Bello, Yohanna Sani Awari
  10.14445/22315373/IJMTT-V67I7P505

MLA

MLA Style: Aishatu Imam Bello, Yohanna Sani Awari"In-Between Forward and Central Difference Approximation for Evenly Spaced Data Using the Shift Operator" International Journal of Mathematics Trends and Technology 67.7 (2021):35-47. 

APA Style: Aishatu Imam Bello, Yohanna Sani Awari(2021). In-Between Forward and Central Difference Approximation for Evenly Spaced Data Using the Shift Operator International Journal of Mathematics Trends and Technology, 35-47.

Abstract
A number of different methods have been developed to construct useful interpolation formulas for evenly and unevenly spaced points. We developed a new interpolation formula obtained through a combination of Newton’s Gregory forward interpolation and a modified form of Gauss backward interpolation formula using the shift operator. This was achieved through the reduction in the subscripts of Gauss’s backward formula by one unit and replacing s by s-1. Comparison of the newly developed method with their counterparts was also carried out and results show that the new formula is very efficient and possess good accuracy for evaluating functional values between given data.

Reference

[1] C. F. Gauss., Theoriainterpolationismethodo nova tractate. In Werke. Gottingen, Germany: Königlichen Gesellschaft der Wissenschaften, 3 (1866) 265–327.
[2] G. D. Birkhoff., General mean value and remainder theorems with applications to mechanical differentiation and quadrature. Trans. Amer. Math. Soc., 7(1) (1906) 107–136.
[3] I. Newton., Methodus differentialis. In The Mathematical Papers of Isaac Newton, D. T. Whiteside, and Ed. Cambridge, U.K.: Cambridge Univ. Press, 8(4) (1981) 236–257.
[4] I. Newton (1960). Letter to Oldenburg., In The Correspondence of Isaac Newton, H.W. Turnbull, and Ed. Cambridge, U.K.: Cambridge Univ. Press, 2 (1676) 110–161.
[5] H. Briggs., ArithmeticaLogarithmica. London, U.K.: GuglielmusIones., (1624).
[6] H. Briggs., Trigonometria Britannica. Gouda. The Netherlands: Petrus Rammasenius., (1633).
[7] E. Waring., Problems concerning interpolations. Philos. Trans. R. Soc. London, 69 (1779) 59–67.
[8] J. Wallis., ArithmeticaInfinitorum. Hildesheim, Germany: Olms Verlag. Kahaner, David, Cleve Moler, and Stephen Nash (1989). Numerical Methods and Software. Englewood Cliffs, NJ: Prentice Hall. , (1972).
[9] Kendall E. Atkinson, An Introduction to Numerical Analysis, 2nd edition, John Wiley & Sons, New York., (1989).
[10] P. S. de Laplace., Mémoiresur les suites (1779). In EuvresComplètes de Laplace, Paris, France: Gauthier-Villars et Fils, 10 (1894) 1–89.
[11] P. S. de Laplace., ThéorieAnalytique des Probabilités, 3rd ed., Paris, France: Ve. Courcier., (1820).
[12] R. C. Gupta., Second order interpolation in Indian mathematics up to the fifteenth century. Ind. J. Hist. Sci., 4(1–2) (1969) 86–98.
[13] R. L. Burden, J. D. Faires., Numerical Analysis, Seventh edition, Brooks/Cole, Pacific Grove, CA., (2001).
[14] S. A. Joffe., Interpolation -formulae and central-difference notation. Trans. Actuar. Soc. Amer., 18 (1917) 72–98.
[15] S. D. Conte, Carl de Boor., Elementary Numerical Analysis, 3rd edition, Mc Graw-Hill, New York, USA., (1980).
[16] John H. Mathews, Kurtis D. Fink., Numerical methods using MATLAB, 4th edition, Pearson Education, USA., (2004).

Keywords : Shift operator, forward difference interpolation, Gauss’s backward formula, Symbolic method, evenly spaced points..