{"title":"Transformations between Bivariate Polynomial Bases","authors":"Dimitris Varsamis, Nicholas Karampetakis","volume":94,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1307,"pagesEnd":1311,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/9999520","abstract":"
It is well known, that any interpolating polynomial
\r\np (x, y) on the vector space Pn,m of two-variable polynomials with
\r\ndegree less than n in terms of x and less than m in terms of y, has
\r\nvarious representations that depends on the basis of Pn,m that we
\r\nselect i.e. monomial, Newton and Lagrange basis e.t.c.. The aim of
\r\nthis short note is twofold : a) to present transformations between the
\r\ncoordinates of the polynomial p (x, y) in the aforementioned basis
\r\nand b) to present transformations between these bases.<\/p>\r\n","references":"[1] D. Hill, \"Interpolating polynomials and their coordinates relative to a\r\nbasis,\u201d The College Mathematics Journal, vol. 23, no. 4, pp. 329\u2013333,\r\n1992.\r\n[2] W. Gander, \"Change of basis in polynomial interpolation,\u201d Numerical\r\nLinear Algebra with Applications, vol. 12, no. 8, pp. 769\u2013778, 2005.\r\n[3] J. Kapusta and R. Smarzewski, \"Fast algorithms for multivariate\r\ninterpolation and evaluation at special points,\u201d Journal of Complexity,\r\nvol. 25, no. 4, pp. 332\u2013338, 2009.\r\n[4] J. Kapusta, \"An efficient algorithm for multivariate maclaurin-newton\r\ntransformation,\u201d in Annales UMCS, Informatica, vol. 8, no. 2. Versita,\r\n2008, pp. 5\u201314.\r\n[5] R. Smarzewski and J. Kapusta, \"Fast lagrange\u2013newton transformations,\u201d\r\nJournal of Complexity, vol. 23, no. 3, pp. 336\u2013345, 2007.\r\n[6] G. M. Phillips, Interpolation and approximation by polynomials.\r\nSpringer-Verlag, 2003.\r\n[7] E. Tyrtyshnikov, \"How bad are hankel matrices?\u201d Numerische\r\nMathematik, vol. 67, no. 2, pp. 261\u2013269, 1994.\r\n[8] W. Gautschi and G. Inglese, \"Lower bounds for the condition number\r\nof vandermonde matrices,\u201d Numerische Mathematik, vol. 52, no. 3, pp.\r\n241\u2013250, 1987.\r\n[9] A. Eisinberg and G. Fedele, \"On the inversion of the vandermonde\r\nmatrix,\u201d Applied mathematics and computation, vol. 174, no. 2, pp.\r\n1384\u20131397, 2006.\r\n[10] T. Sauer and Y. Xu, \"On multivariate lagrange interpolation,\u201d\r\nMathematics of Computation, vol. 64, pp. 1147\u20131170, 1995.\r\n[11] M. Gasca and T. Sauer, \"Polynomial interpolation in several variables,\u201d\r\nAdvances in Computational Mathematics, vol. 12, pp. 377\u2013410, 2000.\r\n[12] D. N. Varsamis and N. P. Karampetakis, \"On the newton bivariate\r\npolynomial interpolation with applications,\u201d Multidimensional Systems\r\nand Signal Processing, vol. 25, no. 1, pp. 179\u2013209, 2014.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 94, 2014"}