Some numerical integration methods based on Bernstein polynomials

In this work we find numerical integration of a function by some approximations of the function using Bernstein polynomials, and we introduce a new fast method to approximate the value of a definite integral and we compare them.

[1]  Hrushikesh Narhar Mhaskar,et al.  Fundamentals of Approximation Theory , 2000 .

[2]  S. M. Hashemiparast,et al.  Numerical integration using local Taylor expansions in nodes , 2007, Appl. Math. Comput..

[3]  Christoph W. Ueberhuber,et al.  Applications of numerical integration , 1994 .

[4]  Peter Deuflhard,et al.  Numerical Analysis in Modern Scientific Computing , 2003 .

[5]  J. Pečarić,et al.  A generalization of general two-point formula with applications in numerical integration , 2008 .

[6]  Bert Jüttler,et al.  The dual basis functions for the Bernstein polynomials , 1998, Adv. Comput. Math..

[7]  Mehdi Dehghan,et al.  Numerical integration using the derivatives , 2006, Appl. Math. Comput..

[8]  Christoph W. Ueberhuber,et al.  Numerical Integration on Advanced Computer Systems , 1994, Lecture Notes in Computer Science.

[9]  T. J. Rivlin.  An Introduction to the Approximation of Functions , 2003 .

[10]  Knut Petras,et al.  Principles of verified numerical integration , 2007 .

[11]  Mircea D. Farcas,et al.  About Bernstein polynomials , 2008 .

[12]  P. Revesz.  Interpolation and Approximation , 2010 .

[13]  József Szabados,et al.  Weighted approximation of functions on the real line by Bernstein polynomials , 2004, J. Approx. Theory.

[14]  David P. Landau,et al.  Numerical integration using Wang-Landau sampling , 2007, Comput. Phys. Commun..

[15]  Birendra Nath Mandal,et al.  Numerical solution of some classes of integral equations using Bernstein polynomials , 2007, Appl. Math. Comput..

[16]  D. Xiu.  Numerical integration formulas of degree two , 2008 .

[17]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[18]  G. Tachev.  Voronovskaja's theorem revisited , 2008 .

[19]  Erwin Kreyszig,et al.  Bernstein polynomials and numerical integration , 1979 .

[20]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[21]  J. Reinkenhof.  Differentiation and integration using Bernstein's polynomials , 1977 .