Abstract
Bu çalışma, n 1 farklı noktadan geçen en fazla n. dereceden bir aradeğer polinomu ile ilgilidir. Aradeğer polinomunun katsayıları doğrusal denklem sistemi olarak yazılmıştır. Bu katsayılardan oluşan denklem sistemi, Vandermonde matrisinin tersinin kapalı biçiminin kullanımı ile çözülmüştür. Aradeğer polinomunun katsayıları, toplam ve çarpım simgelerini kullanarak elde edilmiştir. Geliştirilen formülleri kullanarak aradeğer polinomunun katsayıları için bir algoritma üretilmiştir. Ayrıca, eşit aralıklı noktalar için katsayılar ileri fark ile de formüle edilmiştir. n 1 farklı noktadan geçen, en fazla n. dereceden olan aradeğer polinomunun katsayıları geliştirilen formüller ile hesaplanabileceği ve ara değer hesabına kolaylıkla uygulanabileceği görülmüştür. Bu görüşü destekleyecek örneklere yer verilmiştir.
References
- Eisinberg A. and Fedele G., On the inversion of the Vandermonde matrix, Appl. Math. Comput. (2006), 174 p.1384-1397.
- Martinez J.J. and Pena J.M., Factorizations of Cauchy-Vandermonde matrices, Linear Algebra Appl. (1998), 284 p.229-237.
- Oruç H. and Phillips G.M., Explicit factorization of the Vandermonde matrix, Linear Algebra Appl. (2000), 315 p.113-123.
- Oruç H. and Akmaz H.K., Symmetric functions and Vandermonde matrix, J. Comput. Appl. Math. (2004), 172 p. 49-64.
- Meyer C.D., Matrix Analysis and Applied Linear Algebra, (2000), Siam.
- Björck A. and Pereyra V., Solution of Vandermonde systems of equations, Math. Comput. (1970), Vol.24, 112 p. 893-903.
- Mühlbach G., Interpolation by Cauchy-Vandermonde systems and applications, J. Comput. Appl. Math. (2000), 122 p. 203-222.
- Safak S., On the Trivariate Polynomial Interpolation, Wseas Transactions on Mathematics,(2012), Is. 8, Vol 11, p. 722-730.
THE FORMULAE FOR COMPUTING THE COEFFICIENTS OF THE POLYNOMIAL INTERPOLATION PASSING THROUGH n 1 DISTINCT POINTS
Abstract
This paper deals with the polynomial interpolation of degree at most n passing through n 1 distinct points. The coefficients of the polynomial interpolation are written as a system of the linear equations. The system consisting of the coefficients is solved by the use of the closed form of the inverse of the Vandermonde matrix. The coefficients of the interpolation are obtained by using the sum and product symbols. The algorithm for the coefficients of the polynomial interpolation is developed by generating formulae. Also, these coefficients for equidistant points are formulated by forward difference. It is seen that the coefficients of the interpolation of degree at most n passing through n 1 distinct points can be computed directly by generating special
formulae and can be applied easily to the polynomial interpolation. Numerical examples are represented.
References
- Eisinberg A. and Fedele G., On the inversion of the Vandermonde matrix, Appl. Math. Comput. (2006), 174 p.1384-1397.
- Martinez J.J. and Pena J.M., Factorizations of Cauchy-Vandermonde matrices, Linear Algebra Appl. (1998), 284 p.229-237.
- Oruç H. and Phillips G.M., Explicit factorization of the Vandermonde matrix, Linear Algebra Appl. (2000), 315 p.113-123.
- Oruç H. and Akmaz H.K., Symmetric functions and Vandermonde matrix, J. Comput. Appl. Math. (2004), 172 p. 49-64.
- Meyer C.D., Matrix Analysis and Applied Linear Algebra, (2000), Siam.
- Björck A. and Pereyra V., Solution of Vandermonde systems of equations, Math. Comput. (1970), Vol.24, 112 p. 893-903.
- Mühlbach G., Interpolation by Cauchy-Vandermonde systems and applications, J. Comput. Appl. Math. (2000), 122 p. 203-222.
- Safak S., On the Trivariate Polynomial Interpolation, Wseas Transactions on Mathematics,(2012), Is. 8, Vol 11, p. 722-730.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Review Articles |
| Authors | |
| Publication Date | February 3, 2016 |
| Submission Date | February 3, 2016 |
| Published in Issue | Year 2014 Volume: 4 Issue: 1 |