A Recursive Relationship in Lagrange Interpolating
Received Date: May 04, 2024 Accepted Date: June 04, 2024 Published Date: June 07, 2024
doi: 10.17303/jcssd.2024.3.201
Citation: Edmund A. Chadwick, Ali Hatam (2024) A Recursive Relationship in Lagrange Interpolating. J Comput Sci software Dev 3: 1-4
Abstract
Lagrange interpolation, as described in many numerical analysis books, has a deficiency that by adding a new data, all computations must be recalculated from the beginning. A very simple method that comes here shows that there is no needed to do the calculations from the beginning, but as well as the interpolating by Newton’s divided diffrences, all former computations without any changes, can be extended by adding new data.
Keywords: Numerical Analysis;polynomial Approximation;Lagrange Interpolation;Newton's Divided Diffrences
Introduction
Theorm 1.1 (Theorm 3.2 in [2]) If x0, x1, ..., xn are n+ 1 distinct numbers and f is a function whose values are given at these numbers, then a unique polynomial Pn(x) of degree at most n exists with f(xk) = Pn(xk), for each k = 0, 1, ..., n.This polynomial is given by
where for each k = 0, 1, ..., n,
As it’s seen in both formulas (1) and (2), all La- grange polynomials Ln,k are polynomials of degree n and if a point (xn+1, f(xn+1)) (provided xn+1 differs from xi , i = 0, 1, ... , n) to be added to the previous data, for producing Ln+1,k and therefore Pn+1(x), previous Ln,k are useless and all former com- putations must be recalculated.This is because the numera- tor and denominator of all Ln,k’s are involving with all fixed data and if any new dada to be added, for producing new Ln+1,k, in all factors in the denominator of Ln,k, the number xk must be replaced with the new data xk+1 and this means pre- vious results in denominator can not be extended and all must be completely recalculated, and this is the lack of traditional Lagrange interpolating. Fortunately, interpolating by Newton’s method has no this deficiency because of Newton’s interpolating is based on the following recursive formula (3),
for more analysis see [1,3].
We now introduce the following formula (4),
to modify traditional Lagrange’s interpolating and equipped this method with a recursive formula, such that by adding new data, for extending to higher degree interpo- lation polynomial, all old computations can be used.
New Lagrange Interpolating (Chadwick’s method)
Again we suppose x0, x1, ..., xn are n + 1 distinct numbers, we present the following repeated method to d the polynomial Pn(x) of degree n passing the points (x0, y0), (x1, y1), ... ,(xn, yn).
Equ. (5) shows that Pk has been used to produce Pk+1.
Computer Algorithm
Conclusion
The heart of this article is definig a recurrence formula for Lagrange interpolating polynomials rather than calculating individual formula for each polynomial.This method can also be extended for interpolating polynomials of two variable or multivariate as well
Conflict of Intrest
The authors declare that we have no Conflict of interest. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
This is a joint-authored original work that has not been suported financially by any organisation
- JP Berrut, LN Trefethen (2004) Barycentric lagrange interpolation, SIAM Review, 46: 501-17.
- RL Burden, JD Faires, AM Burden (2016) Numerical Analysis, Cengage Learning, Boston, MA 02210 USA.
- MS Floater, K Hormann (2007) Barycentric rational interpolation with no poles and high rates of approximation, Numerische Mathematik, 107: 315-31.