Lagrange’s Interpolation Formula #numericalanalysis

Lagrange’s Interpolation Formula #maths #eranand

In mathematics, Lagrange's interpolation formula is a formula for constructing a polynomial that passes through a given set of points. The formula is named after Joseph Louis Lagrange, who published it in 1772. The formula is given by: \(f(x)=\sum _{i=0}^{n}f(x_{i})\cdot \frac{\prod _{j=0}^{n}(x-x_{j})}{\prod _{j=0}^{n-1}(x_{i}-x_{j})}\) where \(f(x_{i})\) are the values of the function at the given points \(x_{i}\), and \(n\) is the number of given points. The formula can be used to find the value of a function at any point \(x\), even if the given points are not evenly spaced. This makes it a useful tool forinterpolation, which is the process of finding the value of a function at a point within the range of a discrete set of known data points. There are a few things to keep in mind when using Lagrange's interpolation formula. First, the formula is only valid if the given points are distinct. Second, the formula is not always accurate, and the accuracy will depend on the distribution of the given points. Third, the formula can be computationally expensive, especially if there are a large number of given points. Despite these limitations, Lagrange's interpolation formula is a powerful tool that can be used to solve a variety of problems. It is a particularly useful tool for interpolating data that is not evenly spaced.