Lagrange polynomial

In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points ( x j , y j ) {displaystyle (x_{j},y_{j})} with no two x j {displaystyle x_{j}} values equal, the Lagrange polynomial is the polynomial of lowest degree that assumes at each value x j {displaystyle x_{j}} the corresponding value y j {displaystyle y_{j}} (i.e. the functions coincide at each point).The interpolating polynomial of the least degree is unique, however, and since it can be arrived at through multiple methods, referring to 'the Lagrange polynomial' is perhaps not as correct as referring to 'the Lagrange form' of that unique polynomial. In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points ( x j , y j ) {displaystyle (x_{j},y_{j})} with no two x j {displaystyle x_{j}} values equal, the Lagrange polynomial is the polynomial of lowest degree that assumes at each value x j {displaystyle x_{j}} the corresponding value y j {displaystyle y_{j}} (i.e. the functions coincide at each point).The interpolating polynomial of the least degree is unique, however, and since it can be arrived at through multiple methods, referring to 'the Lagrange polynomial' is perhaps not as correct as referring to 'the Lagrange form' of that unique polynomial. Although named after Joseph Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. As changing the points x j {displaystyle x_{j}} requires recalculating the entire interpolant, it is often easier to use Newton polynomials instead. Given a set of k + 1 data points where no two x j {displaystyle x_{j}} are the same, the interpolation polynomial in the Lagrange form is a linear combination