x the corresponding value {\displaystyle x_{0},...,x_{k}} On other hand, Lagrange interpolation is suitable to use with equal or nonequal step. ) Theorem. j , zeroing the entire product. as above. This has applications in cryptography, such as in Shamir's Secret Sharing scheme. ( {\displaystyle (x_{j},y_{j})} x x x x zeroes... x x Clearly, is the notation for divided differences. {\displaystyle g(x)} The polynomial (2) is the Lagrange interpolating polynomial. = j R i − {\displaystyle L(x)} It is important to notice that the derivative of a polynomial of degree 1 is a constant function (a polynomial of degree 0). When interpolating a given function f by a polynomial of degree k at the nodes ( ≤ = ) between = x InterpolatingPolynomial gives the interpolating polynomial in a Horner form, suitable for numerical evaluation. x ( for z is any value between C and x makes the derivative … i Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears. x One of the most common modern notations for differentiation is due to Joseph Louis Lagrange.In Lagrange's notation, a prime mark denotes a derivative. k − δ On the other hand, if also 2 ( j Note how, given the initial assumption that no two ∏ {\displaystyle {\frac {x_{i}-x_{i}}{x_{j}-x_{i}}}=0} To approximate a function more precisely, we’d like to express the function as a sum of a Taylor Polynomial & a Remainder. with no two F ∑ {\displaystyle k} k I imagine the textbook authors want to show you some of the history of interpolation. 1 O. Kis, Lagrange interpolation with nodes at the roots of Sonin-Markov polynomials (in Russian), Acta Math. {\displaystyle i\neq j} In other words, the user supplies n sets of data, (x(i),y(i),yp(i)), and the algorithm determines a polynomial p(x) such that, for 1 <= i <= n p(x(i)) = y(i) p'(x(i)) = yp(i) Note that p(x) is a "global" polynomial, not a piecewise polynomial. p x j {\displaystyle \xi ,\,x_{0}<\xi