Numberical Analysis --- Interpolation & Polynomial Approximation
Welcome file If a function y = f ( x ) y=f(x) y = f ( x ) is too complicated for calculation or is even unknown, one way to approximate it is to first obtain its values y 0 = f ( x 0 ) , . . . , x n = f ( x n ) y_0=f(x_0),...,x_n=f(x_n) y 0 = f ( x 0 ) , ... , x n = f ( x n ) at a sequence of points x 0 , . . . , x n x_0,...,x_n x 0 , ... , x n , and then construct a relatively simple approximating function g ( x ) ≈ f ( x ) g(x)\approx f(x) g ( x ) ≈ f ( x ) . If g ( x ) g(x) g ( x ) satisfies that g ( x i ) = f ( x i ) g(x_i)=f(x_i) g ( x i ) = f ( x i ) for all i = 0 , . . . , n i=0,...,n i = 0 , ... , n , it is called the interpolating function of f ( x ) f(x) f ( x ) . The most commonly used interpolating functions are algebraic polynomials. Interpolation and the Lagrange Polynomial Find a polynomial of degree n n n , P n ( x ) = a 0 + a 1 x + . . . + a n x n P_n(x)=a_0+a_1x+...+a_nx^n P n ( x ) = a 0 + a 1 x + ... + a n x...