Interpolation. Interpolation. Interpolation is important concept in numerical analysis. Quite often functions may not be available explicitly but only the values of the function at a set of points. . Interpolation. Interpolation is important concept in numerical analysis.
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|ƒ(x) – Pn(x)| <
for all x [a, b].
ƒ(x) = ƒ(x0) + ƒ'(x0)(x - x0) + 1/2! ƒ''(x0) (x - x0)2+..
ƒ(x) = Pn(x) + Rn + 1(x)
where the Taylor polynomial Pn(x) and the remainder term Rn + 1(x) are given by
Pn(x) = ƒ(x0) + ƒ'(x0)(x - x0) + … + 1/n! ƒn(x0) (x - x0)n
Rn + 1(x) = 1/(n+1)! ƒn+1( ξ ) (x - x0)n+1
E(x) = ƒ(x) - Pn(x)
E(x) = 1/(n+1)! (x - x0) (x – x1) … (x – xn) ƒn+1( ξ ); x0≤ξ≤x.
Using Polynomials to approximate a function given discrete points
2. If we desire to add or subtract a point from the set to construct the polynomial, we essentially have to start over in the computations.
The divided difference avoids this.
Fit using a series of 3rd