Polynomial Interpolation

Consider a family of functions of a single variable x: Φ(x; a0, a1, . . . , an), where a0, . . . , an are the parameters. The problem of interpolation for Φ can be stated as follows: Given n + 1 real or complex pairs of numbers (xi, fi), i = 0, . . . , n, with xi 6= xk for i 6= k, determine a0, . . . , an such that Φ(xi; a0, . . . , an) = fi, i = 0, . . . , n. The above is a linear interpolation problem if Φ depends linearly on the parameters ai: Φ(x; a0, . . . , an) = a0Φ0(x) + a1Φ1(x) + · · ·+ anΦn(x). Typical problems in this class include polynomial interpolation Φ(x; a0, . . . , an) = a0 + a1x+ a2x 2 + · · ·+ anx, as well as trigonometric interpolation Φ(x; a0, . . . , an) = a0 + a1e xi + a2e 2xi + · · · + ane (i = −1). Polynomial interpolation will be addressed shortly in length. Trigonometric interpolation is used extensively for the numerical Fourier analysis of time series and cyclic phenomena in general. We will discuss this further along with approximation in the future. The class of linear interpolation also contains spline interpolation. In the special case of cubic splines, Φ is twice continuously differentiable on [x0, xn] and coincident with some cubic polynomial on each [xi, xi+1] for i = 0, . . . , n− 1. Spline interpolation is very useful for representing empirical curves (in graphics and drawing tools), and for dealing with ordinary differential equations (ODEs) and partial differential equations (PDEs). Spline interpolation is described in more details in the notes for optional reading. The class of nonlinear interpolation includes two important schemes: rational interpolation,