- 3.
**Piecewise****polynomial****interpolation**In this section we develop the basic theory of**piece-wise****polynomial****interpolation**within the context de-veloped in Section 2. The viewpoint extends the treat-ment presented in [1,4]. We begin with the space V of bounded functions on Dwhich are C0(U ∪P) and extendable to a contin-uous function of E for ... **Interpolation**works by fitting**polynomial**curves between successive data points. The degree of the**polynomial**curves is specified by the option InterpolationOrder. ... Compare splines with**piecewise**Hermite**interpolation**for random data: The curves appear close, but the spline has a continuous derivative: ...- Theorem 47 (Conditioning of PL
**interpolation**) The absolute condition number of**piecewise****linear interpolation**in the infinity norm equals one. More specifically, if is the**piecewise****linear interpolation**operator, then. (112) ¶. (The norm on the left side is on functions, while the norm on the right side is on vectors.) **PIECEWISE POLYNOMIAL INTERPOLATION**Recall the examples of higher degree**polynomial**in- ³ ´−1 terpolation of the function f (x) = 1+x2 on [−5, 5]. The interpolants Pn (x) oscillated a great deal, whereas the function f (x) was nonoscillatory. To obtain interpolants that are better behaved, we look at other forms of**interpolating**functions. (a)Compute the linear spline swhich**Piecewise Polynomial Interpolation****Piecewise**Hermite interpolants If we are given not just the function values but also the rst derivatives at the nodes: z i = f 0(x i); we can nd a cubic**polynomial**on every interval that interpolates both the function and the derivatives at the endpoints: ˚ i(x i) = y i and ˚ 0(x i) = z i ˚ i(x i+1) = y i+1 ...