數学与应用数学专业毕业论文英文文献翻译.docVIP

Chapter 3 Interpolation Interpolation is the process of defining a function that takes on specified values at specified points. This chapter concentrates on two closely related interpolants, the piecewise cubic spline and the shape-preserving piecewise cubic named “pchip”. 3.1 The Interpolating Polynomial We all know that two points determine a straight line. More precisely, any two points in the plane, and, with , determine a unique first degree polynomial in whose graph passes through the two points. There are many different formulas for the polynomial, but they all lead to the same straight line graph. This generalizes to more than two points. Given points in the plane, ，, with distinct ’s, there is a unique polynomial in of degree less than whose graph passes through the points. It is easiest to remember that , the number of data points, is also the number of coefficients, although some of the leading coefficients might be zero, so the degree might actually be less than . Again, there are many different formulas for the polynomial, but they all define the same function. This polynomial is called the interpolating polynomial because it exactly re- produces the given data. ， Later, we examine other polynomials, of lower degree, that only approximate the data. They are not interpolating polynomials. The most compact representation of the interpolating polynomial is the La- grange form. There are terms in the sum and terms in each product, so this expression defines a polynomial of degree at most . Ifis evaluated at , all the products except the th are zero. Furthermore, the th product is equal to one, so the sum is equal to and the interpolation conditions are satisfied. For example, consider the following data set: x=0:3; y=[-5 -6 -1 16]; The command disp([x;y]) displays 0 1 2 3 -5 -6 -1 16 The Lagrangian form of the polynomial interpolating this data is We can see that each term is of degree three, so the